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THESIS

, f.?
50169530

This is to certify that the

dissertation entitled

Testing for the equality of two
autoregressive and regression functions

presented by
Fang Li i

has been accepted towards fulfillment
of the requirements for

PhoDo degree in StatiStiCS

Major professor

Hira L. Koul

Date May 26, 2004

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6/01 c:/CIRC/DateDue.p65-p.15

TESTING FOR THE EQUALITY OF TWO
AUTOREGRESSIVE AND REGRESSION FUNCTIONS

By

Fang Li

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Statistics and Probability

2004

ABSTRACT

TESTING FOR THE EQUALITY OF TWO
AUTOREGRESSIVE AND REGRESSION FUNCTIONS

by

Fang Li

The thesis comprises of two parts. Both parts deal with comparing the conditional
mean functions of the two independent stochastic processes. The first part discusses
the problem of testing the equality of two autoregressive functions against one-sided
alternatives in the presence of conditional heteroscedasticity in each of the autore-
gressive time series. The second part discusses the problem of testing the equality
of two nonparametric regression functions against two-sided alternatives for uniform
design on [0, 1] with long memory moving average errors.

In the first part, the two time series are assumed to be strictly stationary strongly
mixing, and are allowed to have possibly different heteroscedastic error and station-
ary densities. The proposed class of tests avoids the estimation of the common non-
parametric autoregressive function and is based on the time series differences after
matching the lagged variables. The asymptotic normality under general one-sided
local nonparametric alternatives is derived. The paper also discusses asymptotically
optimal tests against these alternatives within the proposed class of tests.

In the second part, the standard deviations and the long memory parameters are
allowed to be possibly different for the two errors. The proposed tests are based on
the partial sum process of the pairwise differences of the observations. The tests are
shown to be consistent against some general nonparametric alternatives. We also
discuss their asymptotic distributions under the null hypothesis and some general

sequences of local nonparametric alternatives. Since the limiting null distributions

of these tests are unknown, we first conducted a Monte Carlo simulation study to
obtain a few selected critical values of these distributions. Then, based on these
critical values, another Monte Carlo simulation is conducted to study the finite sample
level and power behavior of these tests at some alternatives. This power is found to
be high for small values of the long memory parameters. The paper also contains
a simulation study that asses the effect of estimating the nonparametric regression
function on an estimate of the long memory parameter of the errors. It is observed
that the estimate based on direct observations is generally preferable over the one

based on the estimated nonparametric residuals.

Copyright by
IDADK3iLI

2004

ACKNOWLEDGMENTS

It is great that I have now the opportunity to express my deep gratitude to all the
people who have supported me for the past five years at Michigan State University.

First, I would like to thank my advisor Professor Hira L. Koul for his invaluable
guidance and generous support. He was always there to listen and to give advice when
I needed him. I have benefited tremendously from Professor Koul’s love of statistics,
his perseverance, and his extraordinary kindness.

I also thank Professors Habib Salehi, Lijian Yang, Vince Melfi and Richard T.
Baillie for their services on my thesis committee. Their continuous caring, help and
encouragement are greatly appreciated. Special thanks go to Professor Connie Page
for her help when I was at the consulting service and thereafter. I also thank Professor
Dennis Gilliland for serving as chairperson of my committee in 2000 and his continuing
concern. I thank Professor James Stapleton for his generous help and encouragement
since the very first day I came to East Lansing. I thank Cathy Sparks and Laurie
Secord for being so helpful, especially Cathy for her help on my simulation study.

Many former and present graduate students of the Department of Statistics &
Probability have helped me immeasurably with their friendship and encouragement.
I am also grateful to the Department of Statistics & Probability for offering me
assistantship during the first 4 years of my graduate studies.

This research was also supported by the NSF grant DMS 00-71619, under the PL:
Professor Hira L. Koul.

Finally, I thank my family for their love and faith in me. Their support has helped

to make this dissertation possible.

TABLE OF CONTENTS

List of Tables viii
List of Figures ix
1 Introduction 1
2 Testing for superiority among two time series 12
2.1 Introduction ................................ 12
2.2 Asymptotic behavior of T ........................ 13
2.3 Construction of 23k and f ......................... 18
2.4 Proofs ................................... 30

3 Testing the equality of two regression curves with long-range depen-

dent errors 39
3.1 Introduction ................................ 39
3.2 Asymptotic behavior of T1, T 2 and T1, T2 ................ 40
3.3 Construction of 1:1,, f1 , 6,2 and 62 .................... 47
3.4 Some Additional Proofs ......................... 57
3.5 A Monte Carlo study ........................... 64
4 Bibliography 90

vi

List of Tables

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9

3.10

3.11
3.12

3.13

3.14

3.15

Simulated critical values under a = .1 and .05 .............. 65
Simulated critical values under a = .025 and .01. ........... 66
The choice of m1 corresponding to H1. ................. 73
comparison of H1 to 1:11. ......................... 74
Proportion of rejections under H0 for part I. .............. 75
Proportion of rejections under Ho for part I. .............. 76
Proportion of rejections under fixed alternatives (I) 6(13) 2 1 ...... 77
Proportion of rejections under fixed alternatives (II) 6(x) = a: for Part I. 78

Proportion of rejections under local alternatives (III) 6,,(23) = Tn for

Part I. ................................... 79
Proportion of rejections under local alternatives (IV) (5,,(33) = Tnl‘ for
Part I. ................................... 80
The other choice of m1 = m : m2. ................... 80
Comparison of H to H * which is H with m given in Table 3.11 under
the local alternatives (III) 6,,(23) = Tn and (IV) 6,,(22) = as ....... 81
The comparison of simulated power under local alternative 6,,(rc) = Tn
among these tests based on estimates If and H *. ........... 82
The comparison of simulated power under local alternative 6,,(33) 2 Tag:
among these tests based on estimates [:1 and H *. ........... 83
Preportion of rejections under H0 for Part II. ............. 84

vii

3.16
3.17

3.18
3.19

3.20

3.21

3.22

Proportion of rejections under Ho for Part II. .............
Mean and standard deviation of the estimates 6;", {7% under u1(:c) =

p2(:r) = 2:13, of = 1, a; = 4 and n = 2000. ...............

Proportion of rejections under the fixed alternative (I) 6(x) = 1. . . .
Proportion of rejections under the fixed alternative (II) 6(sr) = x for
part II ....................................
Proportion of rejections under local alternative (III) 6,,(x) = Tn for

part II ....................................
Proportion of rejections under local alternative (IV) 6,,(12) = mm for
part II ....................................
Mean and standard deviation of the estimates H1, H2 under local al-

ternatives (III) 6,,(23) = Tn, (IV) 6,,(23) = Tux for of = 1, 0% = 4 and

viii

87

88

89

List of Figures

3.1
3.2
3.3
3.4
3.5
3.6

histogram of H1 and H1 for n = 300, H = .6, .7, .75. ......... 67
histogram of H1 and H1 for n = 300, H = .8, .85, .9. ......... 68
histogram of H1 and H1 for n = 600, H = .7, .75, .8. ......... 69
histogram of H1 and H1 for n = 600, H = .8, .85, .9. ......... 70
histogram of H1 and H1 for n = 1000, H : .7, .75, .8 .......... 71
histogram of H1 and H1 for n = 1000, H = .8, .85, .9 .......... 72

ix

Chapter 1

Introduction

The first part of the thesis is concerned with testing for the equality of two autoregres-
sive functions against one sided alternatives when observing two independent strictly
stationary and ergodic autoregressive times series of order one. More precisely, let
Y”, Y2,“ z' E Z := {0, :l:1, - - - }, be two observable autoregressive time series such that
for some real valued measurable functions in and p2, and for some positive measurable

functions 01, 02,
(1.0-1) Y1; = H1(Y1,i—1)+ 01(Yl,i—l)51,ia Y2,i = M2(Y2,t—1)+ 02(Y2,i—1)52,ia

for all z' E Z. The errors {514, 2' E Z} and {52,13 2' E Z} are assumed to be two
independent sequences of i.i.d. r.v.’s with mean zero and unit variance. Moreover,
51,” i 2 1 are independent of Ym, and 52,-, 2' Z 1 are independent of I’m. And the
time series are assumed to be stationary and ergodic.

Let I be a compact interval. The problem of interest is to test the null hypothesis:
H0: [11(13): [12(13), VII: 6 I,
against the one-sided alternative hypothesis

(1.0.2) Ha : p1(1:) 2 #20:), V2: 6 I, with strict inequality for some a: E I,

based on the data set Y1,0,Y1,1, - -- ,Ylm, Y2,0,Y2,1, - -- , Yzm.

For example, in China the two rivers, called Yellow and Yangzi rivers, flood often.
A way to overcome this problem may be to build some dams around these rivers. In
order to decide which river to focus on first it may be desirable to compare average
weekly or daily water levels of the two rivers. If, for example, it was found that
the average daily water level of the Yangzi river is larger than the Yellow river, then
one may plan to build dams for Yangzi river first. Since the two rivers are quite far
away from each other it may be reasonable to assume that the daily levels of these
two rivers are independent. In hydrology, autoregressive time series are often used to
model water reservoirs, see, e.g., Bloomfield (1992).

A stationary process with finite variance is said to have long memory if auto-
covariances are not summable. The second part of this thesis is concerned with
testing the equality of two regression functions against the two sided alternatives
when the errors form long-memory moving averages. More precisely, let ,ul and [12
be real valued functions on [0,1], 01, 02 be positive numbers, L1 and L2 be known

positive slowly varying functions at infinity, -;— < H1, H2 < 1, and for i = 1, 2, let

(1.0.3) a“: = L,(j)j(2”i‘3)/2, 3'21,
= 1, i=0,
= 0, j<0.

In the problem of interest, one observes two independent processes Y”- and Y2", 2' =

{0,1, - - ~} such that

. 00
Z 2:
J:

. oo

2

Y2,i = M2(;) + 02U2,i, 112,1” = E a2,j52,i—j:
i=0

where, {51“, z' E Z} and {52m 2' E Z} two independent sequences of i.i.d. standard

r.v.’s Note that

00 00
2 am- 2 00, E 012,- < oo, 2': 1,2.
i=0 i=0

Hence, the error processes “Li and 212,,- have long-memory.

The problem of interest is to test the null hypothesis:

~

H0: [11(33): u2(a:), V2: 6 [0,1],
against the two-sided alternative hypothesis
(1.0.5) Ha : p1(:z:) 7é ,u2(:1:), for some a: 6 [0,1].

based on the data Ym, - -- , Y1,” and Y2,1,- - - , Y2,” where n is a positive integer.

In the one-sample regression setting with independent errors, related testing prob-
lems have been addressed by several authors. Cox, Koh, Wahba and Yandell (1988)
consider generalized smoothing spline and partial spline models. They test the null
hypothesis that the regression function has a particular parametric form against semi-
parametric alternatives. Eubank and Spiegelman (1990) consider the problem of test-
ing the goodness-of-fit of a linear model using a spline-smoothing method. Raz (1990)
considers the randomized model, and tests the null hypothesis of no relationship be-
tween the response and design variable, i.e., regression function is a constant. Hardel
and Mammen (1993) test the null hypothesis that the regression function is of a spe-
cific parametric form, against some smooth nonparametric alternatives. Their tests
are based on some Lg-distances between a nonparametric estimate of the regression
function and the parametric model being fitted. Koul and Ni (2002) test the same
null hypothesis as in Hardel and Mammen (1993), but they base their tests on a class
of minimum Lg-distances. See the monograph of Hart (1997) for more references on
the parametric regression model fitting in the one sample setting.

In the two-sample setting with independent errors, Hardle and Marron (1990)

compared non-parametric regression curves in a parametric frame work. Under the

assumption that there are parametric transformations between the two regression
curves, they test for the values of the transformation parameters. Scheike (2000)
used the squared difference between the cumulative regression functions to construct
tests for fitting a parametric regression model.

In the context of autoregressive time series the problem of fitting a parametric
autoregressive model has been addressed by some authors. The tests of An and
Cheng (1991) and Koul and Stute (1999) are based on a marked empirical process
of the residuals while those of McKeague and Zhang (1994) use integrated Tukey
regressograms. Hjellvik, Yao and Tjostheim (1998) propose tests for fitting a linear
autoregressive function based on weighted mean square residuals obtained from local
polynomial approximation of the autoregressive function. Ni (2004) proposes tests
based on some minimum Lg-distances between a nonparametric estimate of the au-
toregressive function and the parametric model being fitted. All of these papers deal
with two sided alternatives. See the review paper of MacKinnon (1992) for more on
fitting autoregressive time series models.

The papers that address the above one-sided testing problem (1.0.2) in regression
setting include Hall, Huber and Speckman (1997) and Koul and Schick (1997, 2003).
These papers use the idea of matching the covariates to construct their test statistics.
Hall, et al. (1997) prove the asymptotic normality of their statistic under general
alternatives and when the design and error densities are possibly different. They
further show that an adaptive version of their test is asymptotically optimal within
their class of tests against a sequence of alternatives Ha : ,ul — p2 = c/ \/r_z, where c is
a positive constant, and n is the common sample size.

Koul and Schick (2003) propose a test T under heteroscedastic errors and with

possibly different design and error densities. The test T is also shown to be asymp—

totically optimal against the sequence of one-sided alternatives:

71177.2
3
m + n2

 

(1-0-6) #1013) = #2013) + N’%6(x), VIII 6 I, N 2:

where 6 is a non-negative continuous function and n1, 722 are the two sample sizes. In
fact, they discuss general asymptotic optimality theory against these alternatives.
The first part of this thesis extends their test statistic T to the above autoregressive
set up (1.0.1) to test for H0 versus Ha. Some of the analogous asymptotic optimality
theory in the autoregressive setting is also discussed. To describe this test, let u be
a non-negative function vanishing off I and continuous on I, 21) be a kernel density
function on the real line with the compact support [-l,1], a > 0 be sequence of

bandwidths, and let wa(3:) = w(:r/a)/a. Consider

 

 

2\/—(,Y11— 1 )fiO/Zj-l)
'2 ’ li-Y-waY,_—Y._,
7117117122]: 1 91(,Y11— 1))92(Y2,j-1)( 1' 2’3) ( 1’ 1 2" 1)

where 91, 92 are the two stationary densities of Yip, Yao, respectively, assumed to be
bounded away from 0 on I. Under H0, T is approximately a weighted difference of
the two time series residuals matched according to their lag variables.

Using a conditioning argument one sees that under (1.0.1) and the alternatives

(1.0.6),

Em = Has) m(8)-u2(t))wa(s-t)d8dt

= 0(1,) asa—->0,

where
Q = /u(:r)(p1(:t) — ,ug(:r)) dz = N—% /u($)6(:r) dz.
This suggests that large values of T will be significant for H0, in favor of Ha.
But, T depends on the usually unknown densities g1, 92, and thus is unusable

for inference. This suggests to replace gk’s in T by their estimates. Also, as will be

seen later, for a specific 6, the corresponding optimal choice of u that maximizes the

asymptotic power against the local alternative (1.0.6) depends on the densities 91, 92

and the variance functions of, 0%. Therefore, we need to investigate an adaptive

version of T, namely

7:1 n2
1

Z131(Y1,i—1)132(Y2,j—1)(Y1,i — Y2,j)wa(Yl,i—l - Y2,j—1),
71.1712 .

1:1 j=l

 

(1.0.7) T =

where, for each k = 1,2, 13,, is a non-negative estimate of 0;, = fi/gk such that
73,413): 0 for :16 ¢ I.

Under a general set of assumptions, we obtain the asymptotic normality of the
suitably standardized N 1/ 2 (T ~42), under Ho and Ha. We also give an upper bound on
the asymptotic power of these tests and then exhibit a u for which the corresponding
test T achieves this upper bound.

Relatively little is known about fitting a regression function in the presence of long
memory. Koul and Stute (1998) studied a class of tests based on marked empirical
process of certain residuals to fit a parametric regression function when the design
is either fixed or random and when the errors are long memory moving average or
subordinate Gaussian. Koul, Baillie and Surgailis (2002) studied these tests further
when the covariate process is one dimensional and also forms a long memory moving
average.

The papers that address the above two-sided nonparametric testing problem (1.0.5)
with independent errors include Hall and Hart (1990), Kulasekera (1995) and Delgado
(1993). Hall and Hart (1990) test for the difference between the two non-parametric
regression curves using a bootstrap method in the case of common design. Kulasekera
(1995) used quasi-residuals to test the difference between two regression curves, under
the conditions that do not require common design points or equal sample sizes. Del-
gado (1993) used the absolute difference of the cumulative regression functions to test
for the equality of two nonparametric regression curves, assuming the same covariates

in the two samples while the two samples are possibly dependent on each other. The

second part of the thesis investigates Delgado’s test under the above long-memory set
up (1.0.4) to test for Ho versus Ha.

Let

(1.0.8) A(t) :=/ot(;i1(:r)— u2($)) dz, 0 S t S 1.

As in Delgado (1993), assuming pl, #2 to be continuous, a necessary and sufficient
condition for the null hypothesis Ho to hold is that A(t) E 0. Moreover, under Ha,
A(t) aé 0 for some t. This suggests to construct tests of Ho versus Ha based on some

consistent estimators of A(t). One such estimator is obtained as follows. Let

Int]
1
DJ' ::Y1,j_Y2,j, j=1,---,n; Un(t)::;E Dj OStSI.
i=1

Observe that

[M] [M] . .
_ 1 1 J .7
Un(t) — ; ;(01U1,3 — 027420) + ; £01101?)— #2(;))

We shall Show in Remark 3.2.1 below that the long memory linear processes um- for

2' = 1, 2 satisfy

[ml

1
1.0.9 — 1' z"
( ) sup ”2011,,

0<t<1 ._
._ _ ]_1

—)p 0.

 

 

Thus, Un(t) provides a uniformly consistent estimator of A(t). This suggests to
base tests of H0 on some suitable functions of this process. In this thesis we shall
focus on the Kolmogorov—Smirnov and Cramér-von Mises type tests based on the
SUPogtgi IUn(t)| and f01U3(t)dG(t), where G is a d.f. on [0,1].

To determine the large sample distribution of the process Un(t), one needs to
normalize this process suitably. Unlike in the weakly dependent case, the normalizing

sequence here depends on the parameters H,, z' = 1, 2. To make this more precise, let

 

 

a. % pz
1. . i2: 2 ”(1+0i)/2d 1'2: i 6,:2 2 —' 2H,
( 010)p ([0 (9+0) y , 1‘6 H.(2H.-—1)’ ,

and

(1.0.11) 7,3,, = n?0§L§(n)n2H‘_2, 7,, = (731+ r339.

Note that 0 < 6, < 1, i=1,2. Let

1 11,
‘WW’ nzfiflmma

Tn

(1.0.12) T1 : = sup

05151

 

In the case H,’s and a,’s are known, the tests of Ho could be based on T1 or T2, both
being significant for their large values. But, usually these parameters are unknown
which renders T1, T2 of little use. This suggests to replace the parameters in T1, T2 by
their estimates. Therefore, the proposed tests will be based on the adaptive versions

of T1 and T2, namely

 

- 1 . 1 1 ‘
(1.0.13) T12: 5111) TUn(t)l, T2 2: 'Tz'f USU) dt,
05151 Tn In 0
a, = (r3,1 + +33)? +3, 2 k?&?L?(n)n2H‘_2, 1': 1,2,

where, (7,, H,, 2' = 1, 2 are consistent estimates of 0,, H,, and 1%,, z' = 1, 2 are K, given
in (1.0.10) with H, replaced by H,. We also let 6, := 2 - 2H, for later use.

The estimates of 0,, z' = 1,2 will be necessarily based on the residuals Y,”- —
[1,(j/n), j = 1, - -- ,n;z' = 1, 2, where [1,, z' = 1, 2, are some nonparametric estimators
of the regression functions m, i = 1, 2,. The latter estimation problem in the pres-
ence of long memory has been addressed by some authors. Csorgt'i and Mielniczuk
(1995) studied the finite-dimensional distributions of the kernel regression function
estimators and their maximal deviations over an increasing grid in regression models
with non-random designs and long memory Gaussian errors. Csiirgéi and Mielniczuk
(2000) studied similar problems but in regression models with i.i.d. random designs
and errors given by an unknown function of a long memory moving average. Robinson
(1997) gave a general central limit theorem for the finite dimensional distributions

of the kernel type nonparametric regression function estimators for regression models

with fixed uniform design as in (1.0.4) and a broad class of dependent errors including
the long memory moving averages. We use some of Robinson’s results in this thesis
as will be seen later.

The estimation of the long memory parameters H1, H2 has been discussed by
several authors. Fox and Taqqu (1986) and Dahlhaus (1989) discussed the consis-
tency and asymptotic normality of the MLE and Whittle estimators for a stationary
long memory Gaussian process under some regularity conditions on the parametric
spectral density. Robinson (1995a) discussed a form of log-periodogram regression es-
timate of self-similarity and scale parameters under multiple time series models with
long memory Gaussian errors. Robinson (1995b) considered another estimate of the
self-similarity parameter that maximizes an approximate form of frequency domain
Gaussian likelihood in a semiparametric setting where the autocovariance function is
specified only partly. He proved its consistency and asymptotic normality with a rate
of consistency less than n1”. In Robinson (1997), this consistency rate is shown to be
logn under some mild conditions. We shall use this estimation method to estimate
H1, H2.

Under some general assumptions, T1 and T2 are shown to weakly converge to some
functions of a fractional Brownian motion over [0, 1] under Ho. Then it is proved that
the power of the test based on T1 and T2 converges to 1, at the fixed alternative
(1.0.5) or even at the alternatives that converge to H0 at the rate n-C for some
0 < c < 01 /\ 62 / 2. We also obtain an expression for the asymptotic power against the
sequences of local alternatives that converge to the null hypothesis H0 at the rate Tn.

Recall that the long memory parameter H, is related to the residuals 0,11,, =
Y,,, - 11,:(t/n), for 2’ = 1, 2. Based on the suggestions in Robinson (1997), we consider
two sets of estimates of H1 and H2. One set of estimates H,, z' = 1,2 are based

on direct observations Y,,,, t = 1, - -- ,n,, 2' = 1, 2, whereas another set of estimates

H,, z' = 1,2 are based on the residuals 11,, = Y,,, — [Mt/n), 2‘ : 1,2, respectively.
An advantage of H,’s is that their computation does not need the estimation of the
regression functions 11,, 2' = 1, 2. Robinson gave some general sufficient conditions on
semi-parametric spectral density for their consistency.

To assess if there is any finite sample difference between these two methods of
estimation of the Hurst coefficient H1, a comparative Monte Carlo study is conducted.
We found that the average bias and the mean squared error of H1, respectively, is
smaller than that of H1, as long as H1 > .7. For H, 5 .7, the histogram of H1 is
found to be bell-shaped while that of H1 is found to be strongly skewed to the right.
These findings suggest that one should use H,, i = 1, 2 in applying the above testing
procedures. This simulation was carried out for n = 300, 600, 1000, each repeated
2000 times. The kernel used to estimate [11 needed for the computation of H1 was
(3/4)(1 — $2)I(]£L‘] 5 1), with the band width 6 = 11"”.

Under Ho, the sequence of statistics T1 converges weakly to 3111305151 ]Bg(t)], while
T2 converges weakly to f0l B§(t)dt, where Be is a fractional Brownian motion of index
0 := 01 A 02. Unfortunately these limiting distributions are unknown. In the Monte
Carlo study, we first obtained a few selected critical values from the simulation study
based on n = 5000 observations and 2000 repetitions. Then, these critical values were
used in another Monte Carlo simulation study of the finite sample level and power
behavior of these tests. In the study of the finite sample levels of tests T1 and T2,
we made a comparison of the simulated levels of these tests to that of tests T1 and
T2 which are T1 and T2 with H1, H2 replaced by their true values. The fact that
the simulated levels of T1 and T2 tend to be closer to the true levels than those of
T1 and T 2 indicates that the bias in the estimates H1 and H2 does affect the finite
sample level. We also find that the simulated levels can be affected by the bias of the

estimates 6,2 which in turn is mainly due to the estimates of the regression function

10

p, when needed.

In the study of the power behavior of these tests, we considered both fixed alter-
natives and local alternatives. The Monte Carlo power of tests T1 are generally larger
than that of T 2, especially under the local alternatives. Under fixed alternatives, the
simulated power of these tests increase in n and decrease in H1 V H2, and the power
is quite large for small values of H1 V H2. Under the local alternatives, the simulated
power is reasonably stable for all n and the same values of H1 V H2. Also, we found
that the power can be greatly improved by obtaining more accurate estimates of H,.
In these studies 71 = 300, 600, 1000, each repeated 2000 times, the kernel was as men-
tioned above and the bandwidth was taken to be nil/5. The simulation was carried
out when H1 = H2 and when H, 96 H2.

The thesis is organized as the following. Chapter 2 discusses the one sided testing
problem (1.0.6) for the autoregressive functions in model 1.0.1. Theorem 2.2.1 gives
some general assumptions on the estimates 1“), and '02 under which the statistics T of
(1.0.7) are asymptotically normally distributed, under both Ho and Ha. Then, under
some additional conditions, in Lemmas 2.3.6 and 2.3.7, we construct estimates 1‘), and
1‘12 that satisfy these assumptions. Chapter 3 discusses the two sided testing problem
(1.0.5) for the regression functions in model (1.0.4). Corollary 3.2.2 and 3.2.3 give
the asymptotic distributions of T1 and T2 given in 1.0.13 under H0. Lemmas 3.3.1,
3.3.2, and 3.3.4 give the asymptotic properties of the estimates H1, H1, 6%, 622, and
612 + 6%. The last section of this chapter contains the Monte Carlo simulation study

of the finite sample levels and power behavior of the tests based on T1 and T2.

11

Chapter 2

Testing for superiority among two

time series

2.1 Introduction

This chapter is concerned with testing the equality of two autoregressive functions
against one sided alternatives when observing two independent strictly stationary
and ergodic autoregressive times series of order one Y1,,, Y2“, z' 6 Z := {0, :l:1, - - }
obeying model (1.0.1). The problem of interest is to test the equality of two autore—
gressive functions [1,, 112 on a compact interval I against the sequences of one-sided
alternatives (1.0.6). based on the data set Y1,0,Y1,1, - -- , Yum, Y2,0,Y2,1, - - - ,ng.

In this chapter, we will construct a class of tests based on T of (1.0.7). We shall
study the asymptotic behavior of T as the sample size 711 and n2 tend to infinity. To
simplify notation, the dependence on the pair (711,712) will be suppressed and we shall

use the abbreviations

N 712 N n,
— = , q2 = — Z ,
71,1 711 + 77.2 n2 711 + 71.2

 

 

(11:

12

with N as in (1.0.6). Note that under the model assumption (1.0.1),
03(Yk,,_1) = E({0,,(Yk,-1)s,.,,}2|Y,.,,_1), k = 1, 2, 16 z.
Whenever, Esj],1 < 00, we let we denote the conditional fourth moment:
uk(1/,,,,-_1) :2 E({0,,(Yk,_1)gk,}4|Y,,,,-_1) = 02(Yk,,_1)E52,1, k = 1,2, 2'6 2:.

Theorem 2.2.1 of section 2 first gives a result that approximates N1/2(T — Q) by
a sum of martingale differences under a general set of assumptions on the estimates
131 and 1‘22 and under the alternatives (1.0.6). This approximation is in turn used to
deduce the asymptotic normality of the suitably standardized N1/2(T — 9), under H0
and H1. Remark 2.2.2 gives an upper bound on the asymptotic power of the tests
and then exhibits a u for which the corresponding test T achieves this upper bound.
In section 3, under some additional conditions, estimates 131 and 132 are constructed
that satisfy the conditions of Theorem 2.2.1. Section 4 contains some proofs including
that of Theorem 2.2.1.

We end this section by mentioning that tests of H0 could be also based on the
difference of the estimates of f 1111),, k = 1,2. In the regression context, Koul and
Schick (1997) discuss the asymptotics of a large class of such tests, denoted by T2 in
there. A drawback of these tests is that one needs to estimate the common unknown
regression function even to define them, whereas the tests based on T avoid this,
making them relatively easier to implement. For this reason we focus on the tests

based on T in this paper.

2.2 Asymptotic behavior of T

This section investigates the asymptotic behavior of the adaptive statistic T (1.0.7)

under the alternatives (1.0.6) with a non-negative continuous function 6. Note that

13

H

6 = 0 corresponds to the null hypothesis, while 6(23) > 0, for some :1: E I, corresponds
to an alternative. To stress the dependence of the local alternatives on 6, we write
P, for the underlying probability measure and E, for the corresponding expectation.
First, we recall the following definitions from Bosq (1998):

2.1. Definition. For any real discrete time process (X,,z' E Z) define the strongly

mixing coefficients

a(k) :2 sup a(0—field(X,,i 5 t), a-field(X,,z' Z t+ k)); k =1,2,...

tEZ

where, for any two sub a-fields B and C,

a(B,C) = sup |P(B (1 C) — P(B)P(C)|.
BEB,CEC

2.2. Definition. The process (X,,z' E Z) is said to be GSM (geometrically strong

mixing) if there exists co > 0 and p 6 [0,1) such that a(k) S cop", for all k 2 1.

The following assumptions are needed in this paper.

(Al) 11 is a non-negative continuous function on I and vanishing off I.

(A2) The autoregressive functions 111, [12 are continuous on an open interval con-

taining I and p.2(x) is Lipschitz-continuous on I.

(A3) The weight function w(:r) is a symmetric Lipschitz-continuous density on R

with compact support [—1, 1].
(A.4) The bandwidth (1 is chosen such that a2N —> O and 0N” ——) 00 for some 0 < 1.

(A5) The densities g, and g2 are bounded and their restrictions to I are positive

and continuous.
(A.6) Y1,,, Y2,“ 1' E Z are GSM processes.

(A.7) The conditional variance functions of and a; are positive on I and continuous

on an open interval containing I.

14

Rewrite

 

,. 1 711 A 1 712 A
(2-2-1) T = — 71(Y1,z'—1)01(Y1,i—1)51,i — — Z72(Y2,i—1)02(Y2,i—1)52,i
n, i=1 712 1:1
+N‘l/2T3 + T4,
where,
1 "2
(2.2.2) 1101:) == — 2910:) 202%,.-.)wad — 13.-.),
n2 i=1
1 "1
072(1)) 2: a: 172(2)) ;61(Y1,,_1)wa(z — I/l,,_1). III E R,
- 1 "1
T3 : a ZIl(K,i—1)5(Y1,i—1),
i=1
A 1 n1 112
T4 i: R1712 ZZ731(Y1,i—1)172(Y2,j—1)(/12(Y1,i—1)—#2(Y2,j—1))wa(Y1,i—1—Y2,j—1)-
i=l j=l

The statistics &1(z), ’72(z), T3 and T 4 can be thought of as the estimates of u(z)/

91(1), U($)/92(.’L‘), f u(z)(5(z) dz and the remainder term, respectively.

The following definition and assumption are similar as in Koul and Schick (2003)
(K-S).
2.3. Definition. The sequence of estimates m. is consistent for the function 7,, on

I, if

(22-3) 11:11) lids?) — “/k($)| = 0M1)-

An estimate 7; is said to be a modification of ’"yk, if P, (SUPxez I7; (z) — ‘yk(z)| >
0) ——> 0. The sequence of estimates y), is said to be essentially consistent on I for 7),,

if there exists a modification 7; of a, which is consistent for 7), on I .

2.4. Assumption. '7), is essentially consistent on I for '7), = u/gk, k = 1, 2.

The following lemma gives a sufficient condition for the Assumption 2.4 to hold.
It is similar to the corresponding lemma of K-S. However, because of the time series

structure its proof here, given in section 4 below, is necessarily different.

15

Lemma 2.2.1 In addition to the conditions (A3) - (A5), suppose there ezist mod-

ifications v}: of in, such that,
(2.2.4) 0 S u;(z) S K, Vz E I,
for some finite constant K, and

(2.2.5) sup lu;(z) — vk(z)| = 0P5(1)1 k = 1, 2.
zE '

Then Assumption 2.4 holds for the ’71,, k = 1, 2 of (2.2.2).
To state the next theorem let sk(z) :2 0k(z)/g,(z), z E R, k = 1, 2.

Theorem 2.2.1 Suppose the conditions (A.1)-(A.5) hold. In addition, suppose that
there are estimates 131(1):) and 02(z) based on the data Yip, Y1,1,- -- ,Ylj n, , Y2,0, - - - ,

lip/n3] satisfying the Assumption 2.4. Then, as n1 /\ 712 —> oo,
mam NflT—m
1 "1 1 "2
= N1/2(;1' Z ”(Y1,i—l) 51(Y1,i—1)51,i - ‘77:; Z “(Y2,i—1)32(Y2,i—1)52,i) + 0P, (1)-
i=1

i=1

Consequently, Ni(T — Q)/r is asymptotically standard normal, where

(2.2.7) T2 = / 112(1) (q1 s§(x)gl(x) + q2 33(z)g,(x)) dz.
Proof: The proof is given in Section 4.

Remark 2.2.1 The above theorem is still valid if 112 depends on the sample sizes
provided the Lipschitz constant in condition (A.2) does not depend on the sample
sizes. Also, the weight function u can depend on the sample sizes n1, 71.2 but then the
continuity assumption must be strengthened to u being a sequence of equi-continuous

functions. One possible choice for such an u will be

_u z 511
(”8) “’ “ mangle)magma)

 

We shall now show that it is actually the optimal choice in the sense that it maximizes

the asymptotic power of the test.

16

Remark 2.2.2 Optimal choice of u(z). Theorem 2.2.1 suggests a test which rejects

H0 for the large values of T, since under R;

(QIH

N (T — o)/T —+.1 N(0,1),

where r2 is as in (2.2.7). To apply such a test, we need an estimate 7‘2 of 7'2 satisfying
(2.2.9) +2 = 72 + 012(1)-

Lemma 2.3.8 in section 3 below verifies that under some mild additional assumptions,

the following estimator

. 1 "‘ ,. - 1 "2 . .
(2.2.10) 7'2 = qln—l Zaf(}/l,i—l)7f(yl,i—l) + (1211—2 ;0§(Y2,t—1)7§(Y2,i—1)

i=1

satisfies (2.2.9). Consequently, under P5
N%(T‘ — o)/+ —>d N(0,1).
Hence, the test that rejects H0 whenever N iT 2 zai, is an asymptotic level a test.

Here, 20, is (1 — a) quartile of the standard normal distribution. The asymptotic

power of this test is

136(‘N%T Z 20171)

 

P5 (N§(T— (2) > z. _ N59)

_> 1 _ (I) (20: fu(;)f(z)dz3 .

Clearly, maximizing this asymptotic power with respect to u is equivalent to maxi-

 

mizing
f u(z)6 (z) dz

7.

 

7 ==
with respect to 11. By the Cauchy-Schwarz inequality and using the definition of r,

we obtain

 

7 < (/ (52(2)) dz)%
‘ (11 si($)g1($) + (12 83($)92(x) ’
with equality if, and only if, u : u5. Thus the weight function u, of (2.2.8) is an

optimal choice for u.

17

2.3 Construction of 13k and 7“-

To use Theorem 2.2.1 of the previous section we need to have estimates 13),, k z 1, 2,
based on the data Y1,0,Y1,1, - -- 1Y1.[ n, , Yao, - -- ,Ymm satisfying the Assumption
2.4. In addition, to implement the proposed tests at least for the large samples we also
need to construct the estimates 6:, "n, k = 1, 2, so that the estimates i2 of (2.2.10)
are consistent for 7'2 under P5, i.e, satisfy (2.2.9). This section addresses both of these
issues.

We shall first construct estimates of u), = \fu/gk, k = 1, 2, so that under (A.1)-
(A5) and some additional conditions, they satisfy the conditions of Theorem 2.2.1.
For this we also need to estimate the densities gl, g2 and the variance functions of, 0%,
since the optimal choice of u depends on them.

We shall consider estimates of densities based either on the full samples or on

parts of the samples. And some times we will be using different bandwidths. To

unify the presentation, we first define
. 1 "
(2.3.1) gk,,,,a(z) = h- gnaw — Yk,,_1), k 21,2, 71 2 1, a > 0.

Often, we shall write gm for the full sample estimate mm”. The respective estimates
of the densities 91, 92 based on the full samples and bandwidths a1, a2 are 9,, := Slim“

k = 1, 2. The following lemma shows their uniform consistency.

Lemma 2.3.1 Suppose the conditions (A3), (A.5) and (A6) hold. In addition,

assume a1 = a2 = a —> 0 and for some c < 1, aNC —> 00. Then

(2.3.2) sup lgk(z) — gk(z)| = 0P5(1)1 k 21,2.
zEIia

Proof: The proof is deferred to Section 4.

Next, we describe the estimates of of and 03 based on parts of the samples. Let

m1 2 mm, m2 = mn2 be two subsequences of positive integers depending on n1, 712,

18

respectively, with m), _<_ 72),, k = 1, 2. Let b1, b2 denote another set of bandwidths,

and let gm :2 9k,m,,,a, a > 0. Let, for k = 1, 2,

 

1 mk Uk(z)
2' ' I: — Y i ‘ Y i— . A I: ~ ,
( 3 3) Uk(z) mk Z; k,wbk(f€ k, 1) uk(z) gk,b,,($)
1 m . . V z
Vk(:13) 1: —‘ (YIm' - #k(Yk,i—1))2 wa,(z -— Yk,,~_1), oflz) _—_~ "( )

 

Here the bandwidths a), and bk are assumed to satisfy
(2.3.4) a), —> 0, bk ——> 0, mica,c ——> oo, mfcbk —> 00, for some c < 1/2.

Remark 2.3.1 When m1 2 [,/n1] and m2 2 [,/n2], the above estimates are used
to construct the estimates 1?), of 11,, satisfying the condition of Theorem 2.2.1. When
constructing the estimates i of r that satisfy (2.2.9), we use the above entities with

m1 2 721 and m2 = 112.
The next lemma gives the consistency of the above estimates (3,3.

Lemma 2.3.2 Suppose, in addition to (AS), (A.5)-(A.7), and (2.3.4),

(2.3.5) E52,, < 00, k = 1,2.

Then,

(2.3.6) sup lain) — 02(z)| = 0106(1), k = 1, 2.
26

Remark 2.3.2 Note that (A.7) and (2.3.5) imply that V1, V2 are bounded on an open

interval containing I.

To prove Lemma 2.3.2, we need the following three lemmas. Throughout the paper,

for an event A, [A] stands for the indicator of the event A.

Lemma 2.3.3 (Davydov inequality). Let X and Y be two real valued random vari-
ables such that X E L"(P), Y E L'(P) where q > 1, r > 1 and i +% = 1.. i,
then

lCov(X.Y)| S 2p(2a)1/pllelqllYll.-.

19

Here a = a(o-field(X),a-field(Y)) is defined in the Definition 2.1.
Proof: See Corollary 1.1 in Bosq (1998, pp 21-22).

Lemma 2.3.4 Suppose (A3), (A.5), (A.6), (A.7),(2.3.4), and (2.3.5) hold. Then,

2 0P6(1)7

1 ...,,
2.3.7 — Y ,_ 1. _ y 1._
( ) 331:” mkgad 1, 05., wb.($ k, 1)

 

= 0136(1), k =1,2.

 

 

1 "'"
SUP — 201302.14) (5%,.- — 1) we). (13 ‘- Yin-1)
1:61.in ml: i=1

Proof: We shall prove the first claim in (2.3.7) for k = 1 only. The proofs of the
other statements are similar. To keep exposition transparent, in the following proof
we write 5,, Y,, E for 51,-, 1’1, and E5, respectively.

1
Let L denote the length of the interval I and m = [mf]. Divide I :1: b1 into m

 

sub-intervals I1, I2, . . . , Im of equal lengths (L + 2b,) / m and denote their midpoints
by zl, z2, ..., zm. Let, for 1 Sj S m,
1 m‘
A1051) = 771—1 ;01(Y:_1)81wb,($j ‘ Yi—I) .
1 "‘1
A202) = sup —— Zammwwbxx — Y.-.) — wits,- — Y.-.» .
(L‘EIJ' ml i=1

 

Then, the left hand side of the first claim in (2.3.7) is bounded above by supj A1(z,) +
supj A2(zj). So, to prove the lemma, it suffices to show that

(2.3.8) sup A)(z,) : 035(1), 1: 1,2.
1

0, E(52) E 1, and a change of

1

Using a conditioning argument, the fact E5,

variable formula, we obtain

 

1 "‘1
EAflLIIj) = WZE(UI(Yi—l)gi2w§1($j_Yi-l))
1 i=1
1
:: mlbl\/0’?($j—b1t)w2(t)gl($j—b1t)dt
1 .
S. —llw||oo||91||oo SUP Gift), 121,2,---,m.
mlbl eri2b1

20

By the assumptions (A.7) and (2.3.4), supxezflbl 01 (z) = 0(1). This fact, (2.3.4),

and the boundedness of w and g in turn imply

 

ZEAiH ($1) Ilwllmuglum sup ate) -—> o.
<mlbl

zEI:1:2b1

This together with the Chebyshev inequality completes the proof of (2.3.8) for l = 1.

To prove (2.3.8) for l = 2, we need to introduce some more notation. Let

L+2b1

 

B(IL‘) :2 T—n—1b1::]€,][]$—i_1]301+ ], $61K,

31 == 13(561) i=1.---.m.

By the fact that w is Lipschitz continuous on R and vanishes off [—1,1], 310 > 0,

BVTL' EIJ',

lwb,(:v — 31—1) — 101.1(2)- - 31—0)]

 

 

 

 

 

 

 

 

 

 

 

l z—z' . L+2b
Sb? b1] lx_Yi—llsb10rlxj_Yi—1]Sbl]]]$_$jls 2m1]
S lgz-zj Ixj—K—1|351+L+2bl
b1 b1
l_oL+___2_bl L+201
< -— ~ < .
— b—12mb—T [li Y'_ll_b1+ ]
Hence,
ml
Il_0L____+2b1 L+2b,
A - < -—Y,_ <b
2(333) — xeszuiliblalm 112k] b—l b12m [li 1" 1+ ]
(L+21b1)lo 1 [ L+2bl]
= su a z e, z—l”,_ Sb +
IEIigbl 1( ) 2b1m 771101 21;] l l J 1] 1 m
L+2b l ,
(2.3.9) = sup 01(z)(———1—)—0 B], V] = l, - -- ,m.

25112111 2b1 m

By (2.3.4), (L + 2b,)lo/2b1 m = 0(1). Thus, to prove (2.3.8) for l = 2, it suffices to

show

(2.3.10) sup B, = 0126(1).
J

21

To this effect, let

 

 

 

 

 

 

 

_ 1 "‘1 ’ L+2b1'
B' I: '—}/i- <b E i)
J mlbl i221: -|.’II] 1| — 1 + m d l6 I
1 "‘1 ’ L+2b,”
B. 2: E -—Y,-_ <b ,
1.? mlbl i=1 _ '23] 1| — 1 + m J
- 1 m‘ L
B- I: E E -—Y,-_ <b — , '=l,---, .
I] m1b1i=1 ['33] 1| _ 1+m] J m

Note that
81 .<_ IBJ' _ BJ" + “313' “ B1j|+ lBleEIEIIJ j=1a"' ,m.

Thus, to prove (2.3.10), it suffices to proves

 

 

 

(2...?)11) SllplBj—‘le = 055(1),
J
(2-3-12) SHPIBIJ—Bul = 019.5(1),
J
(2.3.13) sup BU = 0136(1).
J
Consider,
- 1 "‘1 L+2bl
E(BJ' — le2 = W; {E(|€i| ‘ E|€ill2E [I13 — Yi—il S b1+ m ]}
1 L + 2b1
S WEI:I$j_)/0|Sbl+ J
1 L + 2b .
S —2 (bl + m l)“91||007 V] :17. H ’m'

7721b?

 

 

m _ m L + 2b
2 E(B,~ — Bj)2 g m1b22 (b1 + m 1) ”ng00 ——+ 0, by (2.3.4).
J=1 1

In view of the Chebyshev’s inequality, this completes the proof of (2.3.11).
Next, consider (2.3.12). Let

L+2b1

 

flidzz [I$j_K—1|Sbl+ :la i=17"'iml;j:1a”'im'

22

Note that for any r > 0,

, L+2b . .
Efl.,.-=Efi1,jsz<bl+ m angina, v2.2.

 

This fact is used repeatedly in the proof below without further mention. Now,

ml

2
- 1
E(Blj — Blj)2 = m E {Z (5133' — Efii,j)}

i=1
1 1 m‘
<——E - ———§:§: .~, -.

1

From Lemma 2.3.3 we obtained that Vq, p, 'y > 1 such that 5 + $ = 1 — p,

Com... 131,.) s 2p(2a(lz' — knfi Ha..- II, || 51.]- ll,
1 L 2b
3 2p(2a(lJ'-k|))5 ((2b+2 f7, ‘)nglu..)

Let K, :2 b1 + (L + 2b1)/m. Now, by (A.6), a(k) 3 Cop", for some Co < 00,

+

o1—
QIH

 

p 6 [0,1), and V]: Z 1. Hence, for all 1 _<_j g m,

K K P

E(Bl,-—Bl,-)2 g C(— + 1" ml— (u(k))%)

mlbf m1b¥

But, Z:;."=‘1“(a(lc))S s filmed“); < 00. By (2.3.4),

1
m K. (11+L—tn2—bL [mf] L+2b1
— = m —— : —
mlbf mlbf mlbl mlbf

 

—>0.

 

 

With 0 < c <1/2 as in (2.3.4) choose a p > max{,3€26, 2_14c,1}. Then

 

 

 

1—l L+2b 1—l 1-1 1—1
rc P b + —L P __ mb " m L + 2b P
mlb1 mlb1 mlb1 mlbfml‘i?

Hence, 2;, E (Bl,- -— B1,)2 —+ 0. Again, in view of the Chebyshev inequality, this
completes the proof of (2.3.12).
Finally, note that sup,- Bl]- g [2(b1 + L—t-nQ—bl)||gllloo]/b1 = 0(1). This completes the

proof of (2.3.13), and hence that of the lemma.

23

Lemma 2.3.5 Under the assumptions of Lemma 2.3.2,

(2.3.14) sup Ifik(:1:)— pk(:z;)| = 0136(1). k 21,2.
$€Iibk

Proof: Again, we shall prove (2.3.14) for the case k = 1 only, and write 5,, Y,- in the
proof below for 51,,- and Y”, respectively.

Recall the definitions (2.3.3). Let

1 m‘
N073) = — #1(Yi—1)wb1($—Yi—1),
m1 i=1
- N016) ~ ._ “1(a)
MCI?) £71.b1(~’17) V77, 1 :1: '_ §I,bl($) Vfl’
DUI) == fi1($)-fl1(:v), D== sup |D(:v)|-
IEIibl

We shall use the decomposition [11 —- 111 = [11 — [21 + [21 — [11 + 111 — 111.
Now, let 7) : iinfxez 91(27). Fix an e > 0. Note that for every J: for which
g1,b,(a:) > 77, D(J:) = 0. Hence, the event D > 6 implies infxezib, g1,b,(a:) < n. This

together with the definition of 17, for all sufficiently small b1,

R; (D > e) 3 P5( inf g1,,,,(a:) < n) 3 P5 ( sup |§1,b,(:z:) — 91(1)] > 17) ——> 0,
xeubl xel’ibl

by Lemma 2.3.1. Hence we have

(2-3-15) SUP ”11(13)‘ [11(11” 2 0P5“).
xEIibl

Next, because

 

 

 

1 1 N(:17)
pun—m) = N<az:)(~ —- ) + .. -m(:v),
gl.b1($) V 77 gl,b1($) 91,51 (III)
for all x, we obtain,
811p |fl1(x)—m($)l
IEIinl
.51.!) (117)
S 811p lu1($)| 811p :———f— — 1, + sup |m($) - #1(y)|-
1613:2111 mEIibl 91,1210?) V 7] IEIib1,|x—y|gb1

 

24

By (A2),

sup |u1(:v) - #101)! = 0(1), 811p |m($)| = 0(1)-
math, lx-ylsbi xEIflb:

And, by Lemma 2.3.1,

glybl (1‘) __ 1

sup

xEIibl

 

 

Thus, we obtain

(23-16) 3215:; MICE) — #1($)| = 0P6(1)'

The identity,

’1‘ 3:11 01(Y,_1)e,-wb, (17 - Yi—l)

[1 1L" _/-—1' 1' : ml ~ 1 V55)
1( ) 1( ) 91,b1($)V77

 

and the Lemma 2.3.4 imply that supxerfl,1 “11(3) — fl1(a:)| = 0p6(1). This together

with (2.3.16), (2.3.15) and the triangle inequality proves the lemma.

Proof of Lemma 2.3.2: Again, it suffices to prove (2.3.6) for k = 1 only, and we

write 5,, Y, in the proof below for 51,,- and Y”, respectively. Let

”(ml—hi: :Zlai(K—1)5?wa1($ - Yi— 1)/91,($)aa1
W) = ‘73— :ol<m_1)(mm-l> — film—1m... (a: — n-l)/§1,..<x),
1 "'1 A 2 ~
vacc) = m—l Ewan--1) — mac--1» waxes — K_1)/gl,..(:c
Then rewrite
(2.3.17) dflx) = V1(:c) + 2 V2(1:) + V3(:I:).

We shall show that

(2-3-18) iglmx x)- 01(13 )l = 012.5(1)
(2-3-19) :21; |V2(m)| = 0125(1), 31:; IV3(2:)| = 0&0)-

25

By (A7),

:1.- 22:3. aflmwm (x - y.-.) _ 02m
g1,a1 (:13) l
S sup l0i($)"01(y)|:0(1)

$61, II-yISal

SUI)
2:61

 

 

This together with Lemmas 2.3.1 and 2.3.4 imply (2.3.18).

Arguing as in the proof of Lemma 2.3.4, one can show that

 

 

 

sug— Eda-l — E6|€1Dwa1(— YH) = 02.0),
me
which implies that
4-27“ IE'I'w (CU-Y )
m z:1 z 01 2—1
sup 1 ~ — E5 5 2 0p 1 .
361' 91,a1(1‘) I 1| 6( )

 

 

This together with (A.7) and Lemma 2.3.5 imply that

SUP |V2(a:)|

$61

_1_ m1
—_ e-w x—Y-
S SUP 01(3) SUP l#1(113)- fi1($)lsup m1 '"1|~" a‘( ' 1)
13611201 12613201 361 91,01 (x)

- 0(1)0P6(1) (Elell + 0136(1)) = 0P5(1),

thereby proving the first part of (2.3.19). Finally,

SUPV3(~”€) S SUP l#1(17) — [MUCH2 = 0106(1),
1:61 xEIibl

by Lemma 2.3.5, thereby completing the proof of the Lemma 2.3.2.
Now, we shall give estimates 13,,(116) )of vk(:c =\/u( )/(gk:1: ,k = 1, 2, based on
Yl,m1 : (Y1,0a YI,13 ' ' ' ) K,m1)a Y2,m2 = (Y2,09 l/2,11 ' ' ' a }/2,m2)

that will satisfy the assumptions of Lemma 2.1 to support the realization of Theorem

2.2.1.

Estimate of 1),, when u is known:

26

In the case u is known, to estimate 21,, we only need to estimate gk. The following
lemma gives the estimate of 1),, that satisfies the assumptions of Lemma 2.2.1 in this

case.

Lemma 2.3.6 Suppose (Al), (A3), (A5) and (A6) hold. Let ck be a sequence of
bandwidths satisfying m}; ck —> 00, for some c < 1. Then the estimate in, = fi/fim

will satisfy the assumptions of Lemma 2.2.1.

Proof: It suffices to consider the case k = 1 only, and write §1 in the proof below for
91,01. Let

1 . ,, ..
77 = 4 3152;9133) > 0, ”1 2 «ll/(91 V 77)-

Note that for every :1: for which §1(a:) > 77, u{(a:) — 231(17) = 0. Hence,

P5 (sup |uf(:1:) — 231(x)| > 0) S P; (igmx) < 17)

1:61

3 P6 (sug |§1(:r) — was)! > n) —+ 0,
2:6

by Lemma 2.3.1. This together with (A.1) shows that of is a modifications of {21
satisfying the assumption (2.2.4) of Lemma 2.2.1.

Next, we have

 

*—ua:=u:c —1 use—{~55 —:1:
ma) 1()| WHWM 91mg ()WIglnvn gm.

This bound together with (A.1) and the fact supxel |§1(:r) Vn— gl(a:)| 2 0p, (1), imply

sug |vI‘(:v) — v1($)| = 0195(1),
x6

thereby completing the proof.

Estimation of 22,, corresponding to optimal u:

In Section 2 it was observed that an asymptotically optimal test against the al-

ternatives (1.0.6) among the above tests T as u varies has the weight function my

27

of (2.2.8) depending on the unknown entities like 0k, 9),, k = 1, 2. Thus there is a
need to construct estimates of 1),, = Mug/9k, k = 1, 2, that satisfy the assumptions of
Lemma 2.2.1. One such set of estimates is given by

uacz:():= (W[$€I]/ 01.01 )+€12~62($) ), uk(:z:) = __ 1168:), k: 1, 2.

91 61( 92,62 (1:)

 

 

 

The following lemma shows that these estimates satisfy the required assumptions.

Lemma 2.3.7 Under the assumptions of Lemmas 2.3.2 and 2.3. 6, m, k = 1, 2, sat-

isfy the assumptions of Lemma 2.2.1.

Proof: By Lemmas 2.3.1 and 2.3.2, and the assumptions (A.5) and (A.7), we can find
modifications 0,3 *(x), 9,:(2: ) of 6,3(22) and gk,c,(:c), respectively, such that c 3 0,3‘ (:17) g

C and c S 9,:(13) S C for Va: 6 I and some c, C > O, k = 1, 2. For example, choose
(2.3.20) 03*(517) = (0,3(22) V c) /\ C, g,:(:c) = (gk,c,(a:) V c) /\ C, k = 1, 2.

Ilere,

c- — inf i111f20,3(:1: ) A gk(a:), C = sup sup 0,3(23) V gk(:z:).
$61k=x61k=1,2

This together with (2.3.20), Lemmas 2.3.1 and 2.3.2, imply that

(2321) sgglaiw—anol=op.<1), sgglg;(x>—gk(z>l=op.(1), k=1, 2.

Let

2*

115(17) = 6W: e 11/ ((11:38)) + (12:22,?)

 

 

Then, 2),: a:()= «in; H)/g,:( ,k = 1, 2, are modifications of 6,,(513), k = 1, 2 satisfying
the assumption (2.2.4) of Lemma 2.2.1.

Next, for all :1: E I,

   

 

 

 

IUZC’E )— UM 6(xg;($)u

 

28

By (2.3.21) and the boundedness of 0,3*(a:), 0,3(23), 9,:(113), gk(:r) and 6(23) over I, ua is
bounded on I,

 

 

 

   

 

 

 

 

 

 

 

 

 

 

325’ Wage) ‘gkml ”all
and
3211) 912(23)
upéwx 11ml] 3/0 ohm) use) _\/of(a:) 03a)
3 gum) 3221’ “91m” 95a) q‘gl(x)+"292(x>

 

 

I: 0135(1), k =1, 2.

Thus, super |u,:(z) — 1111(3)] = 0106(1), thereby proving the lemma.

Now, we shall prove (2.2.9) for the i of (2.2.10) with 13,, based on the entire data
set (Y1,,,,, ng) as follows. Recall that 9,, denotes the kernel estimate of the density
9;, based on the full sample size 711C and the window width ak. Let m1c = n1c in the
definitions in (2.3.3) so that now 0,3 is an estimate of 0,3 based on the full sample.
Similarly in, and &k are now based on the full sample sizes. Use these estimates in
the definition of i2 of (2.2.10). The consistency of such an estimator of 7' is given by

the following

Lemma 2.3.8 Under the conditions of Lemma 2.2.1 and Lemma 2.3.2, “f satisfies

(2. 2. 9).

Proof: It suffices to show

(2.3.22) 5:22 01( (Y1 ,-_1)”y1 (Y1 1- /u2(:1:)s3(x)g11)= (:13) dx + 0126(1).
i=1
Since E03(Y1 ,-_ 1)'yl( (Y1 1. 1) =f3 u g1(:1: )da‘, for all i, by the Ergodic theorem,

it suffices to show that

(2323) ”15:01(Y1i—1)71(Y12-_ _n12012(3/11—1)’71(}11—1)=0P5(1)-

i=1 1'21

29

Now, rewrite the left hand side of (2.3.23) as

1 "I . - .
a" ((’Yi(Y1,i-1) -7?(Y1,.-_1)) Gal/1,91) + (a?(l’1,,~_1)—a?(Y1,,-_1)) 7?(Y1,.-_1)).
i=1

By Lemma 2.3.2 and the boundedness of 01(13) over I,

1 "‘ A .
a: Dam/1,.-.) — ofm,.-_1))[Y1,.-_1 e I] = 0.3.0), Supoim = 0:241)-

i=1 $61
By Lemma 2.1, ’y1(:r) is essentially consistent on I for 71(1‘). This together with the
boundedness of 71 imply
1 "1 A2 2 2
E §(71(Y1,i—1)- 71(Y1,.-_1)) = 0126(1). 21:11) 71 (It) = 0(1)-

These bounds yield (2.3.23) in a routine fashion, thereby completing the proof of the

lemma.

2.4 Proofs

Here we shall give the proofs of Lemma 2.2.1, Theorem 2.2.1 and Lemma 2.3.1. Their

proofs are facilitated by the following result from Bosq (1998; Theorem 1.3, pp 27-30).

Lemma 2.4.1 Let (Xi,i E Z) be a zero-mean real-valued process such that 511131951:
||X,-||00 g b and Sn 2 2le X,. Then for each q E [11%] and each 6 > 0

2

—8 q 4b % n
< —— _
p(|S,,| > ne) _ 4 exp {8v2(q)} + 22 (1+ 6) q a([2q ),

where, the function a is defined in the Definition 2.1, and

2 be n
2.4.1 2 z: — 2 _ = _
( ) v(t1) p20(Q)+2, :9 2g,
02(9) 3: max E{(ljpl + 1 - jplXup1+1 + XUp1+2 + ' ' ' + Xup1+p
0313(20-1)

+((j +1)p - [(J'+1)pl)X[<j+1>P+11}2'

30

Proof of Lemma 2.3.1: The following proof is similar to that of Theorem 2.2 in
Bosq (1998). It suffices to prove (2.3.2) for k = 1. As before, write E in the proof
below for E6.

Note,
(2-4-2) |§1($) - 91($)| S @1017) - E91 (17” + IE91(1=) - 91($)|, V x E R-

First, using a change of variable formula, we obtain

IE91(17) — 91($)l

/g1 (:1: — at)w(t)dt — /g1(a:)w(t)dt' g sup |g1($ — at) — g1(:r)|.

ItIS1

 

By (A.5) and the fact a ——> 0,

(2.4.3) sup |Eg1(:1:) — g1 (11:)I g sup |g1(:1: — at) — 91(1)] —> 0.
IEIia xEIiaJflSl

Next, consider the first term in the upper bound of (2.4.2). We shall use Lemma
2.4.1 to prove

(2.4.4) sup 191cc) — Ema)! = 0P5“)-
xEIia

This with (2.4.3) then completes the proof of the lemma.
To prove (2.4.4), fix an x E Zia, and let X,- = wa(:c—Y1,.~_1)—f w(t)g1(a:—at) dt.
Then,

A A 1 "1 1
g1(a:) — Eg1(a:) = 71—1 2 (watt — Y1,i_1) — /w(t)g1(a: — at) dt) = — 2X1.
i=1 '
By(A.3) and (A.5), for a small enough, suplgg,l |X,-| g Ilwllwa’l, and by the

change of variable formula,

(2.4.5) EX? 3 / a-1w2<t)g.(x — at) dt 3 a-luwnmugulw.

31

Consequently, using the fact that for any square integrable r.v.’s Z1, - -- , 2,., r 2 1,
E(Z1 + - - -+ Z,)2 S r2 mains-S, EZ,2 and the stationarity of X1, 2' E Z, here the 02(q)

of (2.4.1) satisfies
(2.4.6) 02(q) g (p +1)2EX12 g 4p2 a—1||w||ool|g1||oo,
with n 2 n1 and p = n1/2q. Hence, the corresponding 222(q) of (2.4.1) satisfies

_ l
22201) s a 1 (suwuoongluoo + 5 llwllooé) .

Thus, by Lemma 4.1 applied to the above {Xi}’s with n 2 n1, we obtain

 

e2qa
P6(l91(33)—E§1($)| >6) 5 4exp{— }
8(8llwllmllglII.o + g lelooe)

l

4 a
+22(1+ M...) (10.31.
at 2q

 

By assumption (A.6) and the Definition 2.2, a(k) 5 Cop", V k 2 1, for some Co < oo,
1
and some p E [0, 1). If we choose q = [nfa'i] , then, for each 6 > O, the above upper

bound is bounded above by a constant multiple of

 

1

62[n1§a’%]a 4llwll % 1 1 “lei

exp _ + (1+ ..J) [71501—5] Copl 2 ]
8(8leloollgllloo + % ll’wllooél 0‘— 1

l
s flexp {and}.
where fl = we, limit... Ilglloo) and 7 = we, llwlloo. llglloo) are constants. Hence,
1
(24.7) p, (lam — Eileen 2 e) s fiexp {mace}.

Let ||I|| is the length of I. For any set A, let A" denote its interior. Consider
covering B = {:13 : a: 6 I :1: a} by J'nl closed intervals Bjnl, 1 S j S Jnl with equal
length such that Bf,“ 0 B31,“ 2 d, for 1 g j as j’ g Jm, and I231 — $2| 3 W, for

all $1, .732 E Bjn1,1SjS Jm. Let

Ajni ;: 311p @101.) _ E91($)la IS j S Jnr'
:L‘EBjnl

32

Note that

sup Ajn, =2 Am,

SUP |§1($) - E§1($)| =
:rEIzta ISjSJnl
By assumption (A.3), there exists a constant I < 00 such that

|w(u') — w(u)| g ”11' — ul, Va, 11' 6 [—1,1].

Let 30,-”, be a point in Bjn, for 1 _<_ j 3 Jul. Consequently, we obtain, for all :1: E Bjm,

1 S j S Jnla
- . l I +2a . - l I +2a
191(221— 91mm- ———(” '2' l, 1591.). Ema...» s JLLL—Z.
0’ Jul a Jul

Thus if we choose J",1 = [2l(||I|| + 2a)a“2 log2(n1)] + 1, and let Afin = suplsjsjnl

l91(33jn1) — E91($jn1)l, then we have
1

Am 3 A; + .
1 1082011)

 

By (2.4.7) and the assumption that aNc -—> 00 for some 0 < 1, for any 6 > 0,

.1711 l 1
P5(A;11> 6) S 2P6 (l§1($jn1) _ Egl($jn1)l > 6) S Jul 5 exp {—7 120;}
1:1

([2l(||I|| + 2a)a’2 log2(n1)] + 1) flexp {-"ynl%a%} —> 0,

thereby proving (2.4.4), and hence the lemma.

Proof of Lemma 2.1: It suffices to show that ’y1 is essentially consistent on I for

’71. LGt
It 1 * n2 *
71(33) 2 '7; ”1(37) 202(Y2,i—1)wa($ " Y2,1—1)-
i=1
Then,
1 "2
7H1?) - 11(17) = (11W?) ’ 171(fl3lln—2 293(Y2,i—1)wa(33 — 161-1)
i=1
1 ”2
+(@1(:v) — ran—2- Swarm—1)— oz<n,.-.))w.(x — Y2.)
i=1
1 "2
+vi(33)n— 201303.91)— 92(Y2,i-1))wa($ — Y2,i—1)a
2 i=1

33

and

sup |7I(x)—%(:r)| s sup Ivi‘(:r)—2‘21(x)|sur>v5(x)sup§2(x)
er xEI er zEI

+ 811p I’UICT) - 01(Ivll SUP lv3($) - 132(Ivll sup 92(27)
x61 xEI xEI

+ 31:11) 11112:) i212) |v$(w) - i32($)| :13 92(23)-
This bound, Lemma 2.3.1, (2.2.4) and the fact that 111‘, v; are modifications of 131 and
'02, we obtain that '7; is also a modification of ’y1. Thus, it suffices to show that the
modification 7112:) is consistent on I for 71 (:13)
Towards this goal, let

1 "2 1
em) = '73,; ng(n,.—-.)w.(x - 13.-.).

i=1

Note that for all a: E I,

|71‘($)- 71($)| S lvi($)| |171(:v) - x/WJH + lx/5($)| I’UI‘($) - v1($)|.

Therefore, in view of (A.1), (2.2.4) and (2.2.5), it suffices to show that

(2-4-8) :21} |171(x) - V593)! = 010.5(1)-

Let
Um := £2-gem.-.)—v2<Y2,._1))w.(x—1e,.-.>,
Uzo) == 5— M(my...)—..<.))w.<z-1e,.-.).

Then, 171-— \/— = U1 + U2 + v2(§2 — 92) and hence
|171($) — WWI S |U1($)l + |U2(IL“)| + |v2($)(§2($) - 92($))|. V39 6 I-
But,

sup IU1(I)I s sup ruse) — We)! sup 1.2021.
161 xEI xEI

34

Since 222(3) = fi(:r)/gg(:r) is continuous on the compact set I, it follows that

sup|U2(-'II)| S sup |v2(w) - v2(y)|SIéII>|§2($)I = 0(1)S:I;|§2($)|,

:cEI x,y€I lz—ylga
51:11) lv2($)(.512(33) ’ 92017)” S 0(1) 516111) |§2(1‘) — 92W”-

From these bounds, Lemma 2.3.1, and the condition (2.2.5) of Lemma 2.2.1, we obtain

(2.4.8), thereby completing the proof of Lemma 2.1.

Proof of Theorem 2.2.1: Again, we write E for E5 in the following proof. Recall

(2.2.1). To prove (2.2.6) it thus suffices to prove the following.

(2.4.9) N1/2|T4| = 0M1),

 

(2.4.10) N% N-%T3 — 9| = op,(1).
. 1 ""
(2-4-11) N5 a; ZWKYm—l) — ’Yk(Yk,i—1 ))0k(Yk,i—l) 51m = 0195(1), k = 1,2-
i=1

 

By the Assumption 2.4, there exists a '7; such that it is a modification of &k and

is consistent on I for 7k 2 u/gk, k = 1, 2. Hence,
(2-4-12) 511117) |‘lk($) — 7k(17)l = 0135(1), k =13-
1:6

From (A.2), it follows that |u2(:c) — u2(y)| S cla: - yl, for V113,y E I and some
c > 0. Then, by (2.4.12), (A3), (A.4) and the boundedness of 71,

 

 

 

N% m:
N% n1 n2
7. n2 2201(m-1W2Wa-1)“$201.24) — #2(Y2.j-1)|wa(Y1,.-_1 — Y2.-.)
1 i=1 jzl
N% 1 ”2
S n n 2201(Y1,i-1)02(16,j_1)cawa(Yl,,_1 - Y2,j_1)
1 2 1:11.:1
cNia "1 .
2 n1 271(Kvi‘1): 0136(1),
i=1

thereby proving (2.4.9).

35

Next, let
T3,1 1: %:(%(K,i-1)— 71(1’1,,_1))6(Y1,,-_1),
T33 :2 51;: (71(Y1,,-_1)6(Y1,,-_1) - /u(2:)6(:1:) dais) .
Then,
N%(N-%T3 — o) = .731: :&I(n,_l)5m,_l) — / u(:r)6(a:) dz: = T3,1 + T33.
By the Ergodic theorem, T33 = 0P5(1)- The boundedness of 6 on I and (2.4.12) imply

T3,1 S sup lids?) - 71 (1r)| sup 5(23) = 0125(1)-
161 1361'

This proves (2.4.10).

To prove (2.4.11), consider the case k = 1. Let

(«n—1]
1 1 .
W1 1: Nit; ;(’71(Yl,i—1)—71(Y1,i—1))01(Y1,i—1)51,2},
1 1 "1 ,
W2 = N5{"’ 2 (71(Y1,z'—1)'—71(Y1,i—1))01(Y1,i—1)€1,i}-
"1 cum—11H

The left hand side of (2.4.11) for the case k = 1 is equal to the sum W1 + W2. Hence,

it suffices to prove:
(2413) Wj = 0136(1), j=1,2.

But, W1 is bounded above by

1 1 [Wt—ll
51:11) “31013) — 71(3'3))01(173)|N5 a Z l€l,z‘|-
3‘ i=1

Since Nijfif/nl = 0(1), we readily obtain that Ni 51; BET lag) = 0125(1). This

together with (2.4.12) and (A.7) proves (2.4.13) forj = 1.

36

To prove (2.4.13) for j = 2, note that by construction, "u(:c) is a function of

Yllfrfilv Y2”,2 and 2:. Let ZN 2 (Y1) n, , Y2,n2). For simplicity, let

A(Ym—l) 3: (&l(}/l,i—l) — 71(Y1,i—1))01(Y1,i—1),

 

N "I .
E1 2: ('T—zi Z A2(Yl,i-1)€¥,i ZN),
1 i=[‘/n_1]+1
N "1 . .
E2 2: -n—2 Z E(A(Yl,i_1)A(Y1,j_1)El,i 51,.) l ZN ).
1 j>i=[\/n—1+1]

Then,
E(W22|ZN) = E1+2E2.

By (2.4.12), (A.7) and the fact that E511 E 1, and by a property of conditional

 

 

expectation,
N "1 .
E1 = 7—,? Z E(A2(Y1,._1)e¥,.- ZN)
z=wm+1
N . 2 2
S — sup(71(x) - 7106)) 81190109) = 019.0),
"1 2:61 2:61
2N "1 . .
E2 = 33,—. z E(E{A(Y.,._1)A(x,._1)el,.a.,,~ (Y1,.-1,Y2,...)}|z~)
J>1=[\/n_1+1l
= 0.
Hence,
(2.4.14) E (W,2 | ZN) = op,(1).

By the Chebyshev inequality and the property of conditional expectation, for any
e>0andany60>0,

P5 (W22 > ()0)

6 6
z p. (w,2>ao,E(w;|zN) >670) +1». (W22>60,E(W22|ZN) g 670)

_<_ P6(E(W22lZN)>—O)+ ( [ 60 ])

 

s P.(E(W§IZN)>@)+

37

This together with (2.4.14) completes the proof of (2.4.13) for j = 2, and also of
(2.2.6).

By the Martingale Central Limit Theorem, the sum
1 n1 1 n2
:7: ;U(YI,i—1)31(Y1,i—l)5l,i + E ;U(Y'.2,i—l)32(y2,i—l)52,i-
can be shown to be asymptotically normally distributed under P5. This fact together
with (2.2.6) in turn implies the claimed asymptotically normality of N i (T — 9) under

P5, thereby completing the proof of Theorem 2.2.1.

38

Chapter 3

Testing the equality of two
regression curves with long-range

dependent errors

3.1 Introduction

This chapter is concerned with testing the equality of two regression functions against
the two-sided alternatives when the errors form long-memory moving averages. More
precisely, one observes two independent processes Y1.i and Y2,“ 2' = {0, 1, - - - } obeying
model (1.0.3) and (1.0.4). The problem of interest is to test the equality of two
regression functions ”1, 112 on [0, 1] against the two—sided alternative hypothesis (1.0.5)
based on the data Y1,1,- -- ,Y1,n and Y2,1,- ~ ,ng, where n is a positive integer. We
write Ho and H, for the null and alternative hypothese of (1.0.5) in this chapter.

In this chapter, we shall study the asymptotic behaviors of T1, T2 given in (1.0.13)
as the sample size n tend to infinity. Corollary 3.2.1 of section 2 shows that under H0,
T1 and T 2 of (1.0.12) weakly converge to some functions of a fractional Brownian mo-

tion over [0, 1]. Then in Corollary 3.2.2 and 3.2.3, under a general set of assumptions

39

on the estimates 01, 02 and H1, H2, we derived the same asymptotic distributions of
T1 and T2 under H0. Remark 3.2.2 proves that the power of the test based on T1 and
T2 converges to 1, at the fixed alternative ( 1.0.5) or even at the alternatives that con-
verge to H0 at the rate 71“: for some 0 < c < 9%”? We also obtain an expression for
the asymptotic power at the sequences of alternatives that converge to H0 at the rate
Tn. In section 3, under some additional conditions, estimates 61, 62 and H1, H2 are
constructed that satisfy the conditions of corollary 3.2.2 or 3.2.3. Section 4 contains
some additional proofs. Section 5 reports the Monte Carlo simulation results.

Throughout, R 2: (—oo, 00).

3.2 Asymptotic behavior of T1, T2 and T1, T2

This section investigates the asymptotic behaviors of T1, T2 given in (1.0.12) and
the adaptive statistic T1, T2 given in(1.0.13) under the null hypothesis and the al-
ternatives (1.0.5). We write P for the underline probability measures and E for
the corresponding expectations. First recall, say from Beran (1994), the following

definition:

2.1 Definition. For 0 < 0 < 1, a stochastic process 89(3), 3 6 IR, is called a
fractional Brownian motion of index 0 on the real line R, if it is a continuous Gaussian
process with mean 0 and the covariance function COV(B9(t),Bo(S)) = %(|t|2“9 +
|s|2‘9 — It — SIM), t, s e R.

The following 3 lemmas are well known. Recall the notation (1.0.10) and (1.0.11).

Lemma 3.2.1 The long memory linear processes {um}, i = 1,2 of (1.0.4) satisfy

thatforlgt,s§n and |t—s|—>oo,

cov(u,-,t,u,-,s) ~ pglt — s|_9iL?(|t — sl), 2': 1, 2.

4O

Proof: For i = 1, 2, when |t — s| —) 00, by the Karamata Theorem,
00
Z ai.jai,j+It—sl
i=0
_(1+0i)/2 1

_ 2 j j
— t_ 91' l.( +t_ ( +1) —-—+O.’_
l 8' i“ I SI) (It—3| It-sl ) lt—sl It 8'

(3.2.1) ~ p.1t- 8| ”amt — 3|).

 

The lemma is proved, since

COV (ui ,ta “i, s) :15: 0:01' ,jai,j+|t— 3|

Let

(3.2.2) 2'(,- := 0,: u,,, i = 1, 2.

TIT" i j=1

 

Lemma 3.2.2 The long memory linear processes “id, i = 1, 2 of (1.0.4) satisfy that

for0$t,s$1, asn—->oo,

(3.2.3) cov(Z,-(t), Z,(s)) —> (3241' + t2_0i — |t — s|2_9i), i = 1, 2.

l\.’)l|--I

Proof: Consider the case t > 3. Let

 

[nil
N1 = cov(Z,(s), z,(s)), N2— _ cov(n::i Z w, 2(5)).
J=[ns]+1
Then,
(3.24) COV(Z,'(t), 21(8)) 2 N1 + N2.
Rewrite N1 as N11 + N12, where
022 [113] [11810-2 [as]
N112—2Z Z cov(u,-j,u,k),N12=n—2——;2 Zeov( uij,u,-J-)—-)0.

27-1:,1'J-_1j¢k_1T,ni j._1

41

This, together with Lemma 3.2.1 and the fact that for any :1: > 0, L,(na:)/L,-(n) —+ 1,

 

 

 

as n —> oo ,
[n3] [113]
N1 N N11 N n27' 2 2: Z .012 l] kl 0442“] ‘kll
7",” J'=1J'¢k=l
_115: g ,_-__,-..1.1,1<| an)
— 2
2:" j: —1#k= 1 L01)
p2
(3.2.5) ——> 42/ /8 lat-yl-gidxdy = 32")".
“32' 0 0
Similarly,
02 [nt] [n3] 0'2 [nt] [ns]
N2 = ”—27—:‘2: Z ZCOV(U2‘,j,U1,k)N n27 2 2 2.010 fliLEU—kl
J=[ns]+1k=1j:=[ns]+1k l
1
(3.2.) 6) —+ —f [03( :2: — 91' dxdy— — 5 (t2_‘li — 32‘01' — (t -— s)2“9i) .
From (3.2.4)-(3.2.6), we have (3.2.3). [:1

The proof of the next lemma can be found in Davydov (1970), Taqqu (1975),

Avram and Taqqu (1987), Sowel (1990), among others.

Lemma 3.2.3 The long memory linear processes um, i = 1, 2 of (1.0.4) satisfy the

invariance principle: For 0 g t S 1,

Z,(t) Dig] 391.05), 2‘: 1,2,

where Bgi(t) is a fractional Brownian motion defined in Definition 2.1 and=> Dol 1]

stands for the weak convergence of random elements with values in the Skorohod
space D[0,1], with respect to the uniform metric. Moreover, we have SUPogtgi |Z,-(t)|

Remark 3.2.1 Lemma 3.2.3 implies (1.0.9).

Next, let

(3.2.7) T(t) = @2101) — 3‘1 Z2(t), 0 g t g 1.

7'1': 7.11

The following theorem shows the asymptotic distribution of T(t (t,) t 6 [0,1].

42

Theorem 3.2.1 The long memory linear processes um, i = 1, 2 of (1.0.4) satisfy:
(3.2.8) T(t) => BglAg,(t), 0 g t g 1,
in D[O,1], with respect to the uniform, metric.

Proof: First, we observe that

732/73 —+ 1, for 0,- = 61 A 02,

—> 0, otherwise, i z: 1, 2.

This together with Lemma 3.2.2 and the fact that {u1,J-}’s are independent of {u2,j}’s

imply that for 0 g s, t g 1,

 

2 2
Tn,i
c0V(¢’"(1),C’"(S)) = £1: 7,2; COV(Z.-(t). 21(8))
1
(3.2.9) _2 5 (32—19le2 +t2-01A92 _ It _ Slz—omoz), n _) 00

Then, by Lemma 3.2.3, the definition of fractional Brownian motion and the fact that

{v.1 J},S are independent of {u2,,-}’s, (3.2.8), and hence the theorem follows. [I]

Now, note that under the null hypothesis, T1 and T2 of (1.0.12) satisfy

1
T1: sup |T(t)|, T2 =/ T2(t) dt.
0

0951

Hence, we have
Corollary 3.2.1 Under the model (1.0.4) and the null hypothesis,

1
T1 => sup |Bg,A92(t)|, T2 2] B§1A92(t) dt.
0

0931
Next, we need the following additional assumption to obtain the asymptotic dis-

tributions of T1 and T2 given in (2.2.5).

Assumption 3.2.1 Let Hi, [1,-2 be estimators of H,- and of, respectively, satisfying
(3.2.10) (log n)(f1.- — H.) —2p 0, a? — of —1p 0, n —> 00, i = 1, 2,

43

under both null and alternative hypotheses.
Assumption 3.2.2 Suppose H1 2 H2 = H, L1(n) = L2(n) = L(n). Let the

estimators H, (‘72 of H and o2 = 012 + 02 be such that
(3.2.11) (logn)(H—H) —)p 0, [72—02 -—>p O, n—->oo,

under both null and alternative hypotheses.

Note: Under the Assumption 3.2.2, by (1.0.10), we have Isl = 102 = It, 91 = 02 = 0

A A

and so are F01 = k2 = is, 61 = 62 = 2 being their estimates respectively. Also we have

1
(3.212) T1— — sup Un(t) - T2 = . 1 . . / U3(t)dt
03131 FtL(n )n H 10 It2L2(n)n2H‘202 0

Corollary 3.2.2 Suppose that the Assumption 3.2.1 holds. Then under the model

 

 

(1.0.4) and the null hypothesis,

1
T1:> sup '801A92(t)l1 T2 :>/ 331A02(t) dt
0

03t31

Proof: It suffices to prove

(3213) 11:2 sup 131294111.

03:31
since we can obtain the other part similarly. We notice that T1 = (Tn/in)T1. From

Corollary 3.2.1, to prove (3.2.13), it thus suffices to prove

722
(3.2.14) _1; —>p 1.
Tn

By the definition of it? and simple calculation, (3.2.10) implies that

~2
01
2 2
H101

“2
Hi

 

23—11,? —>p0, and —2p 1, i=1,2.

This together with the first part of (3.2.10) gives

 

*2 2 2 2H 2222
- K. o n ‘ K: o 2
n,z=_i_¢__ z 28 210 Hi—Hi -> 1, .21,2,
T3“. K112012722}! K1201? Xpl g(n )( )l P 2
which implies (3.2.14). This corollary is proved. E1

The proof of following corollary is similar.

44

Corollary 3.2.3 Suppose that Assumption 3.2.2 holds, then under model (1.0.4) and

the null hypothesis,

1
T1 => sup IBg(t)l, T2 =>f B§(t)dt.
0

03131

Where, T1 and T2 are give in (3.2.12).

Remark 3.2.2 Testing property of T1 and T2. Under the model (1.0.4), consider

the following alternative that is the same as in (1.0.5):
Ha: [11(03) — u2(:r) = 6(3) ¢ 0 for some J: 6 [0,1],

where 6 is continuous on [0,1] since #1, M are continuous. Theorem 3.2.1 and its
corollaries suggest to reject the null hypothesis for large values of T1 and T2 given in
(2.2.5) under Assumption 3.2.1 or given in (3.2.12) under Assumption 3.2.2.

First, suppose Assumption 3.2.1 is satisfied. Let

 

 

2 l 1 Tn 7;,
13(1) = :- Un(t), T(t) = .21 21(1) — 32 22(1).
Tn 7'1‘: 7'n
Then,
,. A 1 [nt] J
T = T t h t h t — —— —
1(1) (1+ (1. () 1.nj=l“(n)
By the fact that
1 [ml j t
— 2124—2 6(x)d:r, n —> 00
n , n 0
J=1
[nt] . t
1 1 1
(3.2.15) h(t) = 7 — (1) ~ 7— / 6(2) d113, n —> oo
7-,, n j=1 n r" 0

This, together with the facts Tn -—> 0, (3.2.14), and the fact that fat (5(a) dz is not 0

for all 0 3 t 3 1 implies,

(3.2.16) sup |h(t)| —>p 00, n ——> 00.
03131

45'

Hence, in view of (3.2.8),

(3.2.17) T1: sup IT1(t)| = sup |T(t)+h(t)| ——>,. 00, n —> 00.
03131

03131

Let

(3.2.18) 81 = sup lBomo.(t)|,

03131

and define bm such that P(B1 > bm) = 01. Then by Corollary 3.2.2 and (3.2.17),

li_>m P(T1 > bm) = a under Ho and lim P(T1 > hm) = 1 under Ha.

n—>oo

Hence, the test based on T1 is consistent for Ha. Similarly, T2 is also consistent for
Ha. The consistency of the tests T1 and T2 of (3.2.12) under Assumption 3.2.2 follows

similarly.

Note: By using the same arguments as above, we even can claim that under
Assumption 3.2.1 or 3.2.2, the tests based on T1, T2 are consistent for the alternatives
converging to the null hypothesis at rate n‘c for all 0 < c < M, since (3.2.16)

2

is still satisfied when 6 (:13) is replaced by 6($)n‘°. Now consider the limiting power

when
111(1):) - 112(3) = 721503), :1 E [011]-

By (3.2.15) and (3.2.17),

T1= SUP |T1(t)| => SUP lB01A92(t)+g(t)lv
03131 03131

where,
1
g(t) 2/ 6(03)d:1:.
0
Similarly,

T2 2 A (801A02(t) + g(t))2dt.

46

Therefore, the limiting powers of the asymptotic level 01 tests T1 and T2 are
lim P(T1 > bm) = P ( sup |BglAg,(t) + g(t)| > bm) ,
n-—>oo 03131

1
11m P(T2 > b2,a) = P (/ (BOIA02(t) ‘1' g(t))2 dt > b2,a) a
0

“Hr—>00

where 52,0 is defined such that

1
P (/ Bglez (t) dt > (22,0) 2‘ 0!.
0

3.3 Construction of Hi, 1:], (“7,2 and 62

In the previous section, we showed in Corollary 3.2.2 and 3.2.3 that, if Assumption
3.2.1 or Assumption 3.2.2 is satisfied, then the test statistics T1 and T2 are weakly
convergent to some functions of a fractional Brownian motion on [0, 1] under the null
hypothesis. Also in Remark 3.2.2, the tests are proved to be consistent for some
general alternatives.

In this section, under some additional conditions, we shall now construct esti-
mates of Hi, 0,2 that satisfy Assumption 3.2.1 or the estimates of H, 02 that satisfy
Assumption 3.2.2. To estimate of, we need to give kernel estimate of the regression
function 11,-. We will denote the kernel function we are going to use as K and its

bandwidth as b. Consider the following conditions:

(A.1) The regression functions #1, 112 are Lipschitz-continuous on [0,1].

(A.2) The kernel function K is a Lipschitz-continuous density on R with compact
support [-1,1].
(AB) The bandwidth b is chosen such that b —> 0 and nb —+ 00.

(A.4) For 1' = 1, 2, the third and fourth moments of 8,1,1 exist. i.e. 1),-,3 2' E521, and

UM 2 E1311 exist, where am is as in model (1.0.4).

47

(A5) The two slowly varying functions L1(a:) and L2(:z:) in the model (1.0.4) have
positive and finite limits L1 and L2, as a: —) 00, respectively, and satisfy the

following conditions: For some gig—1 < ck < 0, gig—3 3 dk < 0,

(3.3.1) 200: ———L"(j) " [”22”A = O(/\C*),

.1121.
j=1 .7 2
(3.3.2) 3% (Z 1%iew) = 00.“), A —> 0+, 11 = 1, 2.
2:1 2

(AG) For 2' = 1,2, for some 1 > A2 2 H,- 2 A1 > 1/2 and for some 60 > O, as

n—>oo,

2_2Hi ,n 1

i=Hi-— -

 

2 m,- 51 .
(logn) (7;) + 1—2max(5o,H,~-A1) —> 0, m, E [1,

1

Remark 3.4.2 below shows that ARFIMA(O,d,O) model satisfies the assumption (A.5).
First, we will construct the estimates of Hi, 2' = 1, 2 that satisfy Assumption 3.2.1

or the estimate of H = H1 2 H2 that satisfies Assumption 3.2.2. The following

estimators are analogues of the estimators defined at (4.8)-(4.10) in Robinson (1997).
Fori=1,2and k=1, , mie [1,-'21), Let

(3.3.3) (En-(h) = i inh—‘Imou, 151(1) = logé.(h) — (2h — 1% Dog)...
i k=1

' 11:1
where, Ak = 27rk/n and
1 " .
3.3.4 1.2.: .,\ .—)., .,\= 11.“.
( ) Y.Y.( ) wY.( )wY.( ) wY.( ) (237%; ,18

 

Recall, e” always represent the complex value cosa: + isin as, i = \/—1.

Define
(3.3.5) H, : = arg minhE[A1,A2]R,-(h), 2' = 1,2,
(3.3.6) H I : argminh€[A1,A2](R1(h/) + R2(h)).

The following lemma shows the consistency of these estimators.

48

Lemma 3.3.1 Suppose the model (1.0.4), and the Assumptions (A.1), (A.4) - (A6)

hold. Then, the estimates Hi, i = 1, 2 given in (3.3.5) satisfy Assumption 3.2.1, i. e.
(3.3.7) (log n)(H,- — H.) —>P 0, i = 1, 2.

If, in addition, the Assumption (A.7) holds, then the estimates H1 2 H2 2 H given

in {3.3. 6 ) satisfy the Assumptions 3.2.1 and 3.2.2, i.e.
(3.3.3) (log n)(H — H) —+,. 0.

Proof: The proof is given in Section 4 below.

We are now ready to define the estimates of of that satisfies the Assumption 3.2.1.

We first need to give kernel estimates of the regression functions #1, [12. Let

. 1 n nx—t ,
(3.3.9) Mac) — 7—,; ;K(7)m 2—1,2,

where K is a kernel density function on the real line with the compact support [-1, 1],

b > 0 is the bandwidth. Define
(3.3.10) 61-2 :: (2&3)

where, éia’ is am- given in (1.0.3) with H,- replaced by its estimate H,- given in (3.3.5)
or H given in (3.3.6) if (A.7) holds. The following lemma gives the consistency of the

above estimates 6?, i = 1, 2.

Lemma 3.3.2 Suppose the model (1.0.4) and the assumptions (A.1)-(A.6) hold.
Then, (7,2 given in {3.3.10} is consistent for of, i. 6, satisfies the Assumption 3.2.1.

i.e.
(3.3.11) 33—0,? —>p 0, 1:1,2.
To prove Lemma 3.3.2, we need the following lemma.

49

/
11—:

Lemma 3.3.3 The long memory linear processes “id, 2' = 1, 2 of model (1.0.4) satisfy
1 n 00
(3.3.12) ?1' 2113,]. —+p A,, 21,-: Zaij, 1: 1, 2.
'21 j:0
Proof: It suffices to prove (3.3.12) for the case i = 1, since it is similar to obtain the
other part. For simplicity, in the following proof, we will replace “1.)“: 014‘ 51,]- and A1
with u,-, aj e]- and A, reSpectively.
By the Markov’ s inequality, it suffices to prove E ( Ej- 1u2- — A)2 —> 0, or to

J

prove

(3.3.13) E(% 211,2)? = $5 ZZE(u§u§) —> A2.

By the fact that {8], j E Z} are i.i.d standard r.v.’s,

OOWOO

2 2
E(uiuj) : E : E : E : E :E(ak1akzakaahgi-klEi-kng-kagJ—h)

181:0 k2=0 k3: 0k4=0
(3.3.14) = V1 + 2v2 + 14,,

where,

00
. _ 2 2 2 2
V1 - — E , E : E(aklak351—k15j—k3)a

k1=0 i-k1¢j-k3

oo
. _ 2 2
V2 ' “" 2 E : E : E(ak1aki+j-iak2ak2-t-j-i5i—k1Ei-kgla
k1=0i—k19’:i—k2
. _ E 2 2 4
5/3 ' _ E(ak1 akl‘l‘j—iEi—kl)‘

i—kl =i-k2=j—k3=j—k4

By the fact that E8; = 1, aj = O forj < 0, 011.1 < 1 forj large enough, (A.4) and

(3.2.1),
V1 = Z Zogaklak3E(€ 51—2—31)E(5§—k3) _ Zai1agl+1—i(E5?-kll2
k1=0k3= k1=0
k1=0 k1=0
(3.3.15) = A2 + 0 (lj — 2141mm — 21)),

50

oo oo oo
— . . u . _ 2 2
V2 — 2 E : E :aklakl+3‘lak2ak2+J-1 2 E :aklak1+j-i
k1=0k2=0 k1=0

00 00
_ . q 2 __ 2 2
- 2(2 :ak1ak1+lJ-1|) 2 Z ak1ak1+lj-i|
[cl—0 k1=0

m w

_ 2

'_ O (E :akiak1+Ij—i|) + E :aklakiflj—il
klzo k1=0

(3.3.16) = 001—3411110140) ,
and

00
_ 2 2 , 4
V3 - 2:0k10k1+j—1E(51-k1)
111-0

00 (X)
_ 2 2 _ . .
— ”1,4 E :aklak1+j-i — 0 E :aklaIJcHIJ-zl :

[£120 In =0

(3.3.17) = 0 (I1 — 21411301 — 21)).
Then from (3.3.14)-(3.3.17),
E(u?u§) = A2 + 0 (I1 — z‘I“’*L¥(Ij — 21)).
Hence, by an argument similar to the one used in proving (3.2.5), the left hand side
of (3.3.13) is

1 n n
A2 +0 (E Z Z lj —1|-91Lf(|j — 11)) = A2+o(n-9IL§(n)) —+ A2. :1

i=1 1¢j=1

We are now ready to present the
Proof of Lemma 3.3.2: It suffices to prove (3.3.11) for i = 1, since that of the
other part is similar. Let

C”

..2 _ "
Zam- — A1-
i=0

7

From (3.3.10

V

51

By the Cauchy-Schwarz inequality,

%:U1,j (111%) — TIA?)

 

 

i=1
Ifwe show
(3.3.18) A. —n» A1,
1 ” j 2
(3.3.19) — (m > 111(—)) —+po,
n j=1 Tl

then by Lemma 3.3.3, the proof would be complete.
First, consider (3.3.18). Since A1 is convergent series, for any 6 > 0, there exists
a positive integer M,E large enough such that,

A1_A1 2 :L2(]) )j—2H1 U2(H1-—H1)__ )+ Z L2(j j2H1— 3+ 2 L2(j )j—2H1
.7: —Mc j: —M.

j:L2(])] 2H1 3(jHl)—1)+€.

I/\

Then, by the consistency of H 1, guaranteed by (3.3.7) or (3.3.8) in Lemma 3.3.1, and
the fact that the first term of the above bound is continuous in H1 -H1, A1 —A1 —>p 0,
thereby proving (3.3.18).

To prove (3.3.19), consider the decomposition

52

Then, to prove (3.3.19), it is sufficient to prove the following three results:

2
_ 1 " 1 " j—t
(3.3.20) U1- n 32:; (nbgm nb )um) —+p 0,
2
_l n i " 2:1 3 _ _J;
1 n 1 n . t 2 .
_ _ _ 1;. _ 2 2.
(3.3.22) U3 _ n J; (nbgm nb ) 1) “1%) —>0.

First, consider U1. We shall show that EU1 —> 0. Then (3.3.20) will follow by the

Markov inequality. Now, rewrite EUI as E11 + E12, with

uwzzzKfllflwwM>

J=lt1=1t1¢t2=l

 

1 n n

j —t 1
E12— _ ”31,2 Z:K2(—J)E(ul,t)2 = 007%) “l 0-

j=l t=1

On the other hand, Lemma 3.2.1 implies that

EIIN—WZ:

J: lt1=1t175t2= 1

22(2

bti 1 t1¢t2

     

—)P1lt1—t2| 01L2(lt1 “ 1(2|)

t1 j'tz ltl —t2l
< 2
K 711) 3.7:)) [ nb _ :l

plltl - tzl 91111120751 — tzlla

 

 

     

where, [A] denote the indicator of the event A.

By the Assumption (A.2), observe that for 1 g t g n,

2(j-b_n_t) _ 2((j--t+1)/nb 2j—t _ 2
:7:K()—/11(22K)dx+:;/ K(—nb K(:c)dx

(J- W?!"

(l- t)/nb
—/1K22:()d:r—/ K2(:1:)dx
(n—t+1)/nb —1

= 0(1) + 0%) + 0(1) + 0(1) = 0(1).

Then, by the same argument as in the proof of (3.2.5) and the Assumptions (A3),

53

(A.5), we obtain

1 " " t—tl _
E11 3 0(1)—7%: z; [L17532Jp’1’lt1-t21 ”lqutl—tzl)

t1=1 t1¢t2=l

: CG 271—1): [L——t1;bt 2|<2]|t1—132|“"1)

t1=lnbt1¢t2=l

= 0((—7—1—(-1))—él—)—>0,

thereby completing the proof of (3.3.20).

Next, by the Assumptions (A.1) and (A.2), for some C < 00,
U < 0.7121sz 2t) J"_t
2 _ n_b n
Cl t) ' t ' t 2
(._7___ — .7 " .7 —
< — — —— <
- an :1 (11b: nb Hub “4)
02b " t 2
K(j _—
(3 3 23) - 071 .= (nbt:1K nb)

By the Assumption (A.2), we observe that for 1 _<_ j S n,

(j —t) - (j—t+1)/nb j _ t
n_bZK( — / K )dx-l—tz: :/ K(—-———- nb —K(:L‘) dz

 

 

(J- -t)/"b
(j- 70/"b
— l.’L‘K( )das—f K(:c)da:
'/nb -1
1 1 (J 70/”
(3.3.24) 2 1+0(—)— K(a:)d:z:—/ K(:c)da:.
le j/nb —1

Then, by the Assumptions (A.2) and (A3), (3.3.23) —> 0 and consequently (3.3.21)

follows.

)and the Assumption (A. 1),

(j-n)/nb

— x :1:— 1: :1: 21
/an()d /_ K()d)u1()

33

( M 0))
.< < [HD

S 30rgggluflx) (0(n—21123)+ :ng +;11- . Z )

1
= 0(b+W)—+O,

which proves (3.3.22). This also completes the proof of the Lemma 3.3.2. D

Finally, by(

U3 =

l/\

’M: II TM“): n Zita—‘3:

 

5’:
n

J

Now, consider the case when L1(a:) = L2(:c) = L(a:) and H1 = H2 2 H. In this
case, am- 2 012,3- = (13-, say. We are ready to give the estimate of o2 = of + 0% as

below:

p—a

A 1 n— 00
(3.3.25) 0'2 = 271p (Dj+1 — Dj)2, F = Zaj(aj _ aj+1)a
1 j=0

H.
II

where, F is F given above with H in the expression of aj’s replaced by its estimate

H given in (3.3.6). The following lemma shows its consistency.

Lemma 3.3.4 Suppose the model (1.0.4) holds with L1(:r) = L2(:v) = L(:c) and the
assumptions (A.1), (A.4)-(A.7) hold. Then the sequence of estimators of given in
(3.3.25) satisfies the Assmuption 2.2, i.e.

(3.3.26) 32 —>p 02 := of + a;

Proof: First, by the assumption (A.5) and (A.7), L1 2 L2 = L and H1 2 H2 2 H.
Consequently, we have pf 2 p3 = p2 and 01 = 02 = 0. Rewrite
1 11—1
6.2

= ——: {01(U1,j+1 — “1.1) — 02(U2,j+1 — “2.1) + #1(
2n F ,

j+1
n

—(m(’ +1) — #2(%))}2-

 

3 l9-

)

)“M1(

u.
H

 

Since H satisfies Assumption 3.2.2 by Lemma 3.3.1, using the same argument as in
the proof of (3.3.18), we obtain that F —>p F. Thus, to prove (3.3.26), it suffices to

prove the following:

y—a

"—

1
(3.327) 57-2: (11.1041 — ui,j)2 —>p F, 2 = 1, 2,
J=1
n— l . 2
1 1)
(3.3.28) — (”'n L— + —p,-(l)) —> 0, i=1,2,
n J'=1 n
and
n— 1
1
(3.329) '7; (U1j+1 — ul j) (11.2041 — UQJ) —)p 0.
j=l

By the Assumption (A.1), the left hand side of (3.3.28) is less than or equal to

This proves (3.3.28).

Next, the left hand side of (3.3.29) is equal to

n—1 71—1 -1 11-1

1 1 1 1

— U1,j+1u2,j+1 + - E U1,jU2,j — — "E U1,j+1‘u2,j — - U1,ju2,j+1-
3:1 3:1 1:1 J=1

Hence, to prove (3.3.29), it suffices to prove
I n
(3.330) 7—2- :;UIJUZJ' —)p 0
J:

By Lemma 3.2.1, by the assumed independence of the sequences {um} and {UN},

and by an argument similar to the one used in the proof of (3.2.5), we obtain

E(u1.ju2.JU1,kU2,k) = E(u1ju1k)E(u;g jun) ~ p4lJ' - (Crawl/4W - kl),

1 TI
El; ZU1,jU2,j)2 N n21: Z P4IJ "—kI 201140] —kI) *0,
j=l

J=1J¢kl

which implies (3.3.30) by the Chebyshev inequality.

Finally, consider (3.3.27) for 2' = 1. Let

1

m
711.341 = E ék€1,j+1—ka 0k = 72 (01,1: - Clue—1)
k=0

Then the left hand side of (3.3.27) can be rewritten as

11—1 00
1 ~ ..
; u¥1+19PZal2c=Fa
3:1 k=0
by Lemma 3.3.3. This proves (3.3.27), and hence the lemma. [:1

3.4 Some Additional Proofs

Here, we shall give the proof of Lemma 3.3.1. The proof is similar to that of Theorem
4 in Robinson (1997) which in turn makes heavy use of the proofs of Theorem 3 of
Robinson (1997) and Theorems 1 and 2 of Robinson (1995b). For the sake of com-
pleteness and an easy comparison, we shall first recall the assumptions of Robinson

(1997). Consider the model

t “3
(3-4-1) Yt =u(;) +ut, u. = Z 096...), t=1,2,-~
J=-oo
For j = 1, , m E [1, g], and for any numbers 111, - -- ,fln, let A]- = 27rj/n, and

(342) Ifiu(A) :2 w,;(/\)wfi(—)\),

3

       

~ . 1 "‘ 2,.-. ~ _ ~ 1 "‘

G(h) .= R 2‘? A, 1.14),), R(h) _ log G( (2h — 1 E 2 :1 0g A],
J: jl:

. ~ 1

H := arg minhemlAlew), for some 5 < A1 < A2 < 1.

The following are the assumptions as in Robinson (1997). The labels here are the
same as in that paper.

Assumption 2. The r.v.’s {it’s in (3.4.1) are such that 5;" are uniformly integrable
and

E(51Ift—l): 0, E(€t2I]:t—l) =1 3.8, t: 0i1,. . . ,

57

where .7; is the o—field of events generated by {53, s g t}. The constants aj’s in

(3.4.1) satisfy 2“) az- < oo.

j=—°° J
Assumption 4. Either T(x) satisfies a Lipschitz condition of degree 7', 0 < T g 1,
or T(Jc) is differentiable with derivative satisfying a Lipschitz condition of degree 7' — 1,
1 < 'r _<_ 2.

Assumption 7. For finite constants M and ,u4,
E(E?|ft_1) =3 [13 3.8.; E(E?|ft_1)= [14, t: 0, :l:1, ‘ ° ° .

Assumption 8. In a neighborhood (0,6) of the origin, a(/\) = 2;:_oo ajeij" is

differentiable and (d/dA)oz(/\) = O(|a(/\)|//\), as A —> 0+.

Assumption 9. For some fl 6 (0, 2],
f(/\) ~ Gil-2” (1+ 0(,\5)) as A —> 0+, f(/\) = |a(A)|2/27r,

where G > 0 and H E [A1,A2].

Assumption 12. For H 2 A1 > % and some 6 > 0, as n —> 00,

m n2-2H

45
4
(103”) (g) + m1—2max(6,H—A1) _’ 0'

 

Now, we are ready to recall the Theorem 4 in Robinson (1997).

Theorem 3.4.1 For H E (g, 1), let model (3.4.1) and Assumptions 2, 4 with T = 2,

7-9, and 12 hold, and let H be given by (3.4.2) with {it 2 Y1, 1 g t g n. Then
(3.4.3) (log n)(H — H) —>p 0, as n —-+ 00.

Remark 3.4.1 This is Theorem 4 of Robinson (1997). The Assumption 4 above
assumes Lipschitz condition of degree 2. A close inspection of the proof in Robinson
(1997) shows that actually only Lipschitz condition of degree 1 is used there. In
other word his Assumption 4 can be replaced by our Assumption (A.1) in the above

theorem. Also, the Assumption 12 above is implied by our Assumption (A.6).

58

Proof of Lemma 3.3.1: It suffices to prove (3.3.7) for i = 1, as the proof for the
other case is exactly similar. Since #1, 01013” in the model (1.0.4) correspond to u
and 01,- in the model (3.4.1), the claim (3.3.7) will follow from the result (3.4.3) once
we verify the assumptions of the above theorem.

Clearly, (1.0.3) and the Assumption (A.4) imply the Assumptions 2 and 7 above.
Note also that the estimates given by (3.4.2) with ii, = K, 1 _<_ t S n, correspond
to our estimates given in (3.3.3)-(3.3.5). In view of the above Remark 3.4.1, it thus
remains to verify the Assumptions 8 and 9 above with a,- = 01013 of (1.0.3). This is
done in the following lemma. Without loss of any generality assume 01 = 1 and let
ozj = L(j)j‘(‘9+1)/2 with L(j) = L1(j) and 0 = 01 in the remaining part of the proof.
Let

(3.4.4) a()\) = Z akei“, m) = |a(/\)|2/27r.
k=0

Lemma 3.4.1 Under the Assumption (A5), the Assumptions 8 and 9 are satisfied

by the above 01.

The proof of this lemma is facilitated by the following lemma. For 0 < 7 < 1 and

0 < J: < 27r, let \IL,(J:) be the periodic function defined by the formula

_ , 271' " , 7—1 (mm-1 ,7
(3.4.5) ‘11.,(36) —nli+rr.1°-l:(7)- {2(17 + 23%) — 7 n }.

 

i=0
Lemma 3.4.2 For all 0 < 7 < 1, 0 < a: < 27r,
e—éni'ysignU) .

(3.4.6) Zjez’mTenJ-x = @7013)-

Moreover, for some positive real number C, we have

 

 

 

 

°° coij . 1 _.
(3.4.7) I; fl — l"(1-—7)sm(-2-7r'y)x71 _<_C,

J:

0° sinja: 1 _
(3.4.8) I; j, — I‘(1-7)cos(§7r7):ml so,

]:

uniformly in O < :1: 3 Jr.

59

Proof: The proof of (3.4.6) appears in Zygmund (1968: page 69-70). Now we shall
give the proof of (3.4.7) and (3.4.8). Let

<I>,(a:) = lim {:07 + 2j7r)7-1— (27?)? n7}.

 

n—ioo .
J=l

Using the Taylor expansion of the function a: +—> (:1: + 2j7r)”‘1 around :2: = 0 up to

the first term,

 

 

 

 

 

 

I¢7($)I
: - " - 7-. (h.(a:)+2jvr)"'-2 _(2,)._. , _ <,
)ggo{;((2m + 7—1 a: 7 n |, th(:v)|_IJ:|
. n . _ (2707‘1 . n . _ x
S 232%, W)“— 7 "7 I+ ..‘HEXWmlWW 2,—_1-I
3:1 3:1
S C1

for all 0 < |:c| 3 7r and some finite constant C1. This together with the fact that \I',(:z:)

 

 

is a periodic function on 0 < a: < 27r implies, with 02 = C, x F207;? that uniformly in
0 < :1: 3 7r,
27r 27r
‘Ilm— 37-1=___q)$ <C,
|x11 (-:::)| = |x1: (271’ — 3)): |_2i(<1> (—:r) + lim (2(n + 1)7r — $)7—1)|< Cg.
7 7 PM) 7 n-wo ‘
The above two statements in turn imply that
7r

3.4.9 \I' m + \I1 —.’L‘ — —:r"‘1I 3 2C.
( > .( > .( ) m) 2

 

The Fourier series \II.,(:1:) + \II,(—:1:) in the left hand side of (3.4.6) is equal to

1 °° cost:
4cos (5717) 2 v7 .

v=1

 

From this, (3.4.6) and (3.4.9), we thus obtain that

1 °° cosva: 27r
_ E : _ __ 7-1
I4COS (27ml) v=1 v7 P(“/)x

 

S 2C2.

 

60

This together with the relation I‘(7)I‘(1 —7) = 71/ sin 717 implies (3.4.7) with C = 2C2.
Similarly, by considering \II.,(:I:) — \II,(—a:) in the left hand side of (3.4.6), one

obtains (3.4.8). The lemma is proved. [:1

Proof of Lemma 3.4.1: By (A.5), (3.4.7) and (3.4.8), we immediately obtain that

forallO<A§7r,

oo

 

L1 —L1
_ eijA __ eijA
aw — 2,43). +2” film/2 +1
3': _1
1+0 i7r(1—(1+9)/2)/2 (1+0)/2— 1+ "L1 A
= L1F(1———2—)e A A)+Z————L( j—(———,)+,)/2 e” +1,

3': —l

where, Q is a function on (0, Jr] with SUP0<Agn |Q(/\)| < 00. Hence, by (3.3.1) in (A.5),

1 __
(3.4.10) |a(A)| —.— L1r(——2—§)A<9-1>/2 + 00.“), A —> 0+,

1— 0

f(A)= L2r2(—— 2))1” 1/27r +0(A<-" 1”21””)=GlAl-2”I(1+0(Afil)),A—>0+,

where, 0'1 2 Lffzflg—oflkr > 0, and 61 = (1 - 6)/2 + c1 6 (0, 2]. Hence, the
Assumption 9 is verified.

Next, consider the derivative of I‘(7)‘I’,(:c)/27r in Lemma 3.4.2 at 0 < x < 27r, for

 

 

 

0 < 7 < 1,
_(__7) d

(3. 4. 11)——- 2” d? (3;)

- limit—mo (22:1(17 + h + 2,6707-1 — (2%)7—1117/7)

2 11m {
h—+0 h
limn—ioo (22:16?) + 2kfl)7—1 — (27f)7_1n7/fy) (11? + h)’Y-1 _ 1.7-1
T h + h } '

By the fact that 22:1(1: + 2k7r)“"1 — (2a)”7'1n7/7 is uniformly convergent for 0 g
:1: < 27r, the right hand side of (3.4.11) is equal to

11 7—1 __ 7—1 ’1 7—1 _ 7—1
lim lim ((1!) + h + 2k7r) (a: + 2k7r) ) + lim (2: + ) :1:
h—>0 n—>oo h h—>0 h

 

 

n

= 111mb lim (7 — 1)(:1: + hk(:r) + 2kJI)7—2 + 1137—207 — 1), where Ihk($)I S W
-—> n—>oo

= 1/2(:1:) + 357—2” — 1), where |¢b(:c)| is bounded over (0, 27r).

61

Hence, this together with the fact that ‘11.,(23) is periodic function defined for 0 < a: <

27r, we can conclude that

_ 171111). a, 7, 7-2 x ,, _, x
.. PW) (+2) +2/2( +2), S <0.

By considering the derivatives of the Fourier series \II,($) :l: \I/7(—23) in the left hand
side of (3.4.6), and using the relation I‘(7)I‘(1 -'y) = 7r/ sin 7m and (3.4.12), we obtain

the following formulas.

d 0° 3'3:
21“: 0.3)

 

m — 7) meme —1)x7-2 + 101(2),

 

 

 

$313.33 = ru—wcosémm—Drum), o<xs«.
where

101(4) = W)4:02:((::7;r2fl+1“(1—7)sin(%m)(7-1)(-:r+27r)“"2,

902(4) — “(),;jfjgz’rh +I‘(1-7)008(%m)(1-1)(-x+27r)"’2

Then, by the same argument as used in obtaining (3.4.10) and (3.3.2),

—-—1'"/\“’ 3’” +L1|<p1(:r )+ i902(x)|+

 

 

 

d 0° Lk(j)_Lk m
_X (Z j(1+o,.)/2 e’

i=1

 

2

(3.4.13) 2 0 103921), A —> 0”“.
The lemma proved. [:1

Remark 3.4.2 ARFIMA(O,d,O) with d = H — 1/2 is a long memory error with

coefficients satisfying for j —> oo,

_ I‘(j+d) AIL/W, ._ ,.-1— __ 1 1
“"‘I‘(j+1>r(d)"’r(d>’d ’ “(l—a” “M (l

 

62

Also, it is shown in page 186 of Zygmund (1968) that for A1704) = F (j + d —
1)/(F(j)P(d ))

_1 ,
GOA—(“d"J'A— 2"1A‘d 1' Ad 0 A 2
2 j e —( $1115) exp{§1(7r— )}, < < 7r,
j=0
Hence, ARFIMA(0,d,0) with d = H— 1/2 is also a long memory error satisying (3.3.1)

and (3.3.2) in Assumption (A.5).

Now, we have verified all the assumptions in Theorem 3.4.1 for Y; = Yip II = #1
and a, 2 01am. We are ready to give the rest of the proof of Lemma 3.3.1.

Proof of Lemma 3.3.1 continued: For any 6 > 0, Let
N,- = {h : lognlh — H,| > e}, 3.01) = R,(h) — R1(H,-), i=1,2,

where the functions 11,1 2 1, 2 are given in (3.3.3). Define G = {h : A1 3 h 3 A2}.
Then, by the proof of Theorem 3.4.1 in Robinson (1997), we have

P ( inf 31(12) 3 0) = 0(1).

N109

It is similar to prove

P ( inf 3201) g 0) = 0(1).

N209

Hence,
P (lognIH, — 11.12 c) = P (H e N,- n 9)
(3.4.14) 2 P(1&%%R1(h) S 7‘17}:wa R1(h)) S P(1\iirr1‘feS,-(h) g 0) = 0(1), 2 21,2.
This proves (3.3.7).
To prove (3.3.8), let N = {h : log nlh — HI > 6}. By Assumption (A.7), we have
N = N1 = N2. By an argument similar to one used in obtaining (3.4.14),
P (lognlfr — HI 2 e) g P (13.55121) + 5201) g 0)
g P (133231“) 3 0) + P(I$1Ag82(h) g 0) = 0(1).

Hence (3.3.8) is proved thereby completing the proof of the lemma. [:1

63

3.5 A Monte Carlo study

Since the limiting null distributions of the proposed tests are unknown, we first con-
ducted a Monte Carlo simulation study to obtain an estimate of a few selected critical
values of these limiting null distributions. We propose to use these critical values
when performing the proposed tests. Then, another Monte Carlo study is conducted
to assess the behavior of the level under Ho and power of these tests against some
nonparametric alternatives for moderate sample sizes.

The data are generated according to the ARFIMA(0,d,-,O) processes, with d, =
H,- — %, z" = 1, 2, and Gaussian innovations. In other words the data are generated
according to the model (1.0.3) and (1.0.4), with L,- of the assumption (A.5) equal to
1/I‘(d,-), 2' = 1, 2, and with the Gaussian innovations. In our simulation, we used the
code given in Beran (1994) for S-plus to generate the data. This in turn uses the
fast Fourier transform method of Davies and Hart (1987) to simulate a stationary

Gaussian process.

First, we need to estimate the critical values bm and b2,a such that

1
P ( SUP1|B61A02(t)l > b1,a) :: a = P (/ B31A02(t)dt > b2,a) a
O

093
where 61 /\ 02 = 2 — 2(H1 V H2).
Let

1
W1: sup 2.0:), W2 = / 2.3mm, zn(t) ——1— 2%
t€[0,1] 0 i

where u, is an ARFIMA(0,d,O) process with d = H1 V H2 — .5 and 7",, E "rm,- given
in (1.0.11), with H,- replaced by H1 V H2, L,(n) replaced by 1/I‘(H1 V H2 — .5), and
a,- E 1. By Lemma 3.2.3, Zn(t) 1%] Bg,Ag,(t). We use this fact to obtain the values
of [11,0 and b2,a- We use sample size n = 5000 and simulated 2000 replications of
W1, W2, for H1 V H2 2.6, .7, .75, .8, .85 and .9. This was repeated 4 times. The

critical values, the 100(1 — a)th percentiles of W1 and W2 are reported in Table 3.1

64

and 3.2 . The average critical values of 4 repetitions are in parenthesis and they are

used as the actual critical values for these tests.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a level H1 V H2 191,0: 62,0,
.1 .6 1.818 1.795 1.789 1.827 (1.807) 1.134 1.073 1.100 1.150 (1.114)
.7 1.703 1.716 1.767 1.685 (1.718) 1.030 1.044 1.110 0.984 (1.042)
.75 1.702 1.689 1.675 1.714 (1.695) 1.010 0.996 0.991 1.078 (1.019)
.8 1.648 1.697 1.663 1.649 (1.664) 0.945 1.009 0.974 0.995 (0.981)
.85 1.644 1.640 1.639 1.651 (1.644) 0.955 0.979 0.937 0.933 (0.951)
.9 1.697 1.592 1.602 1.637 (1.632) 0.978 0.899 0.880 0.901 (0.915)
.05 .6 2.065 2.042 2.084 2.103 (2.073) 1.493 1.413 1.469 1.576 (1.488)
.7 2.043 2.019 2.052 1.972 (2.021) 1.468 1.517 1.563 1.432 (1.495)
.75 1.993 1.971 1.938 2.000 (1.976) 1.439 1.421 1.410 1.489 (1.440)
.8 1.978 1.967 1.931 1.972 (1.962) 1.421 1.403 1.284 1.355 (1.366)
.85 1.962 1.982 1.916 1.932 (1.948) 1.387 1.326 1.289 1.314 (1.329)
.9 2.038 1.941 1.904 1.956 (1.960) 1.462 1.351 1.256 1.396 (1.366)

 

Table 3.1: Simulated critical values under a = .1 and .05.

 

To investigate the Monte Carlo power of the tests T1 and T2 under some alterna-
tives, we need to estimate Hi, 2' = 1, 2. The estimators 111,-, 2' = 1, 2 given at (3.3.5) do
not use residuals. As a curiosity, we first made a Monte Carlo comparison of the bias
and the MSE of this estimator with the one based on the estimated residuals. More
precisely, we compared the estimate H1 given in (3.3.5) corresponding to the choice
of m1 given in Table 3.3 with the estimate [:11 given in Robinson (1997) and obtained

from (3.3.5) With'Iyly, replaced by 111,121, where K(:r) = 3(1 — x2)[|x| S 1]/4 and

111,1: (Y1,t—("C)_IZK(t;cs)}/L8) [nc < t S n — lncll:
8:1

 

65

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a level H1 V H2 b1,a (12,0

.025 .6 2.290 2.289 2.334 2.384 2.324) 1.919 1.839 1.902 2.014 (1.919)
.7 2.292 2.258 2.338 2.188 (2.269) 1.874 1.920 2.055 1.838 (1.922)

.75 2.285 2.213 2.253 2.255 (2.251) 1.845 1.812 1.881 1.854 (1.848)

.8 2.334 2.160 2.236 2.240 (2.243) 1.887 1.791 1.802 1.715 (1.800)

.85 2.203 2.238 2.170 2.262 (2.218) 1.765 1.697 1.595 1.723 (1.695)

.9 2.316 2.290 2.258 2.284 (2.287) 1.847 1.790 1.695 1.857 (1.797)

.01 .6 2.626 2.572 2.653 2.709 (2.640) 2.499 2.422 2.494 2.476 (2.473)
.7 2.606 2.614 2.574 2.537 (2.583) 2.327 2.458 2.453 2.438 (2.419)

.75 2.605 2.558 2.539 2.587 (2.572) 2.384 2.415 2.442 2.262 (2.376)

.8 2.824 2.582 2.586 2.578 (2.642) 2.624 2.296 2.441 2.373 (2.434)

.85 2.513 2.458 2.543 2.551 (2.516) 2.306 2.115 2.151 2.317 (2.222)

.9 2.578 2.582 2.601 2.570 (2.583) 2.284 2.396 2.435 2.397 (2.378)

Table 3.2: Simulated critical values under a = .025 and .01.
with c 2 71‘3"?”10 and m1 = 724/5 satisfying the assumption 11 of Robinson (1997):

(10g 704 {(flnl-ym1 + (logm1)2 + 1

The data was simulated from the first part of model (1.0.4) with 01 = 1, “1(2) =
211:, a: 6 [0,1], for n = 300,600, 1000 and H1 2 .6, .7, .75, .8, .85, .9. Each simulation
was repeated 2000 times to obtain 2000 values of H1, H1. Then, by comparing their
means of the bias (MB) and MSE in Table 3.4 and their histograms in figure 3.1-
3.6, we can conclude that estimate H1 is better for the parameter values H1 > .7.
Moreover, one also sees from the histograms of these estimators that the Monte Carlo
distribution of I71 is skewed to the right more than that of 1:11, for the true parameter

values H1 3 0.7. Since the behavior of H2 and H2 is expected to be similar, for the

C71
+c+

 

m1 C7711

4 2-2H1
"Ii—2H1

m2H1-1

—c21nZH1 }——> 0, n ——> 00.

above reasons we chose to use Hi, 2' = 1, 2, in our simulations.

66

 

400

200

300

O 100

300

100

Histgram of the estimate of H given in (3.3.3)
0:300, H=.6

Jr 1

t-fl—-fi

 

0.60 0.65 0.70 0.75 0.80 0.85
EH1“, ]

n=300,H=.7

71
.I.E..}!I

0.6 0.7 0.8 0.9

 

EHI[2, 1

n=300,H=.75

'1

0.65 0.70 0.75 0.80 0.85 0.90 0.95

 

51103. 1

Histgram of the estimate of H
based on the estimated residules
n=300, H=.6

1500

500

 

 

 

am. 1
n=300,H=.7
§
§
§ ‘ F-Q‘.
l-flflm=llhnm__

O

0.55 0.00 0.65 0.70 0.75 0.00 0.05
8912.1

n=300,H=.75

1 00 200 300

!!3!I'

 

0

 

0.6 0.7 0.8 0.9

E1913. 1

Figure 3.1: histogram of I71 and H1 for n = 300, H = .6, .7, .75.

Histgram oi the estimate of H given in (3.3.3)
n=300, H=.8

F1

_-IlE-——-—- I-

100 200 300

O

 

0.65 0.70 0.75 0.80 0.85 0.90 0.95

EH04. ]

n=300.H=.85

1' 1

100 200 300

O

 

0.65 0.70 0.75 0.80 0.85 0.90 0.95

511115, 1

n=300,H=.9

100 200 300

 

 

 

o _—:n£mmmumf
0.70 0.75 0.80 0.85 0.“) 0.95
50110. 1

Figure 3.2: histogram of 1:11 and I71

68

Histgram oi the estimate of H based on
the estimated residules for 0:300, Hz. 8

rr "l1

0 IIIIIIIIIIIIIIHIIII:

400

200

 

 

0.0 0.7 0.0 00
EH14. i

n=300,H=.85

11"]

o E----

400

200

 

 

05 0.7 00 0.9 1.0
80151 1

F11

400

200

 

o “

 

00 0.7 0.0 0.9 1.0
£11216, 1

for n = 300, H = .8, .85, .9.

Histgram oi the estimate of H 6given in (3.3.3) Histgram oi the estimate of H
n=600 based on the estimated residules

 

 

 

 

 

 

n=600,H=.6
1!}: ..
110
0 I! 0
0 III-..— I- o -—-____._
0.05 0.70 0.75 0.00 0.00 0.05 0.70
EH10] EH20]
n=600,H=.7 n=600,H=.7
O
8 O
O
o n _1
O . -.
v r Ir 1
§ ‘ - : "1.
o . Ibunmu: o inn-final..—
0.70 0.75 0.00 0.05 0.55 0.00 0.00 0.70 0.75 000
Will! 5911]
n=600,H=.75 n=600,H=.75
i a
g 5.; ‘3‘ E . \

 

"met!

 

 

 

 

o
5 .
~ ‘ Er _0' g .
o _- "fie-.30” I o &£aaéaa££lu_
0.55 0.70 0.75 0.80 0.85 0.90 0.55 0.50 0.55 0.70 0.75 0.” 0.85
EH1[3.] 601213,]

Figure 3.3: histogram of H1 and H1 for n = 600, H = .7, .75, .8.

69

Histgram 0f the estimate of H 8given in (3. 3. 3) Histgram of the estimate of it based on
n=600 the estimated residules tor 0:600, H=.8

 

 

 

O
8 Hz! 8
v 8 [EV I]!
N
O
8 r118 FE _ I!
o _-=---- o __- IIIIIII-III
0.70 0.75 0.00 0.05 0.90 0.0 0.7 00 0.9
WM] mat
n=600,H=.85 n=600,H=.85

fl

 

 

 

 

 

 

 

 

 

O
O
(’)
O
. ' a 8 F a l
'5: _ o
9 ' - ' s
o I teamwfi -_ o -..I IIIIKII' I...
0 80 085 030 0.6 0.7 0.0 0.9
91115, 1 E0215. 1
n=600,H=.9
O
O
o ‘9 Egg
0
v 8 9g
N
g l!
N g [E
o o —IEHu---—nfill-_
0.75 0.80 0.85 0.90 0.95 0.7 0.0 09
511110, 1 510210. 1

Figure 3.4: histogram of H1 and H1 for n. = 600, H = .8, .85, .9.

70

Histgram of the estimate at H given in (3.3.3)

 

 

 

n=1000, H: .6
O
8 E 1
8
o Efl--- -_
0.66 0.50 0.70 0.72 0.74 075 0.70
3011,]
n=1000,H=.7
.8,
8 r J]
o Iiflfihflilgl
0.70 0.75 0.80
9012.1
n=1000,H=.75
.8. '05
i 1
2 l
o __-IEfi----- I.—
0.70 0.75 0.00 0.05
3013.1

Figure 3.5: histogram of H1 and H1 for n = 1000, H =

71

1400

400 800

0

300

100

O

300

 

v v v '

Histgram of the estimate of H
based on the estimated residules
n=1000, H=.6

 

 

0.56

 

fi

0.64

f

0.60 0.62 066 0.68

94211. i
n=1000,H=.7

r1.

, F ,\
[ungu-‘:-_

0.75

056

 

0.65 0.70

50212. l
n=1000,H=.75

0.60

 

 

0.65 0.70 035

mm

0.60 0.75 0.60

.7, .75, .8.

Histgram of the estimate of H given in (3.3.3) Histgram of the estimate of H bases on

 

 

 

 

 

 

 

 

 

n=1000, H=.8 theestimated residulestor n=100, H=.8
.9.:
r1 8
8‘ 1 9
V ', .
o L ‘ q[[g133
a E , s
o _- --—-_ o III-...Il-
0.75 0.80 0.85 0.90 0.6 0.7 0.8 0.9
EH1[4,] 9914.]
n=1000,H=.85 n=1000,H=.85
o
8 F1 8
. r l
o - o _-:-------l
0.85 0.90 0.95 0.85 0.70 0.75 0.30 0.85 0.90
3415.] 9915.1
n=1000,H=.9 n=1000,H=.9
v ;1_ ' ' :33
o I. 3
8 E ' - : ~ g r
o __ - 1“ inn—I’— - o .[fiéiflfiilI-n
0.80 0.85 0.90 0.95 0.65 0.70 0.75 0.5) 0.85 0.90 095
EH1[6.] EMGJ

Figure 3.6: histogram of H1 and H1 for n = 1000, H = .8, .85, .9.

72

 

n\H1 .6 .7 .75 .8 .85 .9

 

 

 

 

 

 

 

 

300 news/.92 n.62/.72 n.54/.62 12.45/52 nee/.42 nee/.32
600 ”82/92 n.64/.72 ”SS/.62 ”AG/.52 ”37/42 ”285/32
1000 ”82/92 ”64/72 ”es/.62 ”AG/.52 n.37/.42 ”285/32

 

Table 3.3: The choice of m1 corresponding to H1.

Next, we consider two parts to investigate the powers of T1 and T2 under some
alternatives. In the simulations, we used p1(:1:) = 2x and [12013) = ,ul(:z:) - 6,,(27), with
6,,(36) = 0, 6,,(13) = 1, 6,,(1‘) = 11:, 6,,(x) = Tu and 6,,(23) = Tux.

In Part I, we take H1 2 H2 2 H and 01:1: 02. Then, L1 2 L2 = L =
1/I‘(H — %). The two test statistics considered are T1 and T2 given in (3.2.12), where
H is as in (3.3.6) with m1 = m = m2 given in table 3.3, it: being is,- given in (1.0.10)
with H,- replaced by 1:1, 62 given in (3.3.25) and L(n) being L with H replaced by
H. The test T,- rejects the null hypothesis Ho if T,- > (),-,0, where bm is given in
Table 3.1 and 3.2, 2' = 1, 2. We consider 6 choices for the value of the parameter
H = .6, .7, .75, .8, .85 and .9. In each case, 3 sample sizes are used, n == 300, 600 and
1000. Under the null hypothesis, for each sample size the proportion of rejections
of the null hypothesis in 2000 replications, for a = .05 and .025, are reported in
Table 3.5 and 3.6. In the same tables, we also compare these simulated a levels of
T1, T2 to those of the tests based on T1, T2, where T,- is like T, with H in is, {72,
I: and 7"" replaced by its true value H. Not surprisingly, The tables show that the
simulated a-levels of T1, T2 are closer to the true levels for most of the chosen values
of H than those of T1, T2 tests. This is most likely due to the fact that H tends to
overestimate H, for H < .85, and underestimate H, for H = .9 as seen in the Table

3.4. The simulated a-levels of T1, T2 tend to be smaller (larger) than the specified a

73

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MB MSE
H1 estimate of H1 /n 300 600 1000 300 600 1000
.6 H, 0.131 0.122 0.120 0.135 0.124 0.121
H1 -0027 -0029 -0.028 0.040 0.036 0.035
.7 in 0.076 0.065 0.064 0.086 0.071 0.067
H1 -0.078 -0.067 -0054 0.098 0.082 0.068
.75 1371 0.054 0.042 0.039 0.069 0.051 0.046
H1 -0092 -0071 -0.058 0.114 0.088 0.072
.8 1311 0.034 0.024 0.022 0.060 0.040 0.033
H1 -0097 -0071 -0.060 0.122 0.090 0.072
.85 in 0.020 0.005 0.005 0.053 0.035 0.028
F11 -0102 -0074 -0.061 0.128 0.092 0.074
.9 H1 -0005 -0013 -0012 0.048 0.036 0.030
H1 -0105 -0077 -0.061 0.130 0.094 0.074

 

Table 3.4: comparison of H1 to H1:

for H < .85 (H = .9).

The Monte Carlo power, i.e., the proportion of rejections, of the proposed tests
for each sample size is reported in Table 3.7 and 3.8 when 111(26) — 112(2) = 6(27), and
(I) 6(23) 2 1, (II) 6(30) 2 2:. It is observed that under both alternatives, the Monte
Carlo power of T1 test is generally larger than that of T2 test, for all chosen values
of H and n, especially under the alternative (II) 6(23) = :13, :1: E [0, 1]. Moreover, the

simulated powers against both alternatives of both tests tend to be decreasing in H

for each n, and increasing in n for each H.

74

 

 

a=.05 a=.025
H Test\n 300 600 1000 300 600 1000

 

 

.6 T1 .0000 .0000 .0000 .0000 .0000 .0000
T1 .0570 .0580 .0560 .0300 .0305 .0320

 

T 2 .0000 .0000 .0000 .0000 .0000 .0000
T2 .0655 .0585 .0540 .0330 .0295 .0300

 

.7 T1 .0020 .0025 .0030 .0005 .0010 .0015
T1 .0520 .0465 .0465 .0285 .0260 .0275

 

T2 .0020 .0030 .0035 .0010 .0005 .0010
T2 .0440 .0515 .0445 .0245 .0270 .0235

 

.75 T1 .0120 .0105 .0115 .0070 .0015 .0020
T1 .0555 .0550 .0550 .0315 .0270 .0295

 

T2 .0110 .0090 .0105 .0060 .0020 .0020

 

 

 

 

 

T2 .0515 .0465 .0575 .0285 .0235 .0305

 

Table 3.5: Proportion of rejections under Ho for part I.

The Monte Carlo power of the proposed tests under local alternative is reported in

Table 3.9 and 3.10 when [11(x)—,u2(:1:) = 6,,(23), and (III) 6,,(112) = Tn, (IV) 6,,(16) = TnIL'.

75

 

 

 

 

 

 

 

 

 

 

 

 

(12.05 5:025

H Test\n 300 600 1000 300 600 1000
.8 T1 .0250 .0260 .0245 .0135 .0145 .0110
T, .0555 .0565 .0475 .0290 .0275 .0270

T2 .0275 .0260 .0275 .0170 .0115 .0125

T2 .0560 .0570 .0520 .0310 .0255 .0265

.85 T1 .0350 .0485 .0500 .0155 .0255 .0250
T, .0495 .0510 .0470 .0215 .0270 .0210

T2 .0335 .0480 .0505 .0170 .0275 .0275

71. .0520 .0545 .0485 .0255 .0295 .0275

.9 7‘“, .0860 .1015 .0850 .0505 .0570 .0480
T, .0600 .0495 .0475 .0235 .0235 .0225

T2 .0815 .0935 .0830 .0525 .0585 .0500

E .0600 .0470 .0460 .0270 .0230 .0235

 

Table 3.6: Proportion of rejections under Ho for part I.

It is also observed that under both alternatives, the Monte Carlo power of T1 test
is generally larger than that of T2 test, for all chosen values of H and 71, especially
under the alternative (IV) 6,,(33) :- Tax, 2: 6 [0,1]. Moreover, the simulated powers
against both alternatives of both tests tend to be increasing in H for each n, and

more stable in n for larger H. And, we find that the powers of these tests for H g .75

are very small. It is actually due to the large bias of the estimates H.

76

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5:05 5:025

Alternative H Test\n 300 600 1000 300 600 1000
(I) .6 T, .9615 1.000 1.000 .8950 .9985 1.000
T2 .9480 .9990 1.000 .8580 .9980 1.000

.7 T, .5595 .8975 .9800 .4220 .8150 .9525

73 .5105 .8635 .9640 .3635 .7535 .9110

.75 7“", .4550 .6825 .8215 .3235 .5420 .7140

T2 .4145 .6385 .7875 .3020 .4975 .6610

.8 T1 .3015 .4605 .5690 .1970 .3355 .4465

T2 .2895 .4255 .5505 .1945 .3105 .4120

.85 T, .1925 .2875 .3845 .1320 .2145 .2945

T2 .1865 .2795 .3775 .1325 .2135 .2965

.9 T, .1780 .2345 .2400 .1220 .1630 .1610

T2 .1695 .2190 .2315 .1215 .1635 .1725

 

Table 3.7: Proportion of rejections under fixed alternatives (1) 6(2) = 1.

Now consider the case H g .75. Let H "‘ be H based on m1 = m = mg given
in Table 3.11. Also, Let T1“, T; be T1, T2 with H replaced by H ‘. First, In table
3.12, by comparing the mean and standard deviation (Stdev) of H with those of H *,
we find that H * is more accurate that H. Then, Table 3.13 and 3.14 show that the
simulated powers of Ti“, T 2* under both local alternatives are much higher than that

of T1, T2. Hence, the various choices of 773 will give different estimates of H, which in

turn will influence the power behavior of these tests greatly.

77

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a=.05 a=.025
Alternative H Test\n 300 600 1000 300 600 1000
(II) .6 T, .3620 .7305 .9260 .2435 .5600 .8370
T2 .1670 .3525 .5875 .0895 .2050 .3895
.7 T, .2015 .3635 .4920 .1370 .2485 .3715
T2 .1135 .1735 .2570 .0655 .1045 .1530
.75 :1“, .1555 .2450 .3375 .0950 .1640 .2390
73 .0930 .1345 .1830 .0600 .0860 .1150
.8 T1 .1355 .1880 .2050 .0860 .1195 .1315
T2 .0945 .1220 .1310 .0580 .0760 .0775
.85 T1 .1235 .1430 .1630 .0775 .1015 .1110
T2 .0995 .1175 .1225 .0695 .0790 .0865
.9 ‘1 .1520 .1715 .1720 .0985 .1070 .1140
T2 .1300 .1455 .1475 .0890 .0990 .0970

 

Table 3.8: Proportion of rejections under fixed alternatives (II) 6(1) = :1: for Part I.

Part II deals with the case H1 31$ H2, 01 = 1 and 02 = 2. The test statistics are T1

and T2 of (2.2.5). The estimates Hi, 6?, 2' = 1, 2, used in these test statistics are given

in (3.3.5) and (3.3.10), with 772,-, z' = 1, 2 chosen according to Table 3.3. Moreover, the
kernel function used is K (2:) = 3(1 -— $2)[|x| S 1] /4, bandwidth b = 13-1/5, and L,- is
replaced by 1/I‘(H,- — .5). The sample sizes, number of replications and alternatives
considered were as in Part I. Table 3.15, 3.16 and 3.18-3.21 report the simulated finite
sample levels and power behavior (proportion of rejections of the null hypothesis) of
these tests under a level .05 and .025 for different values of H1 and H2. As in Part I,

Table 3.15 and 3.16 give a comparison of the simulated a levels of T1 and T2 to that

78

 

 

 

 

 

 

 

 

 

a=.05 a=.025

Alternative H Test\n 300 600 1000 300 600 1000
(III) .6 T, .0010 .0000 .0000 .0000 .0000 .0000
T2 .0025 .0005 .0000 .0005 .0000 .0000

.7 T, .0190 .0165 .0100 .0070 .0060 .0035

T2 .0175 .0145 .0090 .0075 .0055 .0030

.75 T, .0575 .0485 .0370 .0360 .0280 .0115

”I3 .0565 .0445 .0285 .0295 .0260 .0100

.8 '1“1 .1020 .0945 .0910 .695 .0550 .0555

E .1035 .0920 .0880 .0640 .0565 .0535

.85 T, .1450 .1530 .1465 .0955 .0990 .0935

75 .1435 .1465 .1450 .0945 .0960 .1025

.9 7“, .2275 .2265 .2370 .1600 .1640 .1640

re .2200 .2170 .2275 .1635 .1610 .1650

 

 

 

 

 

 

 

Table 3.9: Proportion of rejections under local alternatives (III) 6,,(27) = 7,, for Part

I.

of T1 and T 2, where now T,- is like T,- with H, in 1%,, (if, 1:,- and 7",”- replaced by its true
value H,, z" = 1, 2. The simulated a—levels of T1, T2 are closer to the true levels than
those of T1, T2 tests. But the simulated levels of T1, T 2 are still much larger than the
true levels for H1 V H2 = .9. It is due to the large bias of 6?, 6% shown in table 3.17
for H1 V H2 = .9 which in turn is due to the large bias of the estimators 1‘11, [12 of the
regression functions #1, [.12 for H1 V H2 2 .9. The simulated power behavior under
both fixed alternatives and local alternatives are similar to those of Part I. Moreover,
the simulated power under local alternatives for H1 V H2 = .7 are reasonably high.

This is due to the small bias of the estimates of H2 = H, V H2 = .7 in this case.

79

Table 3.10: Pr0portion of rejections under local alternatives (IV) 6,,(56) = 7,,2: for

Part I.

 

 

 

 

 

 

 

 

 

a=.05 012.025

Alternative H Test\n 300 600 1000 300 600 1000
(IV) .6 1‘", .0000 .0000 .0000 .0000 .0000 .0000
T2 .0000 .0000 .0000 .0000 .0000 .0000

.7 :i"1 .0145 .0060 .0045 .0085 .0015 .0020

T, .0120 .0055 .0040 .0075 .0010 .0010

.75 T, .0245 .0260 .0285 .0100 .0145 .0100

T, .0230 .0240 .0200 .0090 .0080 .0085

.8 T1 .0615 .0540 .0655 .0360 .0290 .0385

T, .0570 .0450 .0545 .0290 .0245 .0285

.85 T, .1070 .0865 .0905 .0725 .0525 .0545

it, .0990 .0730 .0770 .0645 .0470 .0505

.9 T, .1710 .1615 .1660 .1190 .1130 .1050

T2 .1570 .1415 .1435 .1130 .0945 .0970

 

 

 

 

 

 

H

.6

.7

.75

 

 

m

 

 

”9/92

n.7/.72

n.58/.62

 

 

Table 3.11: The other choice of m1 = m = mg.

80

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Mean Stdev

H, H estimate of H \ n 300 600 1000 300 600 1000
(III) .6 H 0.726 0.723 0.720 0.023 0.015 0.012
H* 0.669 .667 0.665 0.017 0.012 0.009

.7 H 0.770 0.766 0.763 0.028 0.019 0.015

H* 0.704 0.704 0.703 0.022 0.016 0.012

.75 H 0.797 .794 0.791 0.030 0.021 0.017

H“ 0.765 0.762 0.762 0.027 0.019 0.016

(IV) .6 H 0.721 0.719 0.717 0.024 0.015 0.011
H“ 0.664 0.664 0.663 0.017 0.012 0.009

.7 F1 0.763 0.761 0.759 0.029 0.019 0.015

11* 0.698 0.699 0.699 0.022 0.016 0.012

.75 H 0.789 0.788 0.786 0.031 0.021 0.017

H“ 0.756 0.756 0.756 0.027 0.019 0.016

 

Table 3.12: Comparison of H to H * which is H with m given in Table 3.11 under the

local alternatives (III) 6,,(3) = Tn and (IV) 6,,(93) = TnIL‘.

81

 

 

a=.05 (1:.025
Alternative H Test\n 300 600 1000 300 600 1000

 

15(36): 7,, .6 T1 .0010 .0000 .0000 .0000 .0000 .0000
T; .0170 .0150 .0140 .0050 .0065 .0030

 

T2 .0025 .0005 .0000 .0005 .0000 .0000
T; .0170 .0215 .0120 .0050 .0060 .0045

 

.7 T1 .0190 .0165 .0100 .0070 .0060 .0035
T; .1745 .1700 .1660 .1180 .1110 .1120

 

T2 .0175 .0145 .0090 .0075 .0055 .0030
T; .1610 .1555 .1520 .1035 .1005 .0985

 

.75 T1 .0575 .0485 .0370 .0360 .0280 .0115
T; .1330 .1460 .1270 .0840 .0955 .0725

 

T2 .0565 .0445 .0285 .0295 .0260 .0100

 

 

 

 

 

 

T; .1210 .1320 .1160 .0835 .0875 .0730

 

Table 3.13: The comparison of simulated power under local alternative 6,,(27) = 7-,,

among these tests based on estimates H and H *.

82

 

 

 

 

 

 

 

 

 

 

 

 

a=.05 a=.025

Alternative H Test\n 300 600 1000 300 600 1000
6(1‘) = Tax .6 T, .0000 .0000 .0000 .0000 .0000 .0000
T,‘ .0070 .0045 .0050 .0010 .0015 .0015

T2 .0000 .0000 .0000 .0000 .0005 .0000

T; .0075 .0060 .0040 .0020 .0015 .0010

.7 T, .0145 .0060 .0045 .0085 .0015 .0020

T,‘ .0880 .0975 .0945 .0515 .0620 .0610

T2 .0120 .0055 .0040 .0075 .0010 .0010

T; .0795 .0790 .0775 .0390 .0475 .0465

.75 T, .0245 .0260 .0285 .0100 .0145 .0100

T,‘ .0785 .0740 .0740 .0445 .0410 .0390

T2 .0230 .0240 .0200 .0090 .0080 .0085

T; .0590 .0610 0.0570 .0365 .0330 .0300

 

Table 3.14: The comparison of simulated power under local alternative 6,, (x) 2 Tax

among these tests based on estimates H and H *.

83

 

 

 

 

 

 

 

 

 

 

5:.05 a=.025

H, H, Test\n 300 600 1000 300 600 1000
.6 .7 T, .0320 .0320 .0270 .0185 .0160 .0120
T, .0645 .0555 .0505 .0355 .0300 .0265

T, .0325 .0340 .0215 .0175 .0195 .0125

T, .0660 .0590 .0470 .0340 .0315 .0220

.6 .8 T, .0975 .0915 .0785 .0560 .0510 .0425
T1 .0745 .0705 .0590 .0430 .0395 .0330

T, .0955 .0930 .0840 .0570 .0500 .0445

T, .0700 .0745 .0665 .0395 .0380 .0340

.7 .8 T, .0865 .0825 .0720 .0505 .0465 .0415
T, .0780 .0580 .0620 .0410 .0325 .0365

T, .0895 .0825 .0755 .0520 .0470 .0430

T, .0830 .0630 .0665 .0435 .0335 .0350

 

 

 

 

84

Table 3.15: Proportion of rejections under Ho for Part II.

 

 

 

 

 

 

 

 

 

 

 

 

 

a=.05 5:025

H, H, Test\n 300 600 1000 300 600 1000
6 .9 T, .1985 .2005 .1800 .1375 .1390 .1155
T, .1155 .1135 .1030 .0600 .0535 .0540

T, .1880 .1950 .1675 .1415 .1370 .1180

T, .1070 .1090 .0995 .0615 .0580 .0585

7 9 T: .1905 .1845 .1855 .1365 .1240 .1225
T: .1150 .1025 .1020 .0625 .0585 .0465

T, .1850 .1750 .1815 .1385 .1290 .1270

T, .1085 .0995 .1005 .0650 .0600 .0515

8 .9 T: .1740 .1725 .1795 .1250 .1125 .1280
T: .1200 .1020 .1090 .0705 .0510 .0590

T; .1685 .1735 .1775 .1200 .1135 .1275

T, .1175 .0970 .1045 .0730 .0510 .0600

 

 

Table 3.16: Proportion of rejections under Ho for Part II.

 

 

 

 

 

 

 

n: 1000 Mean Stdev
\ H .6 .7 .8 .9 .6 .7 .8 .9
6? .946 .932 .893 / .047 .052 .070 /
6; / 3.863 3.727 3.300 / .195 .232 .367

 

 

Table 3.17: Mean and standard deviation of the estimates (7,,

112(27) = 2:12, of = 1, a; = 4 and n = 2000.

85

2

(I; under 111(23)

 

 

 

 

 

 

 

 

 

 

 

 

 

07:05 a=.025

Alternative H, H, Test\n 300 600 1000 300 600 1000
(I) .6 .7 T, .5920 .7865 .9340 .4965 .7040 .8845
T, .5585 .7610 .9105 .4365 .6525 .8420

.6 .8 T, .3620 .4100 .4855 .2865 .3160 .3965

T, .3510 .3905 .4675 .2585 .2890 .3665

.7 .8 T, .3300 .3735 .4645 .2545 .2795 .3680

T, .3120 .3485 .4490 .2330 .2525 .3360

.6 .9 T, .2715 .2070 .2865 .2030 .1905 .2060

T, .2685 .2525 .2685 .1990 .1855 .2090

.7 .9 T, .2615 .2940 .2655 .1955 .2205 .1925

T, .2545 .2780 .2555 .1985 .2195 .1900

.8 .9 T, .2395 .2640 .2605 .1710 .1960 .1795

T, .2305 .2585 .2470 .2000 .1825

.1705

 

Table 3.18: Proportion of rejections under the fixed alternative (I) 6(z) = 1.

86

 

 

a=.05 a=.025
Alternative H1 H2 Test\n 300 600 1000 300 600 1000

 

 

(II) .6 .7 T, .2180 .3380 .4150 .1565 .2485 .3260
T2 .1258 .1935 .2460 .0810 .1280 .1655

 

.6 .8 T, .1895 .2215 .2280 .1295 .1555 .1615
T2 .1570 .1710 .1735 .1000 .1170 .1150

 

.7 .8 T, .1620 .1920 .2150 .1150 .1340 .1430
T2 .1245 .1500 .1550 .0870 .0960 .1025

 

.6 .9 T, .2430 .2345 .2320 .1750 .1635 .1665
T2 .2245 .2110 .2175 .1630 .1565 .1550

 

.7 .9 T, .2110 .2270 .2210 .1465 .1575 .1535
T2 .1940 .2070 .2000 .1435 .1515 .1490

 

.8 .9 T, .2195 .2150 .2090 .1545 .1555 .1530

 

 

 

 

 

 

T2 .1965 .2050 .1940 .1490 .1465 .1420

 

Table 3.19: Proportion of rejections under the fixed alternative (II) 6(T) = a: for part

II.

87

 

02.05 a=.025
Alternative H1 H2 Test\n 300 600 1000 300 600 1000

 

 

(III) .6 .7 T, .1100 .1100 .1010 .0765 .0690 .0565
T, .0935 .0900 .0855 .0550 .0525 .0485

 

.6 .8 T, .2100 .2325 .2040 .1545 .1595 .1405
T2 .1965 .2165 .1840 .1410 .1500 .1270

 

.7 .8 T, .2155 .2250 .2075 .1560 .1555 .1385
T, .2070 .2080 .1990 .1420 .1440 .1220

 

.6 .9 T, .3755 .3815 .3695 .2960 .2970 .2830
T2 .3665 .3710 .3555 .2900 .2925 .2760

 

.7 .9 T, .3755 .3840 .3720 .2975 .2905 .2910
T2 .3685 .3615 .3620 .2900 .2920 .2905

 

.8 .9 T, .3520 .3905 .3840 .2745 .2955 .2975
T, .3340 .3790 .3715 .2660 .2925 .2865

 

 

 

 

 

 

 

Table 3.20: Proportion of rejections under local alternative (111) 6,,(33) = Tn for part

II.

88

 

a=.05 02.025
Alternative H1 H2 Test\n 300 600 1000 300 600 1000

 

(IV) .6 .7 T, .0540 .0615 .0495 .0295 .0390 .0275
T, .0380 .0430 .0410 .0180 .0220 .0215

 

.6 .8 T, .1465 .1325 .1355 .0955 .0935 .0800
T2 .1295 .1190 .1125 .0785 .0795 .0690

 

.7 .8 T, .1290 .1435 .1415 .0900 .0980 .0925
T, .1130 .1310 .1275 .0735 .0890 .0795

 

.6 .9 T, .2620 .2740 .2415 .1920 .1995 .1710
T2 .2315 .2345 .2155 .1730 .1795 .1575

 

.7 .9 T, .2840 .2510 .2485 .2140 .1920 .1830
T2 .2615 .2290 .2295 .1990 .1685 .1605

 

.8 .9 T, .2535 .2410 .2425 .1815 .1820 .1735
T2 .2325 .2200 .2110 .1730 .1705 .1550

 

 

 

 

 

 

 

Table 3.21: Proportion of rejections under local alternative (IV) 6,,(23) = 77,51: for part

II.

 

Mean Stdev
H, \H .6 .7 .8 .9 .6 .7 .8 .9

 

 

(111) H, .719 .762 .821 / .016 .021 .026 /
H, / .711 .790 .872 / .024 .025 .025

 

 

(IV) H, .719 .762 .821 / .016 .021 .026 /

 

 

 

 

 

 

H2 / .705 .783 .866 / .024 .025 .025

 

Table 3.22: Mean and standard deviation of the estimates H1, H2 under local alter-

natives (III) 6,,(2:) = 7'", (IV) 6,,(55) = 7,,27 for of = 1, a; = 4 and n = 1000.

89

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