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THESIS

                                                                              

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310 555 0530

This is to certify that the
dissertation entitled
Parameter estimation in non-linear time series:
Random coefficient autoregressive and self-exciting

threshold models

presented by
Lianfen Qian

has been accepted towards fulfillment
of the requirements for

Ph . D . degree in §__Cl§_t.i§.8__ta

Department of Statistics and Probability

Mei—x

Major professor

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PARAMETER ESTIMATION IN NON-LINEAR TINIE SERIES:
RANDOM COEFFICIENT AUTOREGRESSI'VE AND
SELF-EXCITING THRESHOLD MODELS

By

Lianfen Qian

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

1996

ABSTRACT

PARAMETER ESTIMATION IN NONLINEAR TIME SERIES:
RANDOM COEFFICIENT AUTOREGRESSIVE AND
SELF-EXCITING THRESHOLD MODELS

By

Lianfen Qian

This dissertation studies the parameter estimation in two nonlinear time series
models: Random coefficient autoregressive and self-exciting threshold autoregressive
models.

For the random coefficient autoregressive model of order p (RCAR(p)), we discuss
a class of minimum distance (MD) estimators for the true unknown parameters.
These estimators are defined via certain weighted empiricals as in Koul (1986). The
class of estimators considered includes the least absolute deviation estimator and an
analogue of the Hodges-Lehmann estimator. The dissertation contains a proof of the
asymptotic normality of these estimators and a simulation study. It is observed that
RCAR(2) model with the Hodges-Lehmann type estimator fits the Canadian lynx
data at least as well as with the least square estimator.

For the first order stationary ergodic self-exciting threshold autoregressive model
with single threshold parameter, we show that the maximum likelihood estimators of
the underlying true parameters are strongly consistent under some regularity condi-
tions on the error density. Then, we prove that the maximum likelihood estimator of
the threshold parameter is n—consistent if the threshold parameter is the discontinu-
ity point of the autoregressive function. Further, we derive the asymptotic normality
of the estimators of the coefficient parameters. We also obtain a simple approxima-
tion of a sequence of normalized log-likelihood processes, hence prove the tightness

of the sequence of normalized log-likelihood processes.

 

To: Jenny Yao my threshold daughter

Qingchuan Yao — my beloved husband

iii

ACKNOWLEDGMENTS

I want to express my deep gratitude to Professor Hira L. Koul, my thesis advisor,
for all his continuous encouragement, expert guidance, generous support and extreme
patience during the preparation of this dissertation. His love of statistics and devotion
to research have served as the main source of inspiration to my research.

I also like to thank Professors Mandrekar, Gilliland and Shapiro for serving on
my thesis committee. My special thanks go to Professor Mandrekar for his carefully
reading my dissertation and making positive suggestions that improve the presenta-
tion, Professor Gilliland for training me working for the statistical consulting service,
Professor Shapiro for preparing me the strongly mathematical background.

I appreciate the support of the Department of Statistics and Probability in the
last three years for pursuing my Ph.D. in Statistics. I would like to thank all the
faculty members and staff in the department for all of their help during my graduate
work at Michigan State University. Finally, I want to express my deep thanks to my
husband, Qingchuan, for his continuous support, encouragement and patience. Major

portion of this research was supported by the NSF Grant DMS 94-02904.

iv

Contents

Introduction 1

I Random Coefficient Autoregressive Model

1 Minimum distance estimation 6
1.1 Introduction ................................ 6
1.2 Assumptions and Theorems ....................... 8
1.3 Proofs ................................... 11
1.4 Simulation results ............................. 16

II A Self-Exciting Threshold Autoregressive Model

2 Definitions, assumptions and consistency 22
2.1 The profile maximum likelihood estimation ............... 22
2.2 Assumptions ............................... 24
2.3 Strong consistency of the MLE ..................... 25
2.4 n-consistency of the threshold estimator ................ 30

3 Limiting distribution of am 40
3.1 Uniform consistency ........................... 40
3.2 Asymptotic normality of 91,. ....................... 43

4 Some asymptotic results on log-likelihood process 50

4.1 An approximation In of the normalized profile log-likelihood process in 50

4.2 Tightness of In .............................. 54

4.3 Some problems for future research .................... 58
A Appendix 59
Bibliography 62

vi

List of Tables

1.1 Simulation results ............................. 19

1.2 The estimators of the RCAR(2) model for lynx data .......... 2O

vii

List of Figures

1.1 The graph of the dispersion Mg(u) ......... ' .......... 21

viii

Introduction

Nonlinear time series analysis has achieved a rapid development in the past two
decades. The two main factors expediting nonlinear time series model building are:
Essentially complete theory of linear time series analysis and some complicated dy-
namics phenomena that can not be modeled by linear time series models. Many
different types of nonlinear time series models have been studied in literature (see
Priestly (1980), Pagan (1980), Nicholls and Quinn (1982) and Tong (1983, 1990). In
this dissertation, we will focus on the parameter estimations of two nonlinear time
series models: The random coefficient autoregressive and the self-exciting threshold
autoregressive models.

The first part of this dissertation is concerned with the random coefficient autore-

gressz've model of order p (RC AR(p)) in which one observes {X,-,i E Z} satisfying
X.- = (0 + z.)TY.-_1 + 65, i6 2, (0.1)

for some 9 6 R”, where {e;,i 6 Z} and {Z;, i E Z} are independent sequences
of independent identically distributed random vectors with respective distribution
functions F and G. Here Y0 := (Xo,. ..,X1._,,)T is an observable random vector
independent of {6;}, Y,“..1 := (X;_1,...,X;_p)T, Z.- := (Z51,...,Z,-,,)T, p Z 1 is a
known integer and Z denotes the set of all integers. For the importance of these
models in time series analysis, see the Lecture Notes by Nicholls and Quinn (1982)
and the monograph by Tong (1990). RCAR(p) models include the well known AR(p)
models (take Z,- to be degenerate at 0).

The problem of interest here is to estimate the unknown parameter 9 based on

{Yo, X1, ...Xn}. We will study the Minimum Distance (MD) estimators 9 of 0 which

1

are based on minimizing certain types of distance functions Mg(-), called dispersion,
related to the data and the parameter, for measurable function g from R” to ’R”. The
importance of this methodology in linear models is discussed in Koul (1992, Chapters
5 and 7). These estimators have many desirable properties, including consistency,
robustness against outliers in the error, efficiency and asymptotic normality in AR
models.

Koul (1992) discussed the asymptotic behavior of the estimators 0 under the
AR(p) setup. We obtain the asymptotic normality of 8 under the RCAR(p) setup.
The method of proof is similar to that of Koul (1992) which requires obtaining the
asymptotic uniform quadraticity of Mg(t) and showing Jim - 0) = 0p(1).

This part of the material is organized as follows. Assumptions and statements of
main theorems appear in Section 1.2 while proofs appear in Section 1.3. Section 1.4
contains a simulation study and an application to the Canadian lynx data. The sim-
ulation study shows that the Hodges-Lehmann type estimator is as good as the Least
Square (LS) estimator and Huber estimator (HE). For the additive effects outliers
model, Dhar (1990, 1991) established the robustness of the MD estimator. Later,
Dhar (1993) working on the AR(p) model showed through simulation that the MD
estimator, with H (1:) = a: and g(y) = y, has the smallest absolute bias even for the
small sample size n = 10 and the smallest mean square error for n = 50 and 100,
under the logistic error distribution.

As an application, we fit the RCAR(2) model with MD estimators to the annual
trappings of the Canadian lynx over the years 1821-1934. The result shows that this
model provides an acceptable alternative to the more widely adopted class of AR
models.

A self-exciting threshold autoregressive model is a piecewise linear model. It is
fitted by different linear autoregressive functions in both past variables and param—
eters for different subsets of data. Tong (1977) first mentioned the usefulness of
these models. Later, Tong (1978a, 1978b, 1980) developed these models further in
a systematic way for modeling of discrete time series data. He argued that various

phenomena such as limit cycles, jump resonance, harmonic distortion and chaos can

be modeled by discrete time series that are piecewise linear. He called these models
the self-exciting threshold autoregressive (SETAR) models. See Tong (1983, 1990) for
a comprehensive introduction to general SETAR models.

The second part of the dissertation is concerned with the large sample behavior
of maximum likelihood estimators in a special SETAR model, called SETAR(2;1,1),
defined as follows:

X; = h(X.'_1,0) + 6;, i2 1, (0.2)
for some 0 = (9{,r)T E 725, where 91 = (a0, a1,bo,bl)T 6 R4 and for any a: E ’R,
h(:z:,9) = (a0 + alm)I(:1: S r) + (be + b1$)I(:1: > r).

Here, the errors {6;} are independent and identically distributed random variables
with mean zero, finite nonzero variance and 61 is independent of X0. The parameter
r, the location of the change of the autoregressive function h, is called the threshold.

Define a region 9 of parameters as follows.
9 = {19 = (ao,a1,,60,fl1,s)T 6 R51 (11 (1,,81 <1, 01 ,8] <1}. (0.3)

Petruccelli and Woolford (PW)(1984) proved that the model (0.2) with do = b0 = 0,
r = 0 is ergodic if and only if 0 E ('9. Note that O is much wider compared to
the region of stationarity of AR(I) model. Chan, Petruccelli, Tong and Woolford
(CPTW)(1985) continued PW’s work and found some other sufficient conditions on
the parameters for {X5} in model (0.2) to be ergodic. Note that the process {X,-}
defined in (0.2) is a Markov chain. From ergodicity, one can readily obtain stationarity
if the measure induced by the initial distribution of the Markov chain is the same
as the invariant measure of ergodicity. Since this part of the dissertation discusses
the asymptotic properties of the maximum likelihood estimators, it will be assumed
that the initial measure is equal to the invariant measure. That is, we will work with
stationary and ergodic SETAR (2;1,l) model.

For the case of the threshold r having only finite number of possible values and
assuming Gaussian errors, Tong (1983) constructed a maximum likelihood estimator
of the unknown parameters using Akaike Information Criterion (1973). If the thresh-

old r is known, CPTW (1985) obtained the consistency and asymptotic normality

property of the least-square estimators of the coefficient parameter 91 under some
regularity conditions. But in practice, the threshold parameter r is unknown and
can take infinitely many values in R. In this case, Petruccelli (1986) proved that
the conditional least-square estimator (CLSE) of 9 is strongly consistent for the SE-
TAR(2;1,1) model. Chan (1993) developed the strong consistency of the same CLSE
in a general SETAR model. Furthermore, he claimed that he obtained the limiting
distribution of the CLSE of the threshold under some regularity conditions on the
errors.

We derive the asymptotics of a maximum likelihood estimator (MLE) of the under-
lying parameter 9 in model (0.2), when the errors have a density f, not necessarily
to be Gaussian. Unlike the popular AR model, the likelihood function of the SE-
TAR(2;1,1) model is not differentiable with respect to the parameters. Actually, it
is not continuous in the threshold parameter in general. Thus the routine method of
computing maximum likelihood estimator can not be adopted. Instead, in Chapter
2, Section 2.1 discusses a profile maximum likelihood method to obtain the MLE
9,, = (9;, rn)T of 9 = (9?, r)T. Section 2.2 states assumptions for latter use. In Sec-
tion 2.3, Theorem 2.3.1 shows that the MLE is strongly consistent. If the threshold
parameter r is a discontinuity point of the autoregressive function h, then the maxi-
mum likelihood estimator fn is not only consistent, but also n-consistent as shown in
Theorem 2.4.1, i.e. |n(r,, — r)| is bounded in probability. In Chapter 3, we develop
the asymptotic normality of the coefficient parameter estimator 91,, and some more
byproduct results. In Chapter 4, as a consequence of the n-consistency of 7‘”, the suit-
ably normalized log-likelihood sequence of processes {ln} (see section 4.1) is shown to
be approximated by a sequence of simpler processes which describe the log-likelihood
under known coefficient parameter 91. Through the latter processes, the tightness of
{in} is derived. It is expected that this result will be useful in obtaining the limit-
ing distribution of the standardized maximum likelihood estimator of the threshold
parameter.

Notation. Throughout this dissertation, the symbol 9 is the fixed unknown
underlying parameter, the function f is the p.d. f of 61 and F denotes the distribution

function corresponding to f. The expectation under 9 is denoted by E.

Weak convergence is denoted by =>. A sequence (random) goes to zero (in proba-
bility) is denoted by o(1)(0p(1)) while 0(1) (0p(1)) means that it is bounded (in prob-
ability). The multivariate normal distribution with mean zero and covariance matrix
I‘ is denoted by N(0,I‘). Let 7?, be the real line (—oo,oo), and R = R U {—00, 00},
then the compactness of the set R is under the metric d(-, ) defined by d(x,y) =
larctan a: — arctan yl. A function (,0 satisfies the Lip ( 1) if V at, y E 'R, 3 L 2 0, such
that

Mic) - s0(y)| .<_ le - yl-

For any event A, the complement event of A is denoted by Ac and the indicator
function is denoted by I (A) Throughout, the capital letter C, the symbols 7;, i =
1,2, stand for absolute constants and they can have different values in different
places. The notation a'Ty stands for the inner product of vectors a: and y. For any
matrix M = (mij), “M” = 2,3,- Imijl , MT stands for the transpose of M, det(M)
and adj(M) stand for the determinant, adjoint matrix of M, respectively. Vectors
of dimension more than one are denoted by bold face letters. The index 2° in the

summation varies from 1 to n unless specified otherwise.

Part I

Random Coefficient Autoregressive

Model

Chapter 1

Minimum distance estimation

1 .1 Introduction

This part of the dissertation considers the random coefficient autoregressive model of

order p (RCAR(p)) in which one observes {X.-,i E Z} satisfying
X; = (9 -I- Z;)TY,'_1 + 6;, l6 Z, (1.1)

for some unknown 9 E ’R” and for independent sequences {e,,i E Z} and {Z .-,i E Z}
of independent identically distributed random vectors with distribution functions F
and G', respectively. Also, it is assumed that E61 = 0 and E6? = 012; > 0, EZ1 = 0
and EZIZ$ = 2 Z 0. Here, Y0 := (X0, . . . ,X1_,,)T is an observable random vector
independent of {6,}, Y,_1 := (X;_1,...,X,--,,)T,Z.- := (Z;1,...,Z.-,,)T, p Z 1 is a
known integer and Z denotes the set of all integers. This model includes the well
known AR(p) model (take Z ,- to be degenerate at 0). For the importance of RCAR(p)
models in time series analysis, see the Lecture Notes by Nicholls and Quinn (1982)
and the monograph by Tong (1990) .

The problem of interest here is to estimate the unknown parameter 9 in model
(1.1) based on {Y0,X1, ...X,.}. We study Minimum Distance (MD) estimators of 9
which are based on minimizing some types of distance functions, called dispersion,
related to the data and the parameter. The importance of this methodology in linear

models can be found in Koul (1992, Chapters 5 and 7). These estimators have many

6

desirable properties, including consistency, robustness against outlier in the error,
efficiency and asymptotic normality in AR models.

To describe these estimators, let 9 = (g1, ..., gp)T be a measurable function from
’R” to ’R”, and | - I be the Euclidean norm. For a given nondecreasing right continuous
function H on R, define the dispersion function, for u E 7?”,

MA") = /|n‘1/2:9(Ya-1){1(Xe - ”TYi—l S y) - 1(-Xi+uTYi-1 < y)}|2dH(y)

and a class of MD estimators of 9, one for each 9 and H, to be
9 := argmin{Mg(u); u 6 72”}.

Here I (A) is the indicator function of the event A. The existence of the MD estimator
9 follows from Dhar (1993).

Note that if we take H (9:) == 1:, g(y) = y, then 9 is the Hodges-Lehmann type
estimator. If we denote U.- := Z,.TY.-_1 + 6;, then the RCAR(p) model becomes
X; = 9TY:_1 + U.- and M9 is essentially the same as the K g of Koul (1992) with 6,-
there replaced by U5.

Koul (1992) discussed the asymptotic behavior of the estimators 9 under the
AR(p) setup. A simulation study of Dhar (1993) shows that many of these MD
estimators outperform the least square estimator in an AR(p) model with asymmetric
error. In this paper, we obtain the asymptotic normality of 9 under the RCAR(p)
setup. The method of proof is similar to that of Koul (1992) which requires obtaining
the asymptotic uniform quadraticity of Mg(t) and showing flfi(9 — 9)| = 0p(1).

The material is organized as follows. Assumptions and statements of main theo—
rems appear in section 1.2 while proofs appear in Section 1.3. Section 1.4 contains
a simulation study and an application to the Canadian lynx data. The simulation
study shows that the Hodges-Lehmann type estimator is at least as good as the least
square (LS) estimator and the Huber estimator in the sense of having smaller biase

and mean squared error. For the annual trappings data of the Canadian lynx over

A

the years 1821-1934, it is observed that the RCAR(2) model with 9 estimated by 9

provides an acceptable alternative to the more widely adapted class of AR models.

1.2 Assumptions and Theorems

Throughout this part of the dissertation, we assume that {X;} is strictly stationary
and ergodic satisfying model (1.1). Sufficient conditions for this to happen are dis-
cussed in Theorems 2.1, 2.7 and Corollary 2.3.2 of Nicholls and Quinn (1982). In
particular, when p = 1, then the following two assumptions imply the strictly sta-
tionarity and ergodicity of {X5}.

(i) {65,2' E Z} and {Z,-,i 6 Z} have mean zeros and finite variances of; and 03;,

respectively.
(ii) 92 + 0?; <1.

Now let 7 be a measurable function from 7?.” to ’R, 3], a E ’R, t 6 RP. Define

My; t, a) = / F<y - zTY.-. + n-1/2(tTY.--. + alYi—1|))dG(Z).

V(y;t a) £27m) 1 I(U s y+n“’2(tTY.-_1+aIY.-_1I)),

fi=l

'U(y;t a) =T:7(Yi- -1 )pi(yit 0)

i=1

W(y;t,a) = Wm M) - v(y; ta)-

Write MW), W(y,t), v(y,t), V(y,t) for My; t,0), W(y; t,0), v(y;t,0), V(y;t,0),
respectively. Also define
K,(0 + n-Wt) = /[V(y,t) + V(—y,t) — 11—1/2270/ ,_ -1)]2dH( (y), t e R”.
i=1

Observe that choosing 7(Y.-_1) = gj(Y.-_1) in K,(9 + n'l/Zt) gives the jth summand
in Mg(9 + n‘l/zt). I

We now state the conditions required for the asymptotic uniform quadraticity
of K,(9 + n‘1/2t) in t. In the sequel, all expectations are taken under the true
parameter 9. Moreover, by the symmetry around 0 of the distribution G, we mean
dG(z) = (—1)‘D dG(—z).
(F) Functions F, G and H are symmetric around 0 and F has a Lebesgue density f.

(CI) E 72(Yo) < 00.

(C2) For any t E R” and a E R,

/ E 72(Yo)lp1(y; t.a) — p1(y;0.0)ldH(y) = 0(1).

(C3) There exists a constant k, 0 < k < 00, such that V 6 > 0,V t E R”
lirr},ian(/ [72‘1” ZvP(Y._.)(p.-(y; w) — My; t, -.s))]2.m(y) s 1.52) = 1,
i=1
where 7+ = max(7,0),'y‘ = 7+ — 7.

(C4) For every t 6 R”,
/ {fl/finalise, t) — p.-(y,0)l — A(y)t/2}2dH(y) = 010(1).

where A(y) = 2E 7(Y0)Yg ff(y — zTYo)dG(z).
(C5) lE 72(Y0)F(y — ZfY0)(1— FL?! - ZirYODdHQ/l < 00~

We state two more conditions required for the asymptotic normality of the esti-

mator 9: Let B(y) = Eg(Yo)Yg ff(y — zTYo)dG(z), y 6 R.

(C6) The matrix B(y) is nonnegative definite for each 3; E R, f l3(y)dH (y) and
f BT(y)B(y)dH (y) are positive definite p x p matrices.

(C7) Either
sTg(Y,-_1)Y,-T_,s Z 0, V 1 S 2' g n, V s 6 RP, |s|=1,a.s.

OI'

sTg(Y,-_1)Y;’_,s g 0, v 1 g i g n, v s 6 RP, Isl = 1,a.s.

Remark 1 The above conditions are assumed so that the desired asymptotic
uniform quadraticity and asymptotic normality of 9 are achievable. In the case of
the AR(p) model, i.e., when Z,- E 0, the above conditions (C1)-(C4) correspond
to the conditions (7.4.7a), (7.4.8)—(7.4.10) of Koul (1992) and the condition (C5) is
implied by (7.4.7a) and (5.5.69) of Koul (1992). For the Huber type estimator, i.e.,
for y E R”, g(y) = yI(|y| S c) + kl%ll(|y| > c), for some positive constants k and
c, the conditions (C7) is a priori satisfied. If H is a finite measure, 7(y) = y,, the

10

jth component of y, j = 1, ..., p, g(y) = y and F has uniformly continuous density,
having positive integral with respect to the measure induced by H, then all of the
above conditions (Cl)-(C7) are implied by the strict stationarity and ergodicity of

{Xi}-

Now we are ready to state our main theorems.
Theorem 1.2.1 Suppose that conditions (F), (CU-(C5) hold. Then, V 0 < B < oo,

sup mm + n-‘P’o — 13:19 + n-“Ptn = 0pm,
ItISB

where

16.09 + n-l/Pt) = [Ii/(y. 0) + V(—y,0) — n-1,. in.-.) + A(y) tl’dH(y)-

i=1

Upon using this result p times, the jth time with

7(Yi-1) = gj(Yi-1)a .7 =1: "WP, (1'2)

we obtain the required asymptotic uniform quadraticity of the dispersion Mg(u). For
stating the desired results, we need to clarify the conditions (C1)-(C5) when 7 is as
in the equation (1.2). Condition (C1) is now equal to

(Clg) EgflYo) < 00 for all j = 1,...,p.
Similarly, condition (C2) is equal to V t E R”, a. E R, 1 S j S p, ||t|| S B < 00,
(023) f ng(Yo)|p1(y; t,a) - p1(y;0,0)ldH(y) = 0(1)-

Let (C3g) stand for the condition (C3) after 7*(Y;_1) is replaced by gj-h(Y,-_1),1 S
j S p, in condition (C3), 1 S i S n. Interpret (C4g), (C5g) similarly.
Corollary 1.2.1 Suppose that conditions (F), (CIg)-(C5g) hold. Then
sup |M9(9 + n'1/2t) — Mg(9 + n’l/zt)| = 0p(1), (1.3)
ItlSB

where

we + n-l/Pt) = / I-‘71-5:9(Yi-1){I(Ui s y) — I(—U.~ < y)} + Emma).

11

Theorem 1.2.2 In addition to the assumptions of Corollary 1.2.1, assume that the
conditions (06) and (C7) hold. Then

A

«Hm — 0) => N(0, F), (1.4)
where
r := v-‘E(¢(—U1) - ¢(U;))g(Yo)gT(Yo)(¢(—Ul) — ¢(Uf))TV“.
My) == / Hz 3 y)BT(x)dH(:c), 10(31’) == / I(a: < y)BT(:c)dH(a:),
v == / BT(y)B(y)dH(y).

Remark 2 Least Absolute Deviation Estimator (I.a.d.). If we choose g(y) = y

and H () degenerates at 0, then 9 is the I.a.d. estimator, v.i.z.
910,, = argmin{|n"1/2 Z Y,-_lsign(X,- - tTY,-_1)|2, t 6 RP}.

Because of the importance of the I.a.d. estimator, we summarize all the conditions
on f for the case p = 1. All conditions (F) and (Cl)-(C7) are satisfied when G
is symmetric around zero, F has a uniformly continuous and even density f, and

EX3f(Z1Xo) > 0. Therefore, Theorem 1.2.2 implies that fi(9 — 9) : N(0,&fad ,

 

 

 

where
.2 _ EX3A2(0)lI(—UISO)—I(Ul>0)l2
Glad “' 444(0)
_ EXci _ 570(3)
_ «42(0) _ 4(EX3f(Z1Xo))2°
1.3 Proofs

Notation For any measurable functions f and g from R”+1 to R, define

Ifs - guli; == / [f (y, 8) - g(y, 11)]de (y).

where u, s 6 R”. In the following, Wi, vi stand for W, v with 7 replaced by 7*.
The proof of Theorem 1.2.1 is similar to that of Theorem 7.4.1 of Koul (1992) and

is facilitated by the following two Lemmas.

1-1

12

Lemma 1.3.1 Suppose that the RCAR(p) model (1.1) holds. Then the followings
hold.

(A). The condition (C?) implies that V 0 < B < oo,
E/[Wi(y;t,a) - Wi(y;t,0)]2dH(y) = 0(1),V ItI S B,a E R. (1.5)
(B). The condition (C3) implies that V |t| S B, V 0 < B < oo,

lim inf P( sup Ivi(y; s) — vi(y;t)|2 S 1962) = 1, (1.6)
" ls-tISB

where k and 6 are as in (C3).

(C). The conditions (C1), (C3) and (Cl) imply that V 0 < B < 00,

.2121; My, t) - v(y; 0) - A(y) t/212dH(y) = 0P(1)o (1-7)

Lemma 1.3.2 Suppose that the conditions (C2) and (C3) hold, then V 0 < B < oo,

sup [[W*(y,t) — WP(y,0)1’dH(y) = ..(1), (1.8)
ItIsB
'21:; [W(y,t) - W(y,0)l2dH(y) = 0P(1)- (1-9)

The proofs of the above lemmas are similar to those of Lemmas 7.4.2 and 7.4.3
of Koul (1992) with the following modification: Replace the o—fields used there by
f,- = o{e,~,Z,-,Yo,j S i}, i _>_ 1 and the linear term there by A t/2.

Proof of Corollary 1.2.1. Note that the jth summand in M9 is the same as
that in K, when the function 7 is replaced by g,- in (1.2). Therefore (1.3) follows

from Theorem 1.2.1 easily. El

Before proving the Theorem 1.2.2, we need the following three lemmas.

Lemma 1.3.3 Let {u1,u2,...} be a stationary ergodic stochastic process such that

E{u¥} is finite and E(u,-|u1,...,u,-_1) = 0,V i Z 1, with probability one. Then the

distribution of n‘1/2 ’-‘ 1 u; approaches the normal distribution with mean zero and

variance Euf.

13

Proof. See Billingsley (1961).
Lemma 1.3.4 Suppose that the assumptions of Theorem 1.2.2 hold. Then
W(é - 6l) = 0P0)-
Proof. It suffices to prove that, for every 6 > 0, El B > 0 and integer NE 3
P( inf Mg(u) _>_ Mg(9)) >1— 6, V n > NC, (1.10)

Inl/2(u-0)I>B

because then

P(|,/H(b — 9)) > B) s P( inf M,(u) < M,(o)) < e, v n > N,.
)nP/P(u-0))>B

The following lemma gives (1.10). E]

Lemma 1.3.5 For any 6 > 0,3 B (depending on e ) and NC, 0 < B < 00, such that

P( tinf M,(0 + n‘l/zt) 2 M,(9))>1— e, v n 2 N., (1.11)
l )>8

P( tinf Mgw + n‘l/zt) 2 M,(o)) > 1 — e, v n 2 N.. (1.12)
I |>B

Proof. Write Vj(y, t), Wj(y, t) for V(y, t), W(y, t), respectively, when 7 replaced
by g,- in (2) at jth time. Put Vg(y,t) := (V1(y,t),...,V,,(y,t))T and Wg(y,t) z:
(W1(y, t), ..., Wp(y, t))T. Note that the measure generated by H being o-finite, there
exists a partition {/15} of R such that 0 < [A dH < oo,i = 1,2, Let h = 2:2, 1A,,
then 0 < Ihl}, = fh’dH < 00. For t 6 R”, define

N(t) =-- [[vxy. t) — m—y, t))h(y)dH(y),
Mt) == [nus/,0) — V.(—y.0) + B<y)t1h(y)dH(y).
Then, for any 8 E R”, Isl = 1, by the CS inequality,

)sTN(t)IP = |/sT{V.(y.t)-n(—y.t))h(y)dH(y)|”

/ (sTWy, t) — n(—y.t)))”dH(y)Ih)%.
M,(a + n“1/2t)|h|§,.

|/\

|/\

 

14

ForaO < B< 00, t 6R”, writet= su,|s| = 1, then

TN(t)|2
'f M 0 *1/2 > '8 .
|u|>gl|8|=l g( '1'" 8U)—|UI>BIilsl=l lhlll
Similarly,

. T ‘ 2
inf Mg(9 + n‘l/zsu) Z inf M.
|u|>B,|8|=1 lui>B.|t‘3l=1 lhlif

Note that, by the condition (C5),
E / WP(y.0)tH(y) = Wm) / Fa — zfvoxt — F<y - szondHu) < oo.

It follows easily that Mg(9) is bounded in probability. That is, V e > 0, 3 M. < 00
such that
P(Mg(9) S M.)21— 6/2, for all n 21. (1.13)

Thus it suffices to prove that
|.<1)TN(su)|2

P > c 1 —- , e
(|u|>ia,|s|=1 |h|§, — M) > C (1 14)
(sTN(su)|P
P —— > c — . .
(lul>lBI.lI8I=1 lhlir __ M) > 1 e (1 15)
But, V t E R”,
(t) - 1V (t)|2

IN
/ Inuit) — no.0) - V.(—y,t) + V.(—y,0) — 3(y)t|2dH(y)|h|§;
< Zlhliillwgt — W... + w

|/\

gt ‘1' W-golil ‘l' lvgt " Ugo + ”-gt + v-90 - Btlfq].
By using Lemma 1.3.1(C) p times and Lemma 1.3.2, we obtain that V 0 < B < oo,

sup |N(t) — [\A/(t)|2 = 0p(1).
ItISB

Now rewrite N (t),

Mt) = /1V9(y,0) — V.(-—y,0))t(y)dH(y) + / B<y>h(y)dH(y) t
1: N1 + N2 t.

By the OS inequality,
W s Mg(9)lh|i;-

 

15

Therefore, by (1.13), there exists b = Mel/2|th, such that,
P(|1\71| Sb) 21—5. (1.16)

Denote a, = sTfB(y)h(y)dH(y)s = sTfB(y)dH(y)s, a = inf|,|=1{a,}. Then by
the condition (C6), a > 0. The rest of the proof is similar to that of Lemma 5.5.4.
of Koul (1992). [3

Proof of Theorem 1.2.2. EXpand Mg(9 + n’l/zt) in t,
M909 +0n’1/2t)
1
= M.( )+ 2 tT— W: [m U < y) —I(- U.- < y))BT(y)g(Y.--.)dH(y)
fii=l .
+ tT / BT(y)B(y)dH(y) t.

Let 9 := argmin{Mg(u) : u 6 R”). Then by the same proof as that of the Theorem
5.4.1 of Koul (1992), we have

[(3 — 6V / BT(y)B(y)dH(y)(é — é)| = 010(1).
Therefore it is enough to prove
ma — a) => N(0, r). (1.17)
But 9 must satisfy the following equation

«to? - 0) = (/ BT(y)B(y)dH(y))-‘%. (1.18)

where

5. == Dam—U.) — ¢(U.-‘))9(Yt—1)-

Since {X.-} is strictly stationary and ergodic, so are (¢(—U.-) — tb(U,'))g(Y,_1).
Furthermore, by the condition (F), U1 is a continuous random variable and given Yo,

the conditional distribution of U1 is the same as that of —U1. Thus

El¢(-U1)- ¢(Uf)|}'ol = 0 (1-19)

16

Now, for any p—component vector 9,

E{9T(¢(—Ut) - 1NU?))9(Yi-1)9T(Yi-1)(1/J(-Ut)-- ¢(Ut-))T)3}
= UTE{(t(-U.) — t(U;))g(Yo)gT(Yo)(¢(—U.) — t(U;))T}U.

This expectation exists if Eg}(Yo) < 00, j = 1,...,p and condition (C6) holds.
Then E{flT(I/J(—U;) — 1/2(U,-'))g(Y,-_1)|.77.-_1} = 0 follows from (1.19) and station-
arity of Ug. An application of Lemma 1.3.3 shows that n‘1/22?=1,6T(1,b(—U,-) —
1/J(U,-))g(Y,-..1) converges weakly to the normal distribution with mean zero and

variance

fiTE{(¢(-U1) - 1MU?))9(Yo)£IT(Yo)(1l'(-U1) - ¢(Uf))T}fi.

for all 9 6 R”. U.- Thus, by the Cramer-Wold device, 5,, converges weakly to
the multivariate normal distribution with mean vector zero and covariance matrix
E{(¢(‘U1) — ¢(U1—))9(Y0)9T(Y0)(¢(‘U1) — ¢(Ul-))T}' Hence W(é — 9) 003‘
verges in distribution to the normal distribution with mean vector zero and covariance

matrix

V‘1E{(t(—U1) — ¢(U:))g(Yo)gT(Yo)(tt(—U1) - ¢(Ur))T}V“.

This ends the proof of Theorem 1.2.2. Cl

1 .4 Simulation results

In this section, we investigate the performance of the MD estimators under the
RCAR(I) model for finite samples. A simulation study (100 replications) was per-
formed for samples of size n = 20, n = 50 and n = 100. The samples were generated
by the RCAR( 1) model,

Xi = (9 + Zi)Xi-l + 6i

with the true parameter 9 = .5, and different error distributions and normal distibuted

random coefficients.

17

The comparison is made among LS, Huber and MD estimators. The MD esti-
mators considered for comparison under the RCAR( 1) model are as follows. The
function H, g are taken to be the identity function (i.e., H(y) = g(y) = y,y E R).
The Huber function 43 is defined as follows.

d>(:r.) = { x’ if lxl S C (1.20)

c sign(:c), if le > c.
with c estimated by St = MedianIXgl.

The random coefficient distribution G considered here is a normal distribution
with mean zero and variance .25; This enables to have 92 + of; < 1. The error
distributions considered are the following.

(a) F is the standard normal distribution,
(b) F is the double exponential(1) distribution,
(c) F is the logistic (1,1) .

The RCAR(I) process is generated as follows:

In case (a),

I. generate a vector < w(1), w(2) >T, where the w(i)’s are successivly generated
by a standard normal random number generator. So they are independent.

2. Repeat step 1 (n + 200) times where n is the sample size desired. Let the
w(1), .5w(2) generated in the mth time be em, Zm,m = 1, ...,n + 200, respectively.
Then < 61, ...,cn+200 >T and < Z1, -..,Zn+2oo >T will theoretically be independent,
have zero means and E6? = 1, and EZ,2 = .25.

3. Calculate

Xi = (9 + ZilXi—l + 6i,

where X0 is generated by the normal distribution with mean zero and variance 2. Then
ignore thefirst 200 X values produced. This enables {X;} to reach an equilibrium
since we assume {X;} is stable.

In the case (b) ( or (c)), we independently generate w(1),w(2) from double ex-
ponential(1) ( or logistic (1,1)) and standard normal distribution, respectively. Then

do the same thing as in case (a) for steps 2 and 3.

18

The LS estimator is computed using the formula 2;, X,X,-_1/Z?=1 X?_,. The

Huber estimator is the solution of
T(u) := ZX.’_1¢(X5 — uX;_1) = 0. (1.21)
i=1

with d5 as given in (1.20).

For the Huber estimator, SI =Median IX,- — uX.-_1| is first computed using the LS
estimator 91,, then St is again computed by using the zero of (1.21). This iterative
procedure is terminated when the absolute difference between the consecutive zeros
of (1.21) is less than 10‘6.

To compute the MD estimator, we minimize the dispersion Mg(u) over [—1,1]
and the minimizer is denoted by 9,“. Table 1.1 contains the simulation results of
the averages (Mean) and the mean squared errors (MSE) of the estimators for the
true parameter 9 = 0.5 in the RCAR(I) model with 100 replicates and sample size
n. Notice that the MD estimator computed in Table 1.1 is a local minimization.
According to the paper by Dhar (1993, Lemma 1.1), in the case H (x) = :13, the
minimizer of function Mg(u) can only be one or a convex combination of a pair of
elements from the set

D = { (x. - X.)/(X.-_. — X.-1).(X. + x,)/(x,._, + x.-.) = } .
X,-_1 75 XJ-_1,X-_1 75 —X,--1,1Si,j S n.

Thus the global minimizer can be computed through comparing (9d, Mg(9d)) and pairs
(u, Mg(u)) for u E D starting with (9m), Mg(9md)).

From Table 1.1, we observe that the Huber estimator has the biggest MSEs except
in the case of Dexp(1) error and n = 50 which could be caused by the computing
accuracy. Most of the biases and the MSEs of MD estimator are less than these of
LSEs. Also, the estimated standard deviation of the averages of the estimates can be
computed by S V/ s/TO—(I = S V/ 10, where S V is the sample variance which is related to
MSE by the formula MSE = (n — l)SV/n + (Mean — 9)2. Again, we observe that all
estimators are under estimating. For the sample of size n = 100, the MD estimators
with H (:r) = g(x) = a: are between LSE and Huber estimator. The simulation study

was done by Mathematica.

19

Table 1.1: Simulation results

 

Error distribution

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N(0, l) Logistic (1,1) Dexp(1)

Estimator Mean MSE Mean M SE Mean MSE
n=20

LS .437970 .055660 .432906 .058949 .439949 .068925

Huber .452448 .061695 .447024 .066406 .447882 .075071

MD .441372 .055152 .433595 .061095 .437264 .076194
n=50

LS 0.469420 .020073 .444754 .025691 .477704 .020230

Huber 0.463232 .026582 .458541 .030475 .479045 .026283

MD 0.473069 .020896 .454640 .0263490 .476869 .031124
n=100

LS .475662 .013030 .486802 .010064 .474512 .013748

Huber .488874 .014368 .498575 . 13089 .487110 .013204

MD .483978 .012277 .494234 .11173 .483249 .012016

 

 

 

 

 

20

Table 1.2: The estimators of the RCAR(2) model for lynx data

 

A

 

Estimators él 02 $311 £312 £322 (3%
LS 1.3844 -.7479 .0770 -.0694 .0821 .0364

ML 1.4274 -.8073 .0664 -.0489 .0839 .0300
MD 1.3932 -.7495 .0764 -.0706 .0845 .0367

 

 

 

 

 

 

 

 

An applied example

We now fit a second order autoregressive random coefficient model
X: = (01 'l' Zli)X:_1 + (02 ‘1' Z2i)X:_2 + 6i

to the classical Canadian lynx data. Here X5" = X,- —X, X is the average value of the
Xg’s, and X,- is the loglo of the ith data. We took the first one hundred observations.
The MD estimators 91, 92 of 91,92 are 1.3932, -.749498. Let Z1 = (Z11, Zgl)’ and

2: 2ll 212
212 222

= Ezlz‘f.

 

To estimate the covariance matrix 23 of Z1 and the variance 0%. of £1, substitute
the MD estimator 9 into (3.2.4) and (3.2.5) of Nicholls and Quinn. The estimators
in, 5312, 2‘32. and .9}. obtained thus are .076433, -.070556, .084552 and .03668. The
comparison of the LSE , MLE of Nicholls and Quinn’s and the MD estimator is given
in Table 1.2.

From Table 1.2, we can see that the MD estimator performs at least as well as
the LSE. Also, notice that the ML estimator of Nicholls and Quinn has the smallest
estimated variance 6% and the smallest norm of the estimated covariance matrix of
Z1. The three dimensional graph of the dispersion Mg(u), u E R2 is in Figure 1.1.

The zeros of the characteristic polynomial (1 — 1.39322 + .74949822) are
1.15509 exp {:l:i27r/9.88329}, and so by using RCAR(2) model, it exhibits a period of
9.88329 cycle which is close to the result of Moran(1953).

21

 

Figure 1.1: The graph of the dispersion Mg(u)

Part II

A Self-Exciting Threshold

Autoregressive Model

Chapter 2

Definitions, assumptions and

consistency

2.1 The profile maximum likelihood estimation
Recall that the SETAR(2;1,1) model is defined by
X,- = h(X,-_1,0) + 5,», i 2 1, (2.1)
for some 9 = (9:11‘,r)T E R5, where 91 = (ao,a1,bo,b1)T E R4 and for any .1: E R,
h(:c,9) = (a0 + a1$)I(:c g r) + (b. + blaz)I(a: > r).

Here, the errors {£5} are independent and identically distributed random variables
with mean zero, finite nonzero variance and £1 is independent of X0.

We begin with the definition of the maximum likelihood estimators of the unknown
underlying parameter 9 in model (2.1). Assume 9 is an interior point of 9 defined in
(0.3). Note that O is an open subset of R5. There exists a compact subset K of R4
such that 9 is an interior point of K x R.

Denote Q = K x R, then (I is a compact set. Let 9 = (ao,al,flo,fil,s)T be any
point in (2. Note that {X.-} in model (2.1) forms a Markov chain. Let g0(Xo) be
the initial density of X0 under 19, f be the density function of 61, then the one step
transition densities, starting with X -_1, is f (X.- — h(X,-_1, 19)), i _>_ 1. If one observes

22

23

(X0, - - - ,Xn), then the likelihood function under 19 is H?f(X.~ — h(X.-_1,19))g,9(Xo).
Let 9,, = (a0,,a1,,to,,,131,,,1=,,)T be any measurable function of (X0,X1, ...,Xn) from

R"+1 to 0 such that 9,, maximizes the conditional likelihood function
Ln(19) := H’l‘f(X,~ — h(X,-_1,'9)), over (I.

Write 19 = (19?,3)T, 9 = (9f,r)T. Because of the behavior of the threshold
parameter r in the likelihood function, the maximizing algorithm will be taken in the

following fashion:
Step 1. For fixed 3 E R, denote Ln,(191) = Ln(191,s) = Ln(19). Let 191,,(3) E K

be any value satisfying the following equation:
191,,(3) = argmaxflieKLnAfll).

Step 2. Consider the profile conditional likelihood function 3 -—) Ln(191,,(s),s).
Note that Ln(191n(s), s) has only finite number of possible values. Let in be the any

value satisfying the following equation
1",, = argmam,eRLn(91n(s), s),

and substitute in into 191,,(3) to get

A

oln = 01710311)-

Then

9,, = (9;, 1",.)T is a maximum likelihood estimator of 9. (2.2)

To see (2.2), for any ‘9 = (9?,s)T E Q, by the definitions of 91,, and r", we have
Ln(élna":n) = Ln(191n(7:n)7 7211) Z Ln(‘9ln(3)as) 2 1171(6),

and hence,

L..(91,,,f,,) = sup L..(z9).
1960

This means that 9,, = (9;, f'n)T is a MLE of 9.

24

2.2 Assumptions

In this section, we are listing assumptions and some examples. The following assump-
tions on the density f of 61 will be used in the following chapters of part II.
(Cl) f is absolutely continuous and positive everywhere on R. With the ac. deriva-
tive f’, let (,0 = f’/f and I(f) = f<p2(:1:)f(a:) dz < 00.
(C2) tp is Lip(1).
(C3) cp is differentiable and the derivative cp’ is Lip(1).
(C4) E|61|3 < 00.
To derive the n-consistency, we need to assume the following:

(M) The threshold r in R is the discontinuity point of h, or equivalently,
bo-ao+7‘(bl —01)?i0, TER-

Remark 1. From the invariant equation g9(y) = ff(y—h(a:,9))g9(a:) dz, y E R,
the condition (C 1) implies that 99 is bounded away from 0 and 00 over compact sets.
It is the minimal requirement for obtaining asymptotically efficient estimators of the
coefficient parameters. See Koul and Schick (1995).

Remark 2. For the Markov chain {X,-} in model (2.1), denote its k-step transition
probability by P"(:r, B) where a: E R and B is a Borel set. By the discussions in Chan
and Tong (1985) and Chan (1989), the condition (C1) and 9 E ('9 imply that {X,-}
admits an unique invariant measure G0(~) such that 3 C, p < 1, V a: 6 R, V k 2
1, ||Pk(a:, ) - G9(-)||w S C(1+ |a:|)p", where H - II“, and I 1 denote the total variation

norm and the Euclidean norm, respectively.

Remark 3. The Lip( 1) and the differentiability of (,0 imply the boundedness of

Remark 4. When discussing the CLSE, Chan (1993) assumed the finiteness of
fourth moment of 61 while we need the finiteness of the third moment of 61 only.

Examples satisfying the conditions (Cl)-(C4).

Example 1. If f is the standard normal density function, then (C1)-(C4) hold.

Example 2. If f is the logistic density, i.e. f(:c) E F(.r)(1 — F(a:)), then (C1)-
(C4) hold.

25

Example 3. Let f(:r) = c(m)(1 + xz/m)‘(m+1)/2, —00 < .1: < 00, where m is
a positive integer and c(m) is a constant such that f is a density function. Then,

Ee¥=m/(m—2)<oo,m>2and

 

 

f’(rt) = c(m)( — 171-211) (1 gay—5%. (2.3)
Thus,
so = f’/f = -m;1 1:93:21”,
and
Iron 3 m; 1f(w)- (2 4)

Hence (C1) holds. By (2.4),

 

f’2(:c) m+1 2
f(:r) S (T) “3)

This implies that I ( f ) < 00. The Lip(l) of (p holds because of

+1 2 2
ItI=|—-"'—,,—<t-%;)/(t+%)”

Hence (C2) holds. Note that

Sm+1.
m

 

 

 

and

 

So (C3) holds. Furthermore, for m > 6, E|61|3 < 00 which implies (C4).
Throughout in the following proofs, we use the fact that E lellk < 00 implies
E IXolk < 00, for k = 2, 3, as proved by Chan, Petruccelli, Tong and Woolford (1985).

2.3 Strong consistency of the MLE

We are going to show the strong consistency of the MLE 9... To this effect, let ln be
the conditional log-likelihood ratio:

_ 1 f(X3 — h(Xi-la 19))
W) " Z 21” f(X.- - h(X.--.,o))

 

(2.5)

26

and denote

f(€,' + h(X,'_1,9) — h(X,'_.1, 19))

i-ufifl’ = n
¢(X ) l f(e.-)

 

, ISiSn. (2.6)

Note that
1
ln(’l9) = EZ‘i/J(Xg_1,€g,19). (2.7)

Write i9 = (19?,8)T E Q and h(:r,19) = h,(:c,191). Let

h,(:z:) = (3/3191)(hs($a 191))

: (I(z S s),:rI(a: S s),I(.r > s),xI(:r > s))T, s E R, a: E R.

Observe that for any x E R,

h(x,o) = 1917.43). (2.8)
Also,
|h,(:c)| 3 «1+ 2:2, (2.9)
and for any 3 E R, t E R,
|h,(:c) — ht(:r)| S (/2(1+ $2)I(s At < a: S s V t) (2.10)
S 2(1+a:2)I(|:1:—t| S Is—tl) (2.11)
Thus, by (2.8) and (2.9),
|h(a:,19)l S I191|v1+ :62. (2.12)

Recall that 9 E 6 means the stationarity and ergodicity of underlying process {X5}.

Throughout, we will work on the stationary and ergodic process {X,}.
Theorem 2.3.1 Suppose that the conditions (C1) and {C2} hold. Then,
9,, fl» 9, as n —) 00, {under 9). (2.13)

Before proving Theorem 2.3.1, we need the following lemma. Let U0 denote any open

neighborhood of 19.

27

Lemma 2.3.1 Under the assumptions of Theorem 2.3.1, for any 19 E (I and its open
neighborhood U19,

E sup |w(Xo, £1,19*)— w(Xo, 61,9)I —> O, as U19 shrinks to 19. (2.14)
‘9'eU
19

Proof. Define
U190?) = {19" = (191339?" E 9 = |19'i‘- I91|< n, d(s*,s) < n}, 17 > 0-
It suffices to show that

E sup |¢(Xo,el,19*) — w(Xo, 61,9)I —-+ 0, as 17 —> 0. (2.15)
19.6U1907)

Let 61(9) = X1 — h(Xo,19) and 6(Xo,19') = h(Xo,'9) — h(Xo,19'). For any 19 =
(191T,s)T, recall

h3(Xo, 191) = h(X0,’l91,S) = h(Xo,‘l9), (2.16)

and rewrite

5(X0,o*) = h,(Xo,191) — h,.(Xo,19'{).

For any a: E R, by (2.8) and (2.9),
Imus.) — h.(t,t:)) 3 I01— tax/1 —+ (2.17)
and by (2.10) and (2.11),

|h,($,19’{) — h,-(:r,19'{)| S |9;|‘/2(l + x2) [(3 /\ s“ < x S s V s") (2.18)

S |9;|(/2(1+ 2:2) I(Ix — 3| S ls" — 3|). (2.19)

Thus on U907) and for s E R,

|5(Xo,19")| S lhs(Xo,191)-ha-(Xo,191)|+lhs-(Xo,191)-hs~(Xofl9I)|

S [filfixllflxo — 8| S It" - 3|) + W: — 15"{llx/(1 +X3)
s [filtllluxo — s) s 130(77)- sl) + nl\/(1+X3)

A07, X0), (say): (220)

28

where 30(77) is such that d(so(17),s) = 77.
Note that

9461(9)) = [90(61 + h(X0a9) — 110(0) 19)) — 9461)] + <P(€1)- (2-21)

Condition (C2), EXg < 00 and (2.12) imply that there exists a constant L such that
for any '9 E 0,

Emma) 3 2E [902(61)+ twat) — h<Xo,t))P]
g 21(f) + 4L(|91|2 + |91|2)E(1 + X3) < oo. (2.22)

The absolute continuity of In f, which follows from (C1), and (2.20) imply that

A(niX0)

l¢(X07 61719’“) — t/)(X'0a.€13"'9)l S [- CP(€1(19)+ vll Ch)“ (223)

A(7l1X0) I
Thus, (2.22), (2.23) and Cauchy-Schwarz inequality imply that the

LHS of (2-15) S E{[|2<P(61(t9))|+LA(n,Xo)lA(n.Xo)}
< 2(Etp2(ct(t9)))‘/2(E42(n,Xo))‘/2+LEA2(P7,X0)- (224)

Moreover,
2 2 : 2
EA(mXo)=E(1+Xo)(\/§|t91|1(|Xo-slSlSo(t7)-SI)+PI) ..0, (2.25)

as n —> 0. Therefore, the finiteness of I(f) and (2.25) imply (2.15) for any 3 E R.
In the case 3 = 00, similar to (2.20),
(nmwns(fimmn>m+fl(uxa

g (fi|01|I(Xo > so(n)) + n)\/1+ X3
431(7), X0)

where d(so(17), oo) = 77. Again,
EAi(n.Xo) —> 0. as n _. 0-

Thus the proof goes through for s = 00. The proof is similar in the case 3 z —00,

except one replaces I (X0 > s‘) by I (X0 < 3"). Therefore Lemma 2.3.1 is proved. D

29

Proof of Theorem 2.3.1. Let a(19) = E¢(Xo, 61,19) for 19 E Q. The conditions
(Cl) and (C2), the mean value theorem, the independence of £1 and X0 and Cauchy-
Schwarz inequality imply that E |z/)(Xo, 61,19)| < 00. Thus a is a well defined finite
function from Q to ’R. Note that (1(0) = O and lnx < :c - 1, unless a: = 1. For
any given open neighborhood V of 0 in Q and any 19 6 VC 2 Q \ V, an conditional

argument yields that
61 + h(Xo, 0) — h(Xo, 19))
f (61)

_ n rm + how) — h(Xo,t9))
- E{E[. 2.) P21}

< E {/[m + h(xo,9) — hm, 19)) — 1m] dy} = 0

0(19) = Eln f(

 

(2.26)

 

By Lemma 2.3.1, 0 is continuous and hence the compactness of Vc implies that there
exists 190 6 Vc, such that

sup 0(19) = 0(190) < 0.
19ch

Let 60 = —a(19o)/3. For any 19 6 VC, by Lemma 2.3.1 again, there exists no > 0, such
that
E SUP w(Xo, 61,1?) S E¢(Xoa 61,19) + 5o S 01(190) + 50 = —250- (2°27)
19‘EU0('IO)
Again, the compactness of Vc implies that there exists a finite number M of U191,(170),
191' 6 VC, j = 1,2, ..., M such that U11" U191(170) = V". Then by the ergodic theorem
and (2.27), there exists a no such that for any n 2 no, 1 S j S M,

sup 1.09”) s 1: sup abacus-49*)

0'6Uoj(m) " 19‘ev,9j(no)
S E sup w(Xo,61,19") + 60 S —60, a. s.
19.6U19lno)
J
But,
sup (”(19) _>_ ln(0) = 0. (2.28)
196V

Therefore, for any neighborhood V of 0 in Q, 3 no, st. for all n _>_ no,

sup 1,,(19') S max sup 1,,(19’) S —60 < 0 S sup 1,,(19).
19'eQ\V 1515M (1191(710) 196V

30

This implies that
(line V, a. s.VVanan2no.

By the arbitrary of V, 0,, goes to 0 almost surely. E]

2.4 n-consistency of the threshold estimator

From now on we will invoke the condition (M). The discontinuity of h at 7‘ will give

a stronger result about the estimator 1",, of the threshold 7‘, i.e., the n—consistency of

A

Tn.
Theorem 2.4.1 Suppose conditions (CU-{C3} and (M) hold, then
Infin _ 7')l = OP“)-

The proof of Theorem 2.4.1 is technical and lengthy but interesting. We will begin

with some notation. Let J : 722 —+ ’R. and
p(a:) = EJ(x,q), 191(3) = E|J(:r,el)|, p2(:c) = EJ2(.’E,£1),.’L' E ’R, (2.29)
For u _>_ 0, define
G(u) = EI(r < X0 S r + u), Gn(u) = i210 < X;_1 S r + u),

and

Rn(u)=-71;(XZJ ,-_1,e,)I(r<X;-1Sr+u).

'2 EZP(X,_1)I(T < Xg_1 S 1‘ + U).

Also, let J°(X,-_1,e.-) = J(X,-_1,e.-) —p(X,-..1). For U2 2 ul 2 0, let
Rn(ulau2)=— MEL] i- 1,6: |I(T+U1<X _1<r+u2),

RWI, “2) = ERn(U1, U2)-

31

Lemma 2.4.1 Suppose that (Cl) holds, then there exists constants 0 < m S M < 00
and 0 < C < 00 independent ofn. For any 0 < 5 <1 and V u, ul, M 6 [0,6], V n,

mu S C(u) S Mu, (2.30)
Var(1(r < X0 S r + u)) S CG(u), (2.31)
Var(nGn(u)) S nCG(u). (2.32)

Proof. The facts (2.30) and (2.31) follow from the assumption (C1) and Remark
1. To prove (2.32), without loss of generality, assume r = 0. Note that

Var(nGn(u)) = nVar(I(0 < X0 S u))+z Cov(I(0 < Xk_1 S u), [(0 < Xj_1 S u)).
“U

But, Remark 2 and a conditioning argument yield that for any k 2 2 and u S 6,

|Cov(I(0 < X0 S u), [(0 < Xk-1 S u))|
= |EI(0 < X0 S u)[I(0 < Xk_1 S u) — G(u)]|
g EI(O < X0 g u)|[P"(Xo, (0,u]) — G(u)]| g Cpk'lG(u).

Thus (2.32) follows from the stationarity of {X5} and the fact 2,95]- pk‘j = 0(n). C]

Lemma 2.4.2 Suppose that the functions p1 and p2 are continuous over R. Then,
there exists constant 0 < C < 00 independent ofn such that V 0 < 6 < 1 and

V u3u1?u2 E [016]: ”2 Z ”I: V n)

120.1,...) g C(G(u2) — G(u1)). (2.33)

Var(|Jc(Xo, €1)|I(r + u. < X0 3 r + u2)) s C(G(u2) — G(u1)). (2.34)
var(f2,.(u1,u2)) s C(G(u2) — G(u1))/n. (2.35)

Var(R,,(u) — r..(u)) s CG(u)/n. (2.36)

Proof. Without loss of generality, assume r = 0 again. The continuity of p1

implies that

~

R(u1,u2) E {I(u1 < X0 S U2)E[|JC(X0361)| |X0]}

|/\

2 SUP P1($)E [(111 < Xo _<_ U2)
x€[0,1]

g RHS of (2.33).

32

The continuity of p2 and the Cauchy-Schwarz inequality imply that the

LHS (2.34) g E|J°(Xo,e1)|21(u1 < XO 3 ug)

E{I(u. < X0 3 U2)E[|JC(Xo,€1)|2|Xol}

l/\

4 sup P2($)E1(ul< x. s u.)
2:6[0,1]

g HHS of (2.34). (2.37)

Next, we shall verify (2.35). The argument is similar to the proof of ( 2.32). Expanding
the left hand side of (2.35),

Var(n1~2,,(u1,u2)) = nVar(IJC(Xo,€1)lI(ul < X0 S “2»

+2 COU(|JC(Xk_1,€k)II(U1 < Xk_1 S U2), IJC(XJ'_1,€J')|I(U1 < X'_1 S 11.2)).
k¢j

Note that |p(a:)| S p1(:r). By the continuity of p1, the property of Markov chain and
Remark 2, for any I: 2 2,

|E[|J°(Xk_1,ek)|1(u1 < xk.1 g u2))X1,Xo] — E|J°(Xo,e1)|1(u1 < X0 3 u2)|
|E[ / IJC(X1._1,y)| dF(y)I(u. < x.-. s uglxl] -[’/ Wm)! dF<y>dGa<x>|

s l. [/ |J°(x,y)|dF(y)] (|P*-2(X1.dw) - Go<dm)l)
s 2.:‘I‘opumunlpk-uxu -) - 040mm
S Cpk_2(1+|X1l)S CPk_2(1+ lh(0,Xo)l+ l€1l)- (2-38)

The continuity of p2 and pflx) S p2(:r), Ee'f < 00, the Cauchy-Schwarz inequality
and an argument like (2.37) imply that

E [IJC(X0,€1)€1|I(’U1 < X0 S u2)]
= E{E [IJC(X0,£1)£1|'X0] [(241 < xo 3 242)}
s 2\/ sup p2($)E(€1)2(G(U2)-G(ul))- (2.39)

x€[0,1]

 

Then the definition of the autoregressive function h, the Markov property of {X},
(2.38), (2.39) and Remark 2 yield that

lCov(|Jc(Xo,61)ll(U1 < x. s uz), |J°(Xk-u€k)|1(ul < X.-. S U2>)|

33

= |E(|JC(X0,61)|I(u1 < X0 3 u.) x

(E[|J°(x,._1, ek)|I(u1 < X... g u2)|X1,Xo] —E|J°(Xo,e1)|1(u1 < X0 3 422))“
s Cpk‘2E|J°(Xo,el)ll(u1 < X0 3 u4)(1+ Ih(0.Xo)| + lell)
s Cp"‘”(G(u2) — (3041)).

Therefore (2.35) follows from the stationarity of {X.-} and the fact EM]- pk‘j = 0(n).
The proof of (2.36) follows from the property of the square integrable martingale
Rn(u) — rn(u), for fixed u > 0. That is,

Var(n(Rn(u) — rn(u))) = nVar(J°(Xo, el)I(u1 < X0 S u2)).
This completes the proof of Lemma 2.4.2.

Proposition 2.4.1 Suppose that (Cl) holds and the functions p1 and p2 are contin-
uous. Then, for each 6 > 0, n > 0, there is a constant B < 00, V 0 < 5 < 1 and
V n 2 [13/5] + 1,

 

P ( sup IGn(u)/G(u) — 1| < 77) > 1— e, (2.40)
B/n<uS6
Rut“) - MU)
P (8/1129 G(u) < 17) > 1 — e, (2.41)

 

 

Note. The condition (C1) is for (2.40), the continuity of p; and p2 is for (2.41).

Proof of Proposition 2.4.1. For any B > 0 and 0 < 6 < 1, choose a partition
of the interval (B / n, 6] as follows: Fix a b > 1 and let Mo be the greatest integer less
than or equal to ln(n6 / B) / 1n b. Note that

Mo . .
(B/n,6] = U 1,, 1,: (b'B/n,b‘+lB/n], i: 0,...,M0 — 1, [Mo = (bM°B/n,6].

i=0
Then (2.30) and (2.32) of Lemma 2.4.1 imply that V 171 > 0,
P(sup IGn(biB/n)/G(biB/n) — 1| 2 171)
S 2 VaT(Gn(b‘B/n))/(niG2(biB/n))

3 OZ 1/(mnfBb‘) = C/(mnfBu — 44)). (2.42)

34

For 0 < a: S y S b3: S 6 with IGn(x)/G(x) — 1| < 771 and IGn(bx)/G(b:c) — 1| < 171,
we have,

(1 - 771)G(5'=)/G(b93) -1 S Gn($)/G(b$) - 1 S Gn(y)/G(y) — 1
< Guam/0(4) — 1 g G(ba:)/G(:c)(1 + m) — 1.(2.43)

The strictly increasing property of G and Dini theorem imply that V 17 > 0, one can
choose 771 > 0 and b > 1 sufficiently small such that
G(b‘B/n) G(b‘+lB/n)

 

 

 

 

 

034320 G(bi+lB/n)(l _ ’71)‘ llv I G(b‘B/n) (1+ 771) — 1 < 77. (2.44)
Now let
_ Gn(b‘B/n) Gn(b‘+lB/n)
An, — {09323640 G(b‘B/n) — 1. < "laosflslgg-1 C(b‘HB/n) —1 < Th .

 

 

 

Then on An, (2.43) and (2.44) imply that

sup lGn(u)/G(u) - 1| = @3350 321i)- lGn(u)/G(U) - 1|

 

 

 

B/n<uS6
G b‘B n G bi'HB n
3 63-1234. { wield/ii)“ ‘ ’7‘) ‘ 1| VI ciao/(2))“ + m) _ 1'}

< 77.
And then by (2.42), choosing B sufficiently large and no = [B / 6] + l, for any n 2 no,

P< sup lGn(u)/G(u)-1|Zn)SP(Af.)<6. (2.45)

B/n<u_<_6
Hence (2.40) holds. D
To prove (2.41), for b‘B/n < u S bi+lB/n, i 2 0,

WW) — Tn(u)l S an(u) "' Tn(u) — (12,,(b‘B/n) - 7‘n(b'iB/n)|
+|R,,(b‘B/n) - Tn(biB/n)|
_<_ BARB/72,11) + an(biB/n) " rn(biB/n)l'

Thus, by the increasing property of B." and G,
sup Rn(u) - rn(u)
%§<US bid: 3 0(a)

Rn(b‘B/n,b‘+‘B/n) + |R,,(b‘B/n) — rn(b‘B/n)|
- G(b‘B/n) G(b‘B/n) ‘

 

 

 

 

 

(2.46)

35

Also, note that (2.30) of Lemma 2.4.1 and (2.36) of Lemma 2.4.2 imply that for any
771 > 0)
Rn(biB/n) — rn(b‘B/n)

P (“3" C(biB/n) Z ’7‘)
S 2: Var(R,, Rn(b’B/n) — rn(b'B/n))/(17 302(b‘B/n))

 

 

 

S 021“ mniBb‘) = C/(mniB(1 - b"‘)), (2.47)
and (2.30) and (2.35) yield that

P (sup |B.,,(b"B/n,bi+1B/n)/G(b£B/n) — B(biB/n,bi+lB/n)/G(biB/n)| 2 171)
S 2 Var(I~Z,, Rn(b‘B/n, b'+1B/n))/(nsz(biB/n))

< C(b —1)/ B): 1/b’= Cb/(nfsz). (2.48)
By (2.30) and (2.33),

sup |B(b£B/n,bi+lB/n)/G(biB/n)l S C(biB(b —1)/n)/(mb‘B/n) S C(b-1)/m.
' (2.49)
Thus for any 6 > 0, n > 0, one can choose 171 > 0 and b > 1 sufficiently small such
that 771 + C (b — 1) / m < 17 and then choose sufficiently large B such that

20 v 2b
mn?(1-b“)€ m’nié'

Then, by choosing no = [8/6] + l, (2.46)-(2.49) imply that for any n 2 no,

 

B>

 

 

 

Rn(u) — MU) )
P su > < e. 2.50
(La/«I355 C(11) _ ’7 ( )
This completes the proof of Proposition 2.4.1. C]

Now let
fléi + a + flXi—l)

f(€:') ’

where a = bo—ao, [3 = b1 —al. The functions p, p1 and 122 are defined correspondingly.

 

J(X,'_1, 6,) E 1,0(X5_1, £5) E- in

For u 2 0, define
éw {—1,CI(T<X¢'_1<T+U).

dn(U) 2' EZP(X3_1)I(T < X{-1 S T + U).

36

Corollary 2.4.1 Suppose that (CU-(02) hold. Then, for each 6 > 0 and 17 > 0,
there is a constant B < 00 such that V 0 < 6 <1 and V n 2 [B/6]+1,

1),,(u) — d..(u)
0(a)

 

P( sup

B/n<uS6

 

 

< 17) > 1— e, (2.51)

Proof. The continuity of p1, p2 can be derived from the conditions (C1) and (C2)
(See Appendix). Thus (2.51) follows (2.41) immediately. [I]

Before proving Theorem 2.4.1, we need some more notation. Write

f(Xi — hs(Xi-la t))
f(€4)

 

1,6(X,_1,e,-,t,s) = 1n , t E R4, 3 6 R,

where h, is defined in (2.16). Let

{(Xi—laéiatss) = 2Z'(‘Xi—la€iatas) - 123(Xi-196i9t7r)3 t E R4) 8 E R, 1323 Tl.

Then,for1SiSn,tER4, 36R,

64-1.43») s gas-434,3»
= -cp(X.- — h.(X.-_1, t))h,(X,-_1) - w(Xe - h..(X,-_1, t))h,(X.-_1)
= — [w(x. — h.(X.-_1. t» — w(x. — h.(X.--1,t))] (1.06-1)
—<p(X,- — h,(X.-_1, 4)) [h,(X.-_1) — hr(X,--1)] .

Now we are ready to prove Theorem 2.4.1.
Proof of Theorem 2.4.1. Since 6,, is strongly consistent by Theorem 2.3.1,
without loss of generality, the parameter space can be restricted to a neighborhood

of 0, say,
fl(6)={l9€9:|191-—01| <6,|s—r| <6},

for some 0 < 6 < 1 to be determined later. Then, it suffices to show that 1,,(191, s) —
1,,(191, r) is negative for n|s — r| large enough. More specifically, we shall show that
for every 6 > 0, there is a B > 0 and 'y > 0, 1 > 6 > 0 and no such that for any

”27107

sup

P ( [ln(191,3)-ln(191,r)]
B/n<|s—r|gs,i9en(6) GUS - Tl)

< —7) > 1 — 25. (2.52)

37

For any 19 = (9?,3)T, denote ln,(191) = ln(191,s) = 1,,(19). Now, decompose ln(191,s)-—

1,,(191, 1') into two terms as follows:

[11(1913 3) — [no.9], 7') = [Ins(19l) — l'nr('191) — (Ins(01) _ lnr(01))] 'l" [lns(91) _ lnr(01)]
E 1,1,(19) + [g(s), (say).

We shall prove that there exists a 6 small enough such that
1.1.09)

 

 

SUP -— 2 013(1), (253)
B/"<|3-TIS6.1960(6) GUS - r|)
and
12(3) )
P su _n__<_2 >1“' 2.54
(B/fl<I8I-)r|S6 G(|s _. ll) 7 ( )

We will prove the case s > r only and write 5 = r + u for some a > 0. For the
case 3 < r, the proof will be exactly the same. To prove (2.53), by using the absolute

continuity of 2b,

Til-Z [C(X,‘_1,C{,’I91,S) — C(Xi—laéi,91,5)l
= 1);]; tux-4.6.40. + M. - 91),.)(4. — 0.) dv. (2.55)

n

1.1.(19)

By the Lip(l) of cp, (2.17), (2.18), (2.9), (2.10) and (2.11) imply that there exists a
constant L, for 1 S i S n, t E R4, 3 E R,

|5(X'-1,€i,ta3)|

L h,(X.-_1,t) — h,(X.-_1,t)((/l:—X_E_—l

+[|<p(e;)| + L|h,(X.-_.1,01) — h,(X,~_1,t)|] 1+ x3_, [(3 A r < x.-. s s v r)
(1214717231 [lcp(€e)| + 1491- tum)

x‘/1+X,-2_ll(s/\r<X,-_1 SsVr). (2.56)
Thus, for any 19 6 (2(6), by (2.55) and (2.56), there exists 0 < C < 00, such that

13.09) Gabi) + (312(10- W(U) ]
G(u) C(u) 0(a) ’

with J (9:,y) = |<,o(y)| in the definitions of Ru and 7‘”. Thus, (2.53) follows from the

|/\

 

|/\

 

 

 

sca[

 

 

 

 

 

Proposition 2.4.1 by choosing 6 > 0 sufficiently small.

38

To prove (2.54), recall that a = bo — ao,fl = b1 — a1 and let J = 1/2 in the definition
of p in (2.29). Then 1,2,(3) can be decomposed as follows:

[2(3):l2(r+u)= %Z¢(X'-1’€ )I(r<X,-_1 Sr+u)
_lz[ln( f(6.-+a(+)flX.‘— 1)

 

—p(a + ,BXi-l)] 1(7‘ < Xi-l S 7' + u)

4.; Z[ p(a + flX,-1)-— p(a + fir)]I(r < X._1 S r + u) + p(a + 3706M“)
E [Dn(u) - dn(U)] + EM) + MU)

 

 

 

 

 

 

 

 

 

Then
1,2,(r+u) Dn(u)—dn(u) En('u) Gn (u)
433.2. 0(a) 3 432.24 cm 6(a) + G<u )‘ll'pa ”ml
+p(a + fir). (2.57)

Note that p(a + flr) = f1n[f(:c + a + flr)/f(:c)]dF(:r) < 0 by the condition (M). Fix
an 17 > 0 such that 7 = [—p(a + fir) — 17(2 + |p(a + ,Gr)|)]/2 > 0. Note that

 

 

 

 

sup En(u)
B/n<u<6 1001')
I(r < X;-1 S r+u)
S 8/3336- 12M4 + flX. 1)- P(a + Br)| C(u)
S sup |p(a + fir + fix) — p(a + fir)| M —1 +1] (2.58)
4.30.5] C(1!)

   

 

By the Proposition 2.4.1 and the continuity of p (see Appendix), we can see that

 

goes to zero in probability for sufficiently small 6 > 0. Let

1 Dn(u)—d,,(u) En(u) Cu (14)) _
0(a) 0(a) G<u ll'p(“+fl’")'l

< 17(2 + |p(a + firm}.
< Tl} 9

En u
SUPB/n<ug6 IFH

 

+

 

+

 

A={sup

B/n<u56

 

 

 

 

 

 

 

_ 3.. (u)- duo)

B — {EB/«1:00 G(")
nU( )

{B/n<u<6E 0(a) <77}

D = { sup _(_Gu)_1
B/n<u_<_6l G——2(u)

{sup (G(+)U) < -—27}.

 

C:

 

sup

 

 

 

 

B/n<u<6

39

Observe that [3 0C 0 D C A and (2.57) implies A C 8. Hence, by (2.40), (2.51) and
choosing 6 > 0 sufficiently small,

P(£) 2 P(.A) 2 P(B n c n o) > 1 — e. (2.59)

This completes the proof of (2.54).
Thus, V e > 0, there exists '7 > 0 and 6 > 0 sufficiently small such that with (2.53)
and (2.54),

 

P( sup [171(191, 3) —ln(191,r)] < _7)
g<|s—r|gs,i9en(6) GUS — r|)

 

 

ll('5’) 12(3)
2 P sup —"——— < 7, sup --—'-‘—— < —27
(%<|a-r|56,t9en(6) G(|s ‘ Tl) gas—r56 GUS - 7'l)
13.09)

_>_ — P su -——" Z —2
7) (gds-IZISIS Gas 7") 7)

 

 

> — _—
_ 1 P( sup G([s—r|)

€<|a—r|gs,1960(6)
> 1 — 26.

This ends the proof of (2.52) and hence of Theorem 2.4.1. D

Chapter 3
Limiting distribution of @177,

We now consider the limiting distribution of 91,1. Recall that

f(€,‘ + h(X'._1, 0) — h(X,'_1, 19))
f (G)

 

2,1)(X;_1, 6,319) = 1n , 19 E Q,

and the log likelihood ratio function is
1
[11(19): 5 Z¢(Xi—la€iat9)a ‘9 E Q

In the definition of the MLE 3", the first four components of the parameter point
in Q is treated separately from the last component and we have proved that fin is
n—consistent. Thus, we need some results of 191n(3) uniformly in s in the interval
[r — B / n, r + B / n] for some B, 0 < B < 00 which is given in the following Theorem
3.1.1.

3.1 Uniform consistency
Theorem 3.1.1 Suppose that (C1) and (02) hold. For any 0 < B < oo,

sup |191n(s) — 01| = 0p(1).
ls-rlSB/n

First of all, we need an analogue of the Lemma 2.3.1. Recall that 19 = (19f,s)T
and In,(191) = ("(191,s). Now write

¢‘(Xi‘1’ 65’ 191) = “Xi—1,64, 1913 8).

40

41

Let n > 0, define
(119,07) = {191' € K= |19i-191I < 77}-

Lemma 3.1.1 Under the conditions (CI) and {C2}, for any 191 E. K and its neigh-
borhood U191(17) in K,

E sup |2,Z2,(X0, 61,19?) — 2,1),(Xo, £1,191)| —> 0, as n —> 0. (3.1)
86R.19;€U191(n)

Proof. Let 53(Xo,19'{) = h,(Xo,I9'f) — h,(Xo,191). Observe that by (2.17) on

(119,07).
l5.(Xo,I9I)| S I19? - 191|\/1+ X3 S n\/1+ Xci- (3-2)

By the absolute continuity of In f, which follows from (C1),

in/l-{JK'2
l¢8(X0, 61’ 19”" ¢8(X01 €1,191“ S 77 1+); l90(€1(19)+ v)l dv' (3'3)
‘ o

The condition (C2), EXg < 00 and (2.12) imply that

E sup <p2(cl(191, 3)) < 00. (3.4)
.9612

Therefore, (3.3)-(3.4) and the finiteness of the second moment of X0 yield that the
LHS of (3.1) S 77E[Sllp2|<p(61(19))|\/1+ X3 + LE(1+ X3172]
.967?

1/2
3 217[Esu712>cp2(61(19))) [E(1+ X3)]‘/2 + LE(1+ X3772
36 '

-> 0. awn—>0,

thereby completing the proof of Lemma 3.1.1. C]

Proof of Theorem 3.1.1. The following argument is similar to the one used
in the proof of Theorem 2.3.1, except here we deal with the component 191 of 19 for
all s E ”R. Let a,(191) = a(191,s). By the definition of the function a, for any open
neighborhood V1 of 01 in K and any 191 6 V1", 3 E R, 0(19) < 0.

Given V1 C K, by the continuity of a and compactness of (K \ V1) x 72, there
exists a 1901 E (K \ V1) X R, such that

sup 0(0) = 0(1901) < 0.
196(K\v,)xk

42

Let 601 = —a(1901)/3 > 0. By using Lemma 3.1.1, there exists an 1701 > 0,

E5119 SUP ¢3(Xo, 61119;) S E¢3(Xo,€1,191)+501 S 0(901)+501 = —2501- (3-5)

3672 19iEUfl (7701)
1
By the compactness of K \ V1 again, there exists a finite number M1 of U19 (7701),
1:

191,- 6 K\V1, j = 1,2,...,M1 such that U?" (1191:0701): K\ V1. Then by the ergodic
theorem and (3.5), there exists an n1 such that for any n 2 n1, 1 _<_ j 3 M1,

1
sup sup [w(fl'f) S — Z sup sup ¢,(X,-_1 , 6,, 19;)

SE72 19:6(119110701) n 3612 flieuglj (’701)
S E SUI} 3UP ¢3(Xoa €17 19;) + 501
36R 19:6(119 _(001)
11
S _6013 0.8.,

which implies that

sup sup ln,(19; )_ < sup max sup ln8(19'{) S —601, a.s. (3.6)
.612 fliemv. oeRl<3<Mlt9I11jeU0 (1701)

But,

sup sup ln,(‘l91)__ > sup In,(01) (3.7)
:67; 0161/1 86R

Taylor’s expansion of In f at 6.- yields that 3 7, M < 1, such that,
In. (91) — $290M + 7( h,( X14491) — h,(X-_1,01)))[h.(X.-_1,91) — MOB-1191)]-
Then the Lip(l) condition of cp and (2.19) imply that
1m(91) S 3;: [l‘P(Ci)l + L|h,(X.-_1,01) — h.(X.-_1,01)l]
x [Ian/1+ X?_11(IX.-_1— r1 3 Is — ml
i: [.,.(.,)l + Hallm] 19.1mm.-. - r1 3 Is — r|).(3.8)

Therefore, for any B, 0 < B < 00, (3.8) yields that

SUP llns(01)l

 

Is—rISB/n
l

S E Z [W(Cill + L|91|v1+ X124 |01|VI+X,?_11(|X.-_1 — 1‘l S B/")
1

s ; Shag-11+ Ll01lx/1+(lrl+ B/nv] I91|

 

x\/1+(|r|+ B/n)21(|X,_1— r| _<_ B/n).

43

Take expectation both sides to obtain

E( sup lln,(01)|)=0(n-l).

Ia—rISB/n

Thus, for any 6 > 0, there exists n2, such that as n 2 122, V Is — r| S 8/12,

P(ls—rilréfB/n ("3(01) > —(So) > 1 — C.

By (3.6), (3.7) and the above, there exists a no = n1 V n; such that V n 2 no,

P(sup sup In,(t9'{) < inf sup In,(191)) > 1 — e. (3.9)
3672 19;€K\V1 IS-rISB/n 6161/1
Let
A, = sup sup In,(t9') < inf sup In,(191) .
{36R 19ieK\V1 l ""58” 016V: }
Then, on Ac,

191,,(8) 6 V1, V Is —r| S B/n, V n 2 no.

Thus, by the arbitrary of V1,

sup |191n(s) — 01| :2 0p(1).

l-9--"|SB/n

This ends the proof of Theorem 3.1.1. D

3.2 Asymptotic normality of 91,,

Before stating the next theorem, recall that In,(191) = 1,,(191, s), and for a: E R,

h,(:c) = 6—109:(h’($’191)):(1($ S s),:rI(.7: S s), [(33 > s),a:I(a: > 3))T.

Let
uish’l) = “@(Xz‘ _ hs(Xi—la01))izs(Xi—l)a 1 S 2 S ”-

Denote A,(:r) = h,(:z:)(h,(a:))T.

44

Lemma 3.2.1 Suppose that O < I(f) < 00 and EXg < 00, then

(if 2:21.401) => MO. I‘). (3.10)

where

F = E (P2(€1)Ar(X0)-
Proof. Note that with .75} = o{Xj,j S 2'},
E0 [uer(91)l.7".'_1] = E9 {-90(€i)ilr(Xi-1)|F—1] = hr(Xi-1)E9(—90(€i)) = 0, a- 3-

Therefore, for any vector v E ”R“, by the finite Fisher information of f, vT Z u,,.(01)

is a zero mean square integrable martingale. By ergodic theorem,

£2 vTiE luir(91)("ir(01))Tl'7:‘-1]}v
= Jun-,1; Z AJXi—I)”

——1 vTI‘v, a.s.

Thus, the martingale central limiting theorem of Hall and Hedye (1980) shows that

the sum 71'”2

ZvTugrwl) converges weakly to the normal distribution with mean
zero and variance vTFv for all u E R4. Thus, 12'”2 Z: u;,.(01) converges weakly to the

multivariate normal distribution with mean vector zero and covariance matrix F. 0

Theorem 3.2.1 Suppose that conditions {CU-(C4) hold. Then for any B, 0 < B <
m}

|3_§;1<pB/n «5091743) — 01) => N(0, F—l). (3.11)

As a consequence, for any B, 0 < B < 00,

811p lx/77(191n(8) - 01)| = 0P(1)- (3-12)
Is-rIsB/n

Proof. Note that for any 3 E 77,,

a
'56:(1ns(191)) = Z uia(‘91)-

45

Consider the Taylor’s expansion of (3/8191)(lm(191)) at 01:

0
8191

where 9;,” = 01 + 71(191 — 91), '71 is a function of (X0, ..,X,,,91,s), I71| < 1 and
8
J (t) = Z 5211,,(t) = Zfix. — h,(X,-_1,t))A,(X.-_1), t e 124.
Then, the definition of 191,,(3) and (3.13) yield that
1 Jn8(0‘ns)
7.7; ZU,‘3(01)+ —71——- fi(61n(3) — 91) = 0. (3.14)

For any matrix M = (mij), define ”M” = Z: Imgjl. Then for any finite number of

——(z.., 191) 2211;,(01) + J...(o 1',,,)(191 — 01), (3.13)

matrices {M5} and finite number of real numbers a;, we have

”Em-M." _<_ 2 laelllMill- (3.15)
By the definition of A3,
||A,(a:)|| = (1+ |x|)2, for all s E ’R and a: 6 'R. (3.16)

First, we prove the following: For any B, 0 < B < oo,
J_,,____,(0"

n

+,(_

 

 

 

— 010(1) (3.17)

sup
Is—rISB/n

For any s, such that Is — r| S B/n,

«In 40:...) = Z‘MX: —h(X.-_1,0;,,,))A,(X.-_1)
= Zl‘MXi-hs X.-_1, 9in.))- ‘P’X( X"h ,(X.-_1,01))]A,(X.~_1)
+ )2 [so’(X.- — MUG-1191)) — so'(e.-)]A.(X.--1)
+ Z $0'(€e)As(Xi-1)
a J1,,(0;,,,)+ J2... + J3... (say). (3.18)
For the first term in (3.18), the Lip(l) of cp’, (3.15), (3.16) , (2.16) and (2.17) imply

that

11.11.3013)” 3 Zlcp’Xe—h.Xe-1,9'{...))-<p’(Xa-hs(Xi-1,91))IllAs(Xe—1)||

g LZlh,,(X.-_1,9'{n,)— h.(X.--1.91)| (1 +|X-—1|)2
s LZI191n(s)- 011(1 + X131” (1+ IXi-IIV
S L 8111) I191n(3) - 01|:(1+ |X‘-1|)3. (3.19)

ls-i'ISIB/n

46

Thus (3.19), Theorem 3.1.1, the ergodicity of {X,-} and the finiteness of the third
moment of X0 yield that

Jlns(0ina)

n

sup
Is—rISB/n

Recall that 6,-(191,s) = X.- - h,(X.-_1,191). For has, again the Lip(l) of cp’, (3.15),
(3.16) and (2.19) imply that

l
S L 811p |91n(8) — 91I-Z(1+|Xi—1|)3 = 012(1). (3-20)
I8-'ISB/n n

 

 

 

 

sup H12...” 3 sup Elsa (axons ))—<p’(e.-)||A.(X1-1)II
|s-r|SB/n |s—r|<B/n

S Ll91|Z( 1+X'2— 1) )1/21(|X,-_1 — 7‘l < B/n)(1+|X.-_1|)2
S L|01|Z(1+ |r|+ B/n)3I(|X,-_1 — rI S B/n).

 

The above inequality, the stationarity of {X.-} and Remark 2 imply that
J2ns

n

 

E sup = 0(n-1). (3.21)

Is-rISB/n

For .13.”, observe that for all a: E R,

 

 

 

 

||A,(:1:) - A,(:I:)|| = 2(1+ |:1:|)2I(s /\ r < a: S s V r) S 2(1 + |:1:|)2 I(Ia: — r| S |s — r|).
This implies that
E SUP ||A3(Xo) - Ar(Xo)||

Ia-rISB/n
3 2130+ IXo|)21(|Xo - rl s B/n)

 

 

 

S 2(1+|rfl+ B/n)2EI(|Xo - rl S B/n) = 0(n’1). (3.22)
Also, by (3.15),
sup J37" + F"
|s—rISB/n n
= I.-§i1<1i3/nnl’(zw (6;)[A,( (X14) - X;_1 )H + “-290'(€;)A X._1) )+ F"

 

 

 

E terml + termg, (say).

Thus (3.22), the stationarity of {X.-} and the independence between 6.- and X,-_1 imply
that

E(term1) S n-1E(n|<p'(q)| sup
Ia—rISB/n

= Ewemom-l) = 002“),

     

.(Xo) — Ar(Xo)|l)

47

and by the ergodic theorem, termg —-> 0, as Thus,

 

sup
Is-rISB/n

Therefore, (3.20)-(3.23) and Markov inequality yield (3.17).

J as I
:2 + 1“" = 012(1). (3.23)

 

 

Next, we are going to show that for any B, 0 < B < oo,

= 0(n'1/2). (3.24)

 

E [ sup
ls-v'ISB/n

 

 

% Dam.) - u..(01)]
To prove (3.24),

2341491) - Uta-(91)]
= Zl—<P(Xi - h,(X.-_1, 91))i1.(Xi-1)+ ‘P(€i)9r(Xi—1)l
= zl—Méi + h,(X.-_1, 91) — h,(X.-_1, 91)) + <p(£g)]h,(X,-_1)
- Z 90(6i)[h.(X.-—1) — 5.(X.-_1)]
E [1,” — 12”,, (say).
For any 3, Is — 1'] _<_ 8/12, the triangle inequality, the Lip(l) of (,0, (2.19) and (1.20)
imply that
|I1...| s Lamas--391) - h.(X.-1,91)l Ih.(X.--1)I
S L|91|Z(1+ X3—1)1(|X:'-1 - 1‘| S IS — 7'!)
S L|91|Z(1+(|7‘l+ B/n)2)I(|X,_1 - Tl S 8/12).

Thus, by the stationarity of {X,-} and Remark 2,

Ilns

,/5

Similarly, the independent of c.- and Xi-“ the stationarity of {Xi}, the integrability

 

E( sup
ls-TISB/n

) = 0(n-1/2). (3.25)

 

 

of 4p and (2.19) imply that

12113

J17
Therefore (3.25) and (3.26) yield (3.24).
Note that

 

) = 0(n'1/2). (3.26)

 

 

E( sup
la-rlSBln

EI(X0 S r)EXgI(Xo S r) > (EXoI(Xo S r))2,

48

and

EI(X0 > 7‘)EX31(X0 > 7‘) > (EX01(X0 > T'))2.

Thus EA,(X0) is positive definite matrix and so is I‘.
Denote PM = —Jn,(01m)/n By (3.17) and the positive definiteness of P, I‘m 1S

positive definite eventually for every 3, Is — r| S B / n, and

 

 

 

 

 

 

sup |det(l‘ns) — det(I‘)| = 0p(1). (3.27)
Is—rISB/n
By the Cramer’s rule,
— dj(rns)
I‘m 1 = a .
( ) det(Fm)
Then the continuity property of the adj(I‘n,) in the components of I‘M and (3.15)
yield that
-1 a_d___j(r..)_adj(1‘)
Fifi/J”; P H“ [IT—ea (r...) det(I‘) H
adj(I‘n,) - adj(I‘) . det(Fn,)— det(F)
< d I‘
- l dear...) )II + "a ’( )" det(I‘n,)det(I‘ )="I
That is,
sup "F; l — F 1H = 0p(1). (3.28)
Is—rISB/n

It follows from (3.14), for any 3, Is — r| S B/n,

«H(Ms) - o.) + Flu-”2 2:11.49.)
= —;;—(r F'F)(n-1/22ug,(01) -n-1/2Zu.~,(01))
—r l(11-1/2):tt..,(111)—n-1/2§:u.-,.(31))
—;r ,1— r- F)n'1/2Zu,-r(01). (3.29)

Thus, by (3.24), (3.28) and Lemma 3.2.1,

SUP IVE/(9111(3) - 91) + P—ln-lfl Zair(91)| = 0P(1)-

Ia-rISB/n

Again by Lemma 3.2.1, Theorem 3.2.1 is proved. D

As a corollary of Theorem 3.1.1 and 3.2.1, we have the following uniform conver-

gence rate of 191,,(-).

49

Theorem 3.2.2 Suppose that (CU-(C4) hold, then for any B, 0 < B < oo,

SUP “9171(3) _ I91n(")l : 0P(n_1/2)-
Is—rISB/n

Proof. Consider the Taylor’s expansions of (d/d191)(ln,(191) and (d/dt91)(lm.(191)

at 01 and evaluate them at 191.1(3) and 191,,(1'), respectively. We have

0 = 295.491) + Jns(0ina)(‘9ln(3) '— 91)
= Zita-(01) + Jnr(9;nr)(19ln(r) — 01).

Hence,

Y}? $121.30.) — was] = M16135 — 191.001
+ [rm — r] \/1—1(191n(s) — 191)- [1“... — r]./?1(19.,.(r) — 91).

Therefore, Theorem 3.2.1, (3.17) and (3.24) imply that

sup
ls-rISB/n

 

Mame) — «91m» = can).

This completes the proof. [I]

Chapter 4

Some asymptotic results on

log-likelihood process

In this chapter, we discuss some asymptotic results for a sequence of normalized profile
log-likelihood processes. It is expected that these results will be useful in obtaining
the limiting distribution of the standardized maximum likelihood estimator of the

threshold parameter.

Recall that

Xi—h ,(X;_1,191)) T s T
[(191,s)-1—nf21 f(61) ,(19,, ) 652.

For 2 E R, a sequence of normalized profile log-likelihood processes is

 

ln(z) = —2n[ln(191n(r + z/n), r + z/n) — ln(191n(r), 1)].

Observe that in view of Theorem 3.1.1, 191,,(1' + z/n) is an approximation of 01
uniformly in 2 over bounded sets. Thus a natural candidate for the approximation of
ln is in defined as follows: For 2 6 R,

- hr+z/n(Xi—l 1 01))
. f(€1')

 

1.(2 )=-—-2n[l (01,r+z/n)-l( (01,1) ]nf=—2Zl . (4.1)

4.1 An approximation 1,, of the normalized profile

log-likelihood process ln

50

51

Theorem 4.1.1 Suppose that (CU-(C4) hold. Then for any B, 0 < B < oo,

 

1.(z)—1”,.(z)( = 0,.(1).

sup
IZISB

Proof. Without loss of generality, assume 7' = 0. Decompose the concerned

process in the following way,

 

_g. [ln(z) — 13(2)] = —% [ln(z) — 211. “Xi “ h./.(X.--.,a.))]

 

 

f(€£)
= Z [In “Xi — hz/n(X.-_1,191n(z/n))) _ 1n f(X1' - hz/n(X1—1,91))
f(X1‘ — h0(Xi-1191n(Z/n))) f(X1- (1009—1191))

f(X1- h0(Xi-1191n(Z/n)))
+ 2‘“ f(X.- - lam—1.19.30»)

13.12) +1332), (say).

 

It suffices to show that V B < oo,

SUP |l1n(z)| = 010(1), (42)
IZISB

and
sup |l2n(z)| = 0p(1). (4.3)
IZISB

Actually, we shall prove a slightly stronger result than (4.2). To state this stronger
result, recall that for 1 S i S n,

61(191,S) 2 61(19): X; — h,(X.'_1,‘l91), (I9?,S)T E Q.

 

Denote
_ f(€1(t12/n)) . . a .
pnz,-(t)—ln f(€.'(t,0)) .R R, lSzSn.
Then, for any i, 1 S i S n,
ani(t) = -<P(€i(t12/n))llz/n(Xi-1)+ w(61(t10))llo(Xi-1)

= l— «p(a-(t, mm + 1033509151494)
+[ — 1061610)) + 10(5)] [in/AX >1) — item—1)]
«2(a) [Mm-1) — 532131)]. ‘44)

52

Note that
9171(2) = lenzi(‘91n(Z/n)) _ Pnzi(01)l-
From (3.12), SUPlzISB |\/n(191n(z/n) - 01)| = Op(1). The stronger result that will be

proved is that for any 0 < C < 00,
E{ sup I: [11am - p...(01)1|} = 001-1”). (4.5)
IzISBn/vYIt-91ISC
Denote 913(t) = 01 + s(t — 01) and recall that (,0 = f’/f. By the absolute continuity
of In f,

[01 15.10.11» (t - 01) ds
5 3112(t) ‘l’ 32iz(t) + 3331(9), (say),

pnzi (t) _ pnzl(61)

where, by (4.4),
slam = f baa-(01.0). z/n» + 103101.31. 0))1 13,323-33 — 91) «Is,

82120) = [01 [—cp(6.-(91.(t),0)) + 9(6)] [fig/“(X14 - 90(X1-1)]T (t - 91) 931
s...(t) = —<p(6.-) [baa/X.-.) — 1.(X.-.)]T (t — 0.).
The Lip( 1) of 1p and (2.19) imply that uniformly in all t, i and s,
|90(€1(911(t)1 Z/n)) — 90(61'(911(t)10))l
S Llhz/n(Xi-la 91.0)) - ho(X1—1191.(t))l
S L|911(t)l\/1+ X124 I(lXi—ll S lZl/n)
S L(|91| + |t - 91|)\/1+ X124 I(IX1_1| S IZI/n)-

Therefore, by (2.18) and (2.9),

sup Is1..(t)l
IzIsB.I¢fi(t-01)Iso

Clan-””093 + Cn‘1“)\/(1 + X33311; I(IXt—1l s IzI/n) llzz/n(X1-1)|

g CLn‘1/2[|01|+ (In-W] (1 + (B/n)2)l(|X,--1| 3 8/11).

|/\

53

Thus, the stationarity of {X5} and Remark 2 yield that

E { 811p )2 3112(t)|} = 001-1”)- (4-6)
IzISB.Ifi(t-91)ISC
For 3252(t), the Lip(l) of 90, (2.17) and (2.9) imply that for all t and i,
sup l32i2(t)l
IzIsBJflt-Onlw
s sup / |10(€1(011(t) 0)) — wands mam—1)— bout—nut — 91|

lzl<B W(t— 91)I<C°

S L sup lt—01l(1+(B/n)2)I(IX1—1l.<.B/n)
W(t-Onlsc

S LCzn’l(1 + (B/n)2)I(|X,_1| S B/n).

This together with the stationarity of {X,-} and the boundedness of 90 on compact

set implies that

E { sup l2 32,2(t)|} = 0(n'1). (4.7)
lzlSB1lx/5(t-91)ISC
To s3;z(t), by (2.10),
sup l33iz(t)l
lzISB.l\/77(t-91)ISC
5 Meal sup [lhz/1(X1—1)- ham—nut — all]

Izlsmmt-Omsc
s Cn‘1’2|<p(€1)|\/1+(B/n)21(|X1-1l s 8/12).

The above, the integrability of 1p and Remark 2 yield that
E{ sup IZsa.(t)l} = 001-1”). (4.8)
IzISB.I¢fi(t-01)Isc
Therefore, (4.6)-(4.8) imply (4.5) and hence (4.2).

Now it remains to prove (4.3). To prove (4.3), the result of Theorem 3.2.2 will be
needed. Recall that

_ 50(X._1, 1911(2/n)))
H21 fffX' — ho(X.'_1, 9111(0)»

l2n(z

For any t E R, denote

19lnt(z) = 19111“) )+ t(191n(z/n)- 19171(0 ))

54

Then the absolutely continuity of In f gives

as =2] [—sp(X.~_—ho(X.-1,191n1(z)))h§(X.-_1)]((0191,.(z/n)—191,.())dt

= X31) l—SO( X.- — ho(X.-_1,191m(z))) + <P(€i)l ‘9 h1i(Xi-1)(91n(Z/n) — 9111(0))
- Z 90(61)93(Xi-1)(91n(z/n) — 9111(0))

Thus, the Lip(l) of sp, Theorem 3.2.2, (2.17) and (2.9) imply

sup “The first term of l2n(z)|}
IZISB

S lily; L|191n(z/n) — '191,.,(0)||ho(X.-.1)IEl/O1 Iho(Xi-11‘91nt(2)) - ho(X1'—1191)l dt

1
S 0p(n’l/2)/o |sup |01n1( z)— 01|dtZ(1+X,-2_)

= 0P(n‘1/2)0P(n1/2)0P(n )=0P(1) (4-9)

For the second term of l2n(z), since 2 90(6,)h0(X.-_1) is a zero mean square integrable

martingale process,
I)“. a(..)t.(x-_.)| = 0PM”). (4.10)
But Theorem 3.2.2 implies that
SUP |91n(Z/n) - 1911(0)| = 0p(n'1/2).
IZISB

Therefore (4.3) follows from the above two equations and (4.9). This ends the proof
of Theorem 4.1.1. [I]
Thus, the limiting process of the sequence of processes {ln(z), z E R} is the same

as that of the process ln(z) by Theorem 4.1.1.

4.2 Tightness of in

We need a preliminary result from our model (2.1). Note that I ( f) < 00 implies the

boundedness of f (Koul, 1992, p52). From the invariant equation of go, g9(y) =
ff(y — h(x,9))go(a:)d:c, so is 99.

55

Proposition 4.2.1 Suppose that f is continuous, positive everywhere and bounded
on R, then for any 0 < c < 00, there exists 0 < C < 00, such that for any interval
I C [—c,c] with length 1(1), V k 21,

P(X0 e I,X,. e I) s 00(1))”. (4.11)

Proof. We shall use the mathematical induction and the boundedness of f. Let

:1: be any point in R. For k = 1,

P(III,1) =3 P(Xl E 1|X0=$) = P(h(Xo,0)+€1 E I|X0=$)
= [11(1) — h(:vs0))dy s 01(1).

If the assertion holds for k = m, then for k = m + 1,
Pm+1(o,1) = / P(o.dy)Pm(y, 1) s 01(1) / f(y — h(o.01)dy s 01(1).

Thus,
P(x0 e I,X,. e I) = /1 P"(:1:, I)gg(:c)d:c g C(l(I))2.

Now for z 2 0, recall (4.1), let a = a2 — a1, fl = B2 — )31,

 

ln(z) = -2Zln f(€1' +;(:)flXi-l) I(r < X;_1 S r + z/n).

Note that in E D[O, co), the following discussion will involve weak convergence on

D[0, 00). To that effect, we need to introduce the modulus of a function in the space

D[O,oo). For any 1b 6 D([0, 00)), define rab(¢)(t) = d(t), a S t S b, a < b.

Theorem 4.2.1 (Whitt, 1980) Let P”, n 2 1, and P be probability measures on
D([O,oo)). Then Pn => P if and only if Pnr'l => Pr'l on D([sk,tk]) for all k

5ktk 5ktk

and some sequence {[sk,tk], k 2 1} with Uiidshtk] = [0, oo).

-1
Oath

Corollary 4.2.1 Pn is relatively compact if and only if Pnr is relatively compact

for all k and some sequence [shtk] such that UE‘3__1[sk,tk] = [0, oo).

 

56

Thus, take [shtk] = [k,k + 1], it is enough to work on D[k,k + 1], for every 11: Z 0.
That is, for every 45 6 D[0, 00), define

fl§°(¢) = sup [min{|¢(U') - ¢(u)|s |¢(U") - ¢(u)|}l

kSu—bsu’SuSu”Su+65k-{-l

+1 SUP” I44") - ¢(k)| + 8UP WU) - (WV +1)|s k 2. 0-

_<_u5k k+l-6SuSk+l

Theorem 4.2.2 Suppose that f is continuous, positive everywhere and bounded on

R, then ({ln(-z),z Z 0}, {ln(z),z Z 0}) is tight. That is, V k 2 0, V e > 0,
15111531111 P(flflln) > c) = o. (4.12)

Proof. We shall only show the tightness of {ln(z), z 2 0}, since the proof is the
same for {ln(—z), z 2 0}.

The following argument is similar to the proof of Lemma 3.2 in Ibragimov and
Has’minski (1981, pp. 261). Let A.- = A;(u,u + 5] be the event that a trajectory of
in possesses at least i discontinuities on the interval (u,u + 6]. We shall prove the

following inequalities:

P(Al) g 05, P(Ag) s 052. (4.13)

A trajectory of in has at least one discontinuity on (u,u + 6] only if at least one
X14 6 (r + u/n, r + (u + 15)/n]. Denote C.- = {X.-_1 6 (r + u/n, r + (u + 6)/n]}, then
by the boundedness of f and Remark 2,

P(Al) g 2 P(C.) g 06. (4.14)

A trajectory of in has at least two discontinuities on (u, 11 +5] only if at least one pair
of (X;_1,X,-_1) 6 (r + u/n,r + (u + 6)/n]2, i 71 j. Hence, by Proposition 4.2.1 and
the stationarity of {X1},
P(Az) S ZP(X.-_1 E (r+u/n,r+(u+6)/n],X,-_1 E (r+u/n,r+(u+6)/n]) S 062.
#9
This and (4.14) prove (4.13).
Now let B be the event that on the interval [11,]: + 1], there exists at least two

points of discontinuities of in such that the distance between them is less than 25.

57

Let us divide the interval [19, 11+ 1] into m = [6“] subintervals 6.- of length m‘l. Each
interval with length less than 26 is totally contained in 6,- U (6.4.1 U 6,4,2). Therefore,

n1 1n—2
B C U 142(65) U U A2(5;+1 U 6.4.2).
{:1 i=1
Hence,
n1 7n-2
M?) s 2 mass» + )3 P(A2(91‘+1 u a...» s cme s ca. (4.15)
i=1 i=1

Furthermore, as long as the event B does not occur (i.e., the complement of B, say
B", occurs), any interval of the form [u - 6,u + 6] possesses at most one point of
discontinuity of l... So that this function is continuous on either [u, u + 6] or [u — 6, 11].
For example, suppose that l" is continuous on [u,u + 6]. Then in has no jump on

[u, u + 6]. Note that in is a step function, so in is a constant on [u,u + 6], i.e.

sup Iln(u) - ln(u”)| = 0.

uSu”Su+6

Finally, on BC, there is at most one discontinuity point of in and V e > 0,
{ sup (1”,.(11) — 12.01)) > e/2} (1 BC c 41(1 + k/n, r + (k + 6)/n],
kSuSk+6
thus, by (4.13),

P({Ki‘iiss (1",.(11) — 12.01)) > e/2} n B) (4.15)
s P(Ioh-(r + k/n, 1 + (k + 15)/11]) s 05,

and

P({ sup (1”,.(11)—i..(k+1)|>e/2}n136) (4.17)
k+1-6SuSk+l

gimmu+(k+1-Qhaw+w+iyfl)£03
Therefore, by (4.15)-(4.17), for every 6 > 0,

P111113») 3 P(B)+P(Bcn{o§(i.)>o})

g 05 + P({ sup (1.(1.) — i,(k)| > e/2} n B“)
kSuSk+6

+P({ sup lln(u)—ln(k+1)|>c/2}nB°)
k+l—6Su$k+l

305. (4.13)

|/\

58

It follows from (4.18) that, V k 2 0, V e > 0,
girré sup P(Bflln) > e) = 0.

Therefore, {ln(z), z 2 0} is tight. CI

4.3 Some problems for future research

Note that for each n, the process in is a jump process with finite number of possible
jumps at nX,-_1,i = 1, ..., n. It is thus reasonable to expect that the limiting process
of in will asymptotically behave like a compound Poisson process with rate 99(r)
whose left and right jump distributions are given by the conditional distribution of
(1 = —2ln[ f (61 — a — 6X0) / f (61)] given X0 = r“ and the conditional distribution of
(2 = —2 ln[ f (61 +a+ 3X0) / f (61)] given X0 = r+, respectively. The former conditional
distribution is the limiting conditional distribution of (1 given r—61 < X0 S r—62 and
the latter is that of (2 given r +61 < X0 S r+62 as 6110, 621,0 and 61 Z 0, 62 > 0.

I am presently working on the above problem.

After obtaining the limiting distribution of in, it will be easy to obtain some
inference on the limiting distribution, which will be related to the compound Poisson

process, of the standardized maximum likelihood estimator of threshold parameter r.

Appendix A

Lemma A.0.1 Suppose that p1(y) = f|1n[f(:c + y)/f(:c)]|dF(:r) < 00, V y E R.
Then (i) and (ii) below are equivalent.

(i). p1(y) is continuous at a + fir,
(ii). f I ln[f(a: + a + fly)/f(:r)] — ln[f(a: + a + ,Br)/f(a:)]|dF(:r) is continuous at r.

Proof. Suppose that (i) holds and yn —+ r. Let

 

 

f($+a+fiya) f($+a+flr)
9,, a: = ln , :1: = 1n .
( ) mo) 9‘ ) mo)
Then, the continuity of f implies that,
ya —1 g, point—wise
and which implies
92f ->g+s g; —’g‘- (A-l)
Thus, by (i),
/ lgnldF —+ / |g|dF. (A.2)
Combining (A.1) and (A2) implies that
[1,de —+ f 9* (IF. (A.3)

Since 0 S (9* - 9:? S 9i: (g:h — gf)’: —-> 0 and fgidF S |g|dF < 00, by dominate
convergence theorem,

[(gi — g:)+ dF _. o. (A.4)

59

60

The result (AA) and f((g* — 9,, 35) dF —1 0 imply that

[(91 -g?f) dF-+0-

Thus,

/ 11* — 1:1 or —» 0 (4.5)
which implies

fm—aera

The fact that (ii) implies (i) is obvious. CI

Lemma A.0.2 Let p(y) = fln[f(:c+y)/f(:r)]dF(x), then the continuity ofpg implies
the continuity of p1 and p.

Proof. Let yn —1 r, denote

 

 

_ f($+a+flyn) _nf(w+a+flr)
hn(:1:) — 1n f(z) , h(:I:) —1 f(x) .
Then
h, _. h, / (5.12111? —-> [521113. (A.6)

A convergence theorem in Hajek-Sidak (1967, pp. 154) and (A.6) imply that
/ |h,, — hlzdF —-1 0. (A.7)

By Lemma A.1, (A.7) is equivalent to the continuity of p1. The continuity of p follows

from (ii) of Lemma A.0.1. D

Lemma A.0.3 The conditions (CI) and (C2) imply the continuity of

102(1)) = E {1n[f(61 + y)/f(61)l}2 on R-

Proof. For any :1: and y in R,

{<———)<—)}|
11::11 (nmnnwl

f(€1 + y) f(€1) f(€1)
S 2 la: - yl (El<p(61 + 17(rc - y))|’)‘/2/

nf(<51+113) lnf_(___€1+y)
{(Ell f(€1)21/2)w( f(€1)21/)2},

 

SEln

 

 

 

61

where In] < 1, which is between a: and y. The conditions (C1) and (C2) imply
the finiteness of E]cp(cl + 17(2: — y))|2 and E lln Affillz for any a: and y. Therefore,
E {ln[f(61 + y)/f(el)]}2 is continuous in R. D

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