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This is to certify that the

dissertation entitled

Weak Convergence of Weighted Empirical Processes
Under Long Range Dependence with Applications
to Robust Estimation in Linear Models

presented by

Kanchan Mukherjee

has been accepted towards fulfillment
of the requirements for

PhD. degree in SLan§LjCS

 

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oddlcm‘pmS-pn

 

 

WEAK CONVERGENCE OF WEIGHTED EMPIRICAL PROCESSES
UNDER LONG RANGE DEPENDENCE WITH APPLICATIONS TO
ROBUST ESTIMATION IN LINEAR MODELS

By
Kanchan Mukherjee

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

1993

 

ABSTRACT

WEAK CONVERGENCE OF WEIGHTED EMPIRICAL PROCESSES
UNDER LONG RANGE DEPENDENCE WITH APPLICATIONS TO
ROBUST ESTIMATION IN LINEAR MODELS

By

Kanchan Mukherjee

A discrete time stationary stochastic process is said to be long range
dependent if its covariances decrease to zero like a power of lag as the lag tends to
infinity but their absolute sum diverges. In this dissertation, a uniform closeness

result of weighted residual empirical process to its natural estimator is derived
under the LRD setup. These are then used to prove the asymptotic uniform
linearity of a class of linear rank statistics and the asymptotic uniform quadraticz'ty
of a class of L2- distance statistics. These results, in turn, are applied to
investigate the asymptotic behavior of the above estimators in a linear regression

model when the errors are function of LRD Gaussian random variables.

Some intriguing phenomena are observed in connection with the inherent
nature of the limiting distributions of the above estimators. Unlike the weakly
dependent case the limiting distributions may not be normal always. Moreover,
when the errors are LRD Gaussian and the design matrix is centered, the
asymptotic covariances of the class of rank and minimum distance estimators
become those of the least square estimator- a phenomenon which is in complete
contrast with the i.i.d. error case. Similar statement applies to LAD and a large
class of M- estimators. These results are proved under some conditions on the
design matrix that are very similar to those under the i.i.d. setup

The dissertation also considers the asymptotic behavior of regression
quantiles and regression rank scores which are natural generalization of the notion
of order statistics and regression rank scores processes from the one sample model
to the linear model. Under the LRD error setup the aforementioned uniform

closeness result is used to obtain the asymptotic representations of regression

'-

 

 

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To my parents and grandparents

 

 

ACKNOWLEDGMENTS

I would like to express my deepest regard to Professor Hira Lal Koul for his
careful guidance, encouragement and affection throughout my graduate study at
Michigan State University. His erudition, dedication and contribution to statistics
have been my main source of inspiration to work in the area of nonparametric

statistics.

I would like to thank Professors Joseph Gardiner, James Hannan and
Habib Salehi for serving on my thesis committee and for their suggestions leading
to the improvement of my dissertation. 1 cure my special thanks to Professors
Dennis Gilliland, Connie Page and James Stapleton from whom I learned many

aspects of good teaching and statistical consulting.

The dedication of my parents and grandparents to the pursuit of my higher
education is greatly appreciated. The support and encouragement of my sisters,
Baishakhi and Mousumi, my friend Dr. Suman Majumdar, and his wife .layasri
during my doctoral study were invaluable. The careful guidance provided by my
teachers at UPANISHAD during the early stage of my education deserves special

mention.

 

TABLE OF CONTENTS

Chapter
0. Introduction

1. Preliminaries

1.1 Introduction

1.2 Uniform closeness of weighted empiricals
1.3 Proofs

2. Rank estimation

2.1 Introduction

2.2 Asymptotic uniform linearity of linear rank statistics
2.3 Proofs

3. Minimum distance estimation
3.1 Introduction

3.2 Asymptotic representation of minimum distance estimators

4. Regression quantiles and related processes
4.1 Introduction
4.2 Theorems and proofs

4.3 L-estimators and regression rank scores statistics

4.4 Asymptotic uniform linearity of linear regression rank-scores statistics.

5. Bibliography

vi

Page

24
24
26
29

33
33
35

42
42
45
54
57

61

 

CHAPTER 0
INTRODUCTION

A discrete time stationary stochastic process is called a long range
dependent (LRD) or a long memory process if its correlations decay to zero like a.
pOWer of lag as the lag tends to infinity but their absolute sum diverges. Quite
often, econometric and time series data appears to be stationary and exhibit
strong correlation between observations separated by large lag which does not
decay to zero at a fast enough rate to be absolutely summable. Similar phenomena
have been observed in hydrology in connection with the construction of the Aswan
dam over the river Nile, Egypt, when hydrologist Hurst noticed that the annual
volume of river-flow shows long term behavior over time (Mandelbrot and Van
Ness, 1966). Mandelbrot and Van Ness proposed fractional gaussian noise to
model observations with such strong correlation. Later, Granger and Joyeux
(1980) and Hosking (1981) independently came up with fractional ARIMA model
to include more processes with non-gaussian marginal distributions. The salient
features of these processes are that their spectral density diverges at zero and their
correlations are not absolutely summable and that creates considerable

mathematical difficulties for its analysis.

The usefulness of LRD processes in modeling a wide variety of physical
phenomena heralded an upsurge of interest among many researchers who explored
different probabilistic aspects of LRD processes in the last two decades.
Investigation of the behavior of different statistics and estimators based on LRD
observations are also carried out by some authors. Taqqu (1975) obtained the
weak convergence results of partial sum processes based on random variables

(r.v.’s) that are a measurable function of LRD Gaussian r.v.’s.Taqqu characterized

 

2
the limiting distribution of the partial sum process for r.v.’s having Hermite rank

(see Remark 1.2.1) one and two. Later, Dobrushin and Major (1979) and Taqqu
(1979) independently characterized the limiting process (called Hermite process)
for r.v.’s having arbitrary Hermite ranks through a multiple Weiner-Ito integral
representation. It was observed that the limiting process is non-gaussian if its

Hermite rank is more than unity.

Along with these technical results, parallel research has proliferated on the
estimation of some parameters describing the correlation structure. Fox and
Taqqu (1986) and Yajima (1985) proposed maximum likelihood estimation of the
index of the LRD 0 (see (1.12)) based on LRD Gaussian sequence and Yajima

(1985) considered the least square estimation (l.S.e.) of 9 based on LRD r.v.’s.

In linear regression model with LRD errors, Yajima (1988, 1991) obtained
the strong consistency and the asymptotic distribution of the l.S.e. of the
regression parameters under some conditions on the cumulants of the marginal
error distribution function. Koul (1992a) derived the asymptotic uniform linearity
(AUL) 0f M-statistic and limiting representation of normalized M-estimators in a
regression model when the errors are a function of LRD Gaussian r.v. and the

score function is absolutely continuous.

Motivated by the seminal work of Koul (1992a), in this dissertation we
derive the asymptotic representation of some more robust estimators of the
regression parameters in a linear regression model when the errors are a function
of LRD Gaussian r.v.’s. In particular, we consider the behavior of a class of rank
estimator (R—estimator) proposed by Jureckova (1969) and Jaeckel (1972), and

minimum distance estimators (m.d.e.) proposed by Koul and Dewet (1983) and

 

3
Koul (1985b). Finally, we also consider regression quantiles (RQ) proposed by

Koenker and Basset (1978) of which the special case is the least absolute deviation
estimator (LAD).

The investigation of the behavior of robust estimators based on dependent
observations started with the work of Gastwirth and Rubin (1975). Gastwirth and
Rubin studied the behavior of R—, M— and L-estimators in a location model under
A-mixing errors. Koul (1977) generalized their results to linear regression model
with strongly mixing errors which contains A-mixing class. In all of the above
weakly dependent cases the correlations are absolutely summable and thus the
effect of dependency becomes relegated, at least asymptotically. Consequently, the
limiting distributions of the suitably normalized estimators is Gaussian as in the
case of independent identically distributed (i.i.d.) errors. But, in the case of the
LRD errors, the limiting distributions of these estimators are quite different from
their weakly dependent counterparts in two fundamental ways. First of all, the
normalizing factors are different and secondly the limiting distribution is not

always normal. For more on these see Remark 2.2.1.

The crux of proving AUL theorems in linear models is unified in the work
of Koul (1991, 1992) in the form of a uniform closeness theorem for a weighted
residual empirical process to its natural estimate. Hence the fundamental tool for
proving most of the results in this dissertation is the, uniform closeness result of
weighted residual empirical process to its natural estimate in a linear regression

setting when the errors are a function of LRD Gaussian r.v.

The technical difficulties for proving the uniform closeness result in this

setup is surmounted by using a modification of an ingenious chaining argument

.. .a

4
of Dehling and Taqqu (1989). Dehling and Taqqu came up with a chaining

argument to prove the uniform weak reduction principle of ordinary empirical
process. Theorem 3.1 of Dehling and Taqqu (1989) obtains an upper-bound for
k _ _
Pl ..,. Ira‘En ”Quesx) - Fm - Jm(x> (m!) ‘Hmoim > 6],
kgn, xEI 121
which converges to zero V6 > 0. Here m, Jm, Tn and Hm are defined in Section

1.2. In Theorem 1.2.1 of this dissertation, we invoke a similar chain to prove that
- n -
Pl :2Pll7nli§l7ni{1(5i 5X +6110 ‘ le + Eni) " Jmlx 'l‘ Enilim!) lle(7li) > 6i

converges to zero. Then, this result is used to derive the uniform closeness result.
In other words, we obtain a partial generalization of the. uniform weak reduction
principle of Dehling and Taqqu from the ordinary empirical process to a very

general weighted empirical processes with nonzero 5 lgi $11. This partial

ui’
generalization allows us to prove the uniform closeness result, which along with
some other results in Chapter 1, is used to obtain the asymptotic representations
of the rank estimators, minimum distance estimators and regression quantiles. As
a byproduct of Theorem 1.2.1, we also obtain the weak convergence results of
weighted empiricals. In our case the novelty in proving Theorem 1.2.1 lies in
choosing the chain suitable for weighted empirical process which, of course,

reduces to the Dehling and Taqqu chain when the weighted empirical reduces to

the ordinary empirical.

Notation. In this dissertation. I(A) denotes the. indicator function of an

event A. The index i in the summation varies from 1 to n unless specified

t

otherwise. For a vector u E R", u ( null ) denotes its transpose (Euclidian norm).

If D is an nxp matrix, then diii’ 1 g i _<_ n, denotes its ith row and d.j’1 g j 5 p, its

    

Warm constants whoa: 3:5 row is x3“. 2

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CHAPTERI
PRELIMINARIES
1.1. Introduction.

We consider the following multiple linear regression model in this
dissertation. Let (ni, i2 1} be a stationary, mean zero, unit variance Gaussian
process with correlation p(k) :2 E(n11]1+k), k _>_ 1. Suppose 6i := G(ni), i2 1,

where G is a measurable function from R1 to R,1 and let X denote the n x p design

t

matrix of known constants whose ith row is xni,

l _<_ i 5 n. Consider a linear model

where one observes the response variable {Yni}, 1 5 i g n, satisfying,

(1.1) Yni = xxtfifi + 6i, 1 g i g n, for some fl 6 RP.

The long range dependence of the r.v.’s {111} is implied by assuming that

for some 0 < 0 < 1,
(1.2) pm = k-9 L(k). k 31,

where L(k) is a slowly varying function at infinity, i.e., L(tx)/L(t)——> l as t—>oo for

every X > 0. We assume that L(k) is positive for large k . From Lemma VIII.8 of

Feller (1968, vol 2), it follOWS that 337(k) z 00. Examples of such functions L are
1

positive constants or L(k) :: log k.

In this dissertation, we derive the asymptotic representation of some
familiar estimators of the regression parameter [1 that are known to be robust in
the linear models with independent errors. In particular, we consider a family of
R-estimators, m.d.e., and regression quantiles. In this chapter, we describe the

basic probabilistic results and their proofs that are needed throughout the rest of

 

this dissertation.

The technique of obtaining asymptotic representation of suitably
normalized estimators defined as a solution of a system of equations goes back to
Cramer (1946). The basic idea is to derive an asymptotic Taylor type expansion of
a suitable score function and to ensure the existence of stochastically bounded
solutions. Therefore using the same technique with suitable modifications one can
obtain the asymptotic representation of R-estimators and M-estimators, defined as
a solution of a system of equations. In linear regression models, different authors
have used different techniques to derive a Taylor type expansion, see, e.g., Koul
(1969) and Jureckova (1971) for R-estimators, among others. Koul (1991)
envisaged a unified approach to these problems as a consequence of uniform
closeness of some weighted residual empirical processes to its natural estimate,
centered at its expectation. Using uniform. closeness and the smoothness of the
error distribution function one can obtain an one step Taylor Type Expansions of
the nonsmooth empirical processes and asymptotic uniform linearity is a

consequence of that.

1.2. Uniform Closcness of Weighted Empiricals.

Let 6 (n) have the same distribution as that of the marginal distribution of
{€i, izl} ({ni, i21}). Let F be the distribution function of c and I := (x:
0<F(x)<1}; {quqzl} denote the Hermite polynomials (see Remark 2.1 for
definition). It has the following properties:

(2-1) EHq(n) = 0, (121, EHq(nJ-)Hq(nk) = @qu - k) Vq20, Vi, k 21, and

EHq(’lj)Hr(7lkl : (l Vq ¢ r, Vj, k _>_ 1.

 

8
A quick proof of the last fact with j¢k can be given as follows. Suppose that

q < r and write 775 = pnk + 71*, where p:= p(j - k). Observe that Hq(pnk + rf") is
a polynomial of degree q in 77k with coefficients involving 17*. Since 77k and 17* are
independent the result follows from E nkH (17k) = 0 V0 3 c<k which is a
consequence of the fact that the k th degree Hermite polynomial is obtained
through the Gram—Schmidt orthogonalization of the square integrable r.v.’s
{17L 1 s c _<_ k}-

Define,

Jq(X) == EI(G(TI) SX)Hq(77)J (TI—(X) — E1(G( fll<xllHq(fl ll X EL (121,
and let m be the Hermite rank of the class of functions {I(G(n)gx), x61}
introduced by Dehling and Taqqu (1989), i e

(2.2) m := minimum{m(x): x 61}
where

m(x) :2 minimum{q 2 1; Jq(x) ¢ 0}.

In View of (2.2) We have,

(2.3) manage-ax): 2.1.,<x)/q!H.,<n.). vx e 1, vial,
qun

where the above equality is in the Liz—sense.

For L and 0 in (1.2) define Tn :2 nil‘mel/‘ZLm/‘ZML 1121. Throughout

this dissertation, we assume that 0 < (9 < l / 1n.

Let {yup 5m; 1 5i 3 n} be arrays of real numbers and define, for x E 1,
WW == Til 27ni1(€i SK + 6",), Vli(x)== Tii‘ZZ'rmlki 5X).
1 l

nn(X) == TL‘Zrninc + 6“,), MAX ) — Tn ZimF (x )

 

Un(x) == Vn(x) ' #n(xl, WK") 3: Vn*(X)' #n*(x)
Sn(x) : TifZ’lnifIiEi 5x + {ni)~F(x + {nil'JmO‘ + €nil(m!)-lle(’7i)a

530‘) == Til Zvnilflq S x)-1“(X)--1m(x) (lllll'lemll-

The following theorem obtains the uniform closeness of the processes Un

and U3. Here, 6i :2 C(ni) and (1.2) is satisfied. Also Tn is defined as above.

Theorem 2.1. Suppose that the weights hm; 1_<_ign} satisfy the following

conditions:
2 " 2
(A.1) nlmsaiirSn’yni: 0(1). (A2) 21:711i21’ V n21.
Then,
(2.4) sup (S;(X)l = 0P(1).
x El

Moreover, assume that (A3), (A4), (A.5) and (A.G) hold, where
(A3) 1m<a§< n léml = 0(1). (A4) izwmsnil = 0(1).
(A.5) The d.f. F ofe has uniformly continuous densityf on I, f>0 a.e. (Lebesgue) on I,
(A.6) The functions Jm and .ll‘lf, are continuously differentiable with respective

. . I +I I +I
derivatives .1", and Jm. Moreover, .lm(x) and .lm(x) converge to zero as x

converges to c 2: ian and d := sup I.

Then,

fi

10
(2'5) xsuepllsnbcll = op“) a
(2-6) xsgpllUMX) - U300! = 0p(1),
and
(2-7) :gpllrfi‘FilrndIk; 3" + {nil-HE; lel'TrilZ7niénjftxll = 0p(1)'

The proof of the above theorem uses a chaining argument similar to
Dehling and Taqqu (1989) and appears in the next section. An analog of (2.6)
when the errors are i.i.d. appears in Koul (1969, 1970) and was further generalized

to include the case of strongly—mixing errors by Koul (1977, Proposition A1).

Using the technique similar to the proof of Theorem 2.1 one can obtain
pointwise and uniform convergence over compact versions of the above theorem.

This is stated below in a form that is useful in Chapter 4.
Theorem 2.2. Suppose 6i := C(ni), i 3 1 and (1.2) is satisfied.

If (A.1), (A2) and (A3) hold, and F is continuous at x E I, then
(2.8) we - stn = 0,.(1).
If, in addition, F has (I. continuous density f at x 61 and (AA) holds, then
(2.9) lrt‘jflniflm 5x + 6”,) — 1(6, 5x)} - T;.'i27m-£,,if(x)l= 0,,(1).
In addition to (A.1) - (AA), suppose that the following hold:

(A.7) f is continuous and positive on I

 

—fi—

11
(A.8) Jm, J; are continuously differentiable.

Then for every b e (0, oo),

(zulieiuixlgb }slTfi‘ZiI7ni{I(6i X+£m)- 1(6-< <X)}-Tn Zrniénif(X)l=0p(1)

Finally, as a byproduct of (2. 4) and (2 (6) we obtain a weak convergence
result of weighted residual empirical processes based on LRD observations. Note
that from (A. 1) and (H2) of Section (1.3), the sequence of r. v. s { Tn ;7niH m(’lil
} is bounded, uniformly in LA2 norm and hence has a subsequence converging to a
r.v. Zm(7), say. The determination of Zm(7 ) foi general weights hm, 1<i<n}
with m > 1 is still an open p1oblem. In the following corollary, the weak
convergence is understood in D[—oo, 00] equipped with the a-field generated by the
open balls of the sup metric.

Corollary 21. Under (A 1)- (A.)6 the process {Tn ;7n1[1(5i5 - + {nil—
F(. + 5m)“ is tight and converges weakly to the process {(1111) 1Jm( (.)Zm(7)}

along a subsequence. //
Some remarks about the Hermite ranks and the assumptions of the above
theorem are now in order.

Remark 2.1. Let 7, be a standard Gaussian r.v. and Q: R1411 be a
measurable function such that EQ2(7]) < co. Recall from Sansone (1959) that the
Hermite polynomials {Hk, k>0}, defined by :7: d)(x) = (-l)k Hk(x)¢(x) (
anal-Daisey, e< “ 2tx/=5Ojtk11k(x)/1a ) have the property that {HR/(1:01”) is

an orthonormal basis for L“ (R1, Bl, (1(1)). The index of the first nonzero coefficient

in the Fourier expansion of the r.v. (2(1)) with respect to this orthonormal basis is

fi

12
called its Hermite rank (see Taqqu, 1975). Clearly, if Q is an odd (even) function

then its Hermite rank is 1 (2). Also integration by parts shows that if Q is
monotone and right continuous such that the function qu vanishes at -oo and 00

then the Hermite rank of Q is 1. //

Remark 2.2. Here we consider some examples of the Hermite ranks of a
class of functions, see (2.2). If Q is strictly monotone and continuous, then the
Hermite rank 111 in (2.2) is equal to 1. To see this, consider the case when Q is
strictly increasing. Then using the fact that ¢(x) Hr(x) dx = - d{¢(x) Hr_l(x)}, we
obtain that for ye I, r21, sue gy)H,(n) = E10, 5 o-1(y))H,(r,)
= — ¢(Q'l(y))Hr_l(Q'l(y)), which is nonzero for r = 1. The same is true when Q is

strictly decreasing.

Now let Q be an odd function with the additional property that
{x E R1: Q(x) 5 0} equals either [0, 00) or (~30, 0]. Then also 111 is equal to one. To
see this, consider the case {x 6 RI: Q(x) g 0) 2 (~00, 0]. Note that this implies that
Q(x) Z 0 for X > 0- Then for y S 0. y 6 Ir EI(Q(7I) S")?! = EI(Q(7I) SYM 1(7) S 0) < 01
since the range of Q is I. Similarly, for y 2 0, y E I,

EI(Q(77) S y)n = 131(c)(7)) S WK?) 5 0) + EM 5 Q(U) S y)771(72>0)

<-¢(0) + ¢(0) = 0.

The proof is similar in the case {x e R1: Q(x) g 0} = [0, 00).

An example of Q for which in = 2 in (2.2) and conditions (A.5) and (A6)
are satisfied is given by Q(x) = [XII/6, 5>1. Dehling and Taqqu (1989) showed
that for any m3 1, there is a Q for which the Hermite rank of the class of

functions {I(Q(17) g x), x E I} is 111. //

fi

13
Remark 2.3. Note that Jm and J; are functions of bounded variation and

hence are differentiable almost everywhere. Also, under (A.6), sup lJ;n(x)l and
x61

sup IJ$'(X)| are finite. //
x e I
If G is strictly monotone and continuous with d.f. F, then from Remark

2.2, m=1. The following proposition states that in this case (A.6) is satisfied if the

Fisher information 1(f) of the density fis finite.

Proposition 2.1. Assume that G is strictly monotone, continuous and (A.5)
holds. If, moreover C(11) has an absolutely continuous density f and
I(f) := f[f’/f]2dF <oo, then (A.6) holds. //

Proof. Consider only the case when G is strictly increasing and continuous.
Note that this entails G = F'ld). Hence from Remark 2.2, J’l(x) =
¢’(G'l(x))/G’(G‘1(x)) = -f(x)<I)'1F(x). Note that for any ae(0, 1) there is a
Ka e (0, 00) such that )‘I>'l(u)l g Ka[u(1-u)]'a. 0 g u _<_ 1. Fix a b e I such that F(b)
> 0. Then for x > I),

W) @“(thm s K. ax) [PM (more
5 K. [Punt-'1 jut) (It/[l-F(x)l“
s K. [F(b)l'a El-f’(t)/f(t)l/[l-F(t)l"‘dF(t)
s K. [Foot-a {it -f’(t)/f(t)l"’ arm”? {in-Fare awn”?

Upon choosing 2a < 1 in this inequality, it follows that )f(x)(l"l(F(x))l -> 0 as x —.

(1. Similarly one can prove that |f(x)(l"l(F(x))l —> 0 as X —> c. //

A “9-931

fi,

14
1.3. Proofs.

In order to prove Theorem 2.1, we need some preliminary facts about
Hermite expansions and ranks which are summarized below. These can be found
in Taqqu (1975, 1979) and Dehling and Taqqu ( 1989). In what follows, L, with or
without suffix, is a generic notation for slowly varying functions and c is a generic

constant.

Let {7711 12 1} and p be as in Section 1.1 and p > 0 be a fixed integer. Then
the following facts hold:
(111) zzhpo-rh = 0(112"’9[L(n)]”) if1>9 <1.
= 0(11L0(n)) if pd = 1,
= 0(11) if p6 > 1.
(11.2) For any measurable function h E L2(Rl. Bl. d‘I’) with Hermite rank p < 1/0,
Variance< i2 hot) i = (xi: glam-r) I) = ()(112"’9[L(n)l")-
(11.3) (i) Reciprocal and product of slowly varying functions are slowly varying.
(ii) For any slowly varying function v, V(u)ll6 o oo (0) for all 6>O (<0).
Now, we shall give the proof of (2.5). That of (2.4) is similar and simpler.
To begin with we obtain an upper bound for the expected. value of the square
increment of the Sn process. Throughout the rest of this chapter, for a function h :
Rl—tRl, h(x, y) stands for h(y) — 11(x), x gy.
Lemma 3.1. Suppose fi := C(ni) , 13 1, and (1.2) is satisfied. Then
(3.1) ra2E1 Sex. M2
5 Z: 2'7"inth [F(x+g“i, yam) F(x+5nj, y+§nj)]l/2 lp(i-j)]m+l.
1
Proof. The Heerite expansion of {HQ 5 x+£ni) - F(x+£ni)} and (2.3)

yields that

fi

15
TanI Sn(x1 Y)l2

=i2 Emmot- q 2‘12““ rngrlqecfini. Y+E,,i)/q! Jr(x+£,,-. y+c,,,-)/r!
xEHq(nj)Hr(7/k)-
Using (2.1), the above is equal to
It? Jinn“,- q 2%“ Jq(x+£,,,, y+£,,,) Jt.(x+£.,J-. y+£,,,-)/q! p“(i—j)l
5 2i: :Ll'rnnnjl q 2%“ qu(X+E,,ir y+£,,i) Jt,(x+£,,jr y+£,,j)/q-'l lp'“+l(i-i)l
5 i2 phenol L sz'lii(x+f,tii Y+E,,;)/<1!ll/2 ( ZEN-Jamey. y+£,,,-)/r!J’/2 lp"‘+l(i-j)l.

where the last step follows from the Cauchy-Schwarz inequality applied to the

sum involving q. But the Hermite expansion of
{l(x+£ni < 6i gy+£ni) - F(x+£“i, y+£m)} and (2.1) yield that
q >ZmJ(2](x+Enll y+£ni)/ql : E [1(X+€lll<€l S y+£ni)-F(X+€ni, y+£ni)]2
S F(X+€ni’ y+£ni)’ V 121.
Hence the lemma is proved. //
Proof of (2.5). Without loss of generality, assume that 7"]- 3 0, 1 g i 5 n.
For the general {711i} the result follows from 711i: 7m - 7;” and the triangle
inequality. In what follows, we use a 111mlification of the chaining argument of

Dehling and Taqqu ( 1989). Let
(3.2) /\(x) :2 F(x) + .l$(x)/ ml, XE I.

Note the following facts:

 

—‘—

16
(3.3) /\ is strictly increasing, A(d) < co, and by (A.5), (A.6), A is differentiable

with uniformly continuous derivative A' = f + Jay/111! satisfying /\’(x) _.0 as x—» c
and d.

(3-4) FRY) S Mer), XSy-

(3-5) lJm(xa Yi/m” SJHXtYV 111’ S /\(Xr Y)r XSY-

Fix a 6> 0 and an 11 3 1. Recall that r" :: 11(l‘mm/2Lm/2(n) and let

(3.6) K, = x(6, n) := integer part of log2{/\(d)§: 7"] (6rn)'l} + 1.
1:]

By (A.1), (A2) and the inequality

_ n 02 — /2” ,2 I/‘-’ .
“Ely“, 2 o'“/ [L(u)] Ema/(n (1:ng“ 7m):
it follows that
. _ n
(3.7) fill-i517,“ —» 00 and /\(d)/2'r1 z 6/(Tnligl’7ni).
I: _

Next define refining partitions of I as follows. Note that A is invertible and

define
(3.8) a, k ;= x1) 1(a)) 2-k],j = 0, 1, 2k, k=0, 1, 2, r.
Clearly,
c = 7r”, k < ”I, k < < 7r__)k_l, k < ”2k, 1.- = d,

and
(3.9) hr”, k) - he, k) = Aha/2k.

For an x E I and a k E {0, l, ..., I17} define jff by the relation
(3.10) Wjfi,k gx< Fjfi+l,k .

Now define a chain linking each point x e I to c by

 

 

(3.11) C = ij
Then,
(3'12) Sn(x)=snj(7rx 0 17rjX 1)+Sn7rj)l(( ,j,l17r)2( 2)

<7r. < ..... <7r. <x<7r.
J’l‘rl— ‘1,’§rx j§+lrh

+ ...... + Sn(7rj:_l ’ IC—l ,ij. , K) + 5,4713% , IC 1X).

First consider the last term in the right hand side of (3.12). By the triangle

inequality, its absolute value is bounded by .An(x) + iBn(x) + Cn(x), where

no == 7:511... IUD-1’s, + 6", <e< _ x + 6”,).

‘an(x) 3: 711227..) Fffljé ’ “it E”;- X + 5”,).

Cn(x) 3: Th] li§17111']"‘(7rj§.s + 51111 X + £111) H1-11(7li)l /1n!, x E I.
By the monotonicity of the indicator function, triangle. inequality and (3.4), (3.5),
.An(x) g Thigh“ 1(7rjze1h: + a”, <ei s eh,“ . h. + 5”,)

_ 11
S lSn(7r-X . 17r-x+1 Nil + 7111,217111 PM"): + €111?7r jfi+1 , x + é.ni)
7 1:

JK ’ h in J
11
+ T1.11 (21'7“) J111f7l'ji’fl‘t {man-15+] ,1: +5 511i) l /111'
I:
-1 n . .
S lSnffijfi , x s Wj§+l , x )l + Tllig711i(l+lHln(7llll ) A (”J-i: .‘l' {M175 1,1: + {111)
= lSn(7rjé ’ K1 Wjfi+1 , N)‘ + bufxiv say

Similarly, €3,1(x) + Cn(x )_ <b,,(x) and hence

(3-13) lSn(7r'X

J,“

1c 1 X)‘ S lsnfflj)’: , 7rJ-X+ 1N)l+ 2 1),](X )

N

 

 

We now show that

(3.14) sup bn(x) = 0p(1).
x E I

To that effect, by the mean value theorem,

bn(X) = Tali§7ni (1 + 1Hm(7li)1 ) [A (W1:- 1 h‘ ’ 7riii-+1 , N) + £ni{’\l(unix) ' ’V(vnix)}]’

where, unix (Vnix) is a number 111 [njx , 7rx + Enil ( [njx x , 7r.x +
N7

-,+1N JK+I K me

{nil ). Therefore by (3.9),

1mmsm§nm+mumouow

_ II
+ Sllp 1/‘I(unix) - AI(Vnix)1 Tll'i;17]]i (1 + 1Hm(7lill ) 1Eni1'

i, x
But by (A.4), the stationarity and the Gaussianity of {1“},
(no qgmu+unm111hn=ow1a§mwi1=mi
Hence by (3.7)
(3.16) b,,( x) <{6 + .1... 1m ulix) - .\’( v“,\)1 } oI (1 ).
Now to show that sup |/\’(u“ix) - /\’(v”,x)) : 11(1), note that,

1,x

7r.x + 2 max 16111"

(71' ~
i J\+1 11;!” 131511

[1.1--

l r<n€1x< 11 ”IX VIIIX

The uniform continuity of X1 011 compacts and the fact

_ _ ‘ _1 11
M15?“ , x) ' “\(Wj’é . x) _ Md) 2 h S 6/(Tniglni) To

imply that for every b 6(0,oo),Su}.){1rjié_{_l ,1.- - 75.: , K ; lxl Sb, x E I} —r 0. Hence

 

19
from (3.3) it follows that

sup { |/\’(uIIIx) - A(v (Vnixlli 1<i<n, x61} = 0(1).
This together with (3.16) and the arbitrariness of 6 readily yields (3.14).
Now from (3.12) and (3.13) we obtain
(3.17) sxupllSn(x)1<supx 1811(7TjXI0 ,7rjx 1)] + supx [Sn(1rj,l( l ,7rjx 2)1-1- .....

+ supx ’S"(7r.l,)f-1 I ”-1 1 733,11 '1‘ supx Sn(7fj§ , N 1 Wjé—f—l , n)

+ 2 sup bn(x).
x61
Fix a k E {0, l, ..., It-l}. Observe that

sup (S (”-x _t7r-x . ll
x61 11 Jk’l‘ Jk+l’l‘+1

max sup 1511(7rx

rTF-x )1-
V ' 7 , 1k 1
0953”” XE["1 k+1~”1+1 k+l) J“ “+1 +

If 7rI k+l = 7rr k for some re {0, 1, 2,.. “211} then for xe [7r-

7

1 k+l 17ri+l, 11+1)

= . = , .' rtt ,'1 1' _' 1111 1for the 't'hiit val'
”if: 1 k "1,k+l Flii+l ‘ k+l’ so tln he a)0\e sup1ei 1n 1 1 er lS

zero. If 7rI k+1 9t 7r k for all 1‘6 {1, 2, ..., 21‘} then i is odd. Therefore for

r!

X6I7rhk+l , ”1+1 k+l),71' xk=71'i_17 l\’+l alulfljf+ltk+l= W1, 1x+l SO that
. . = S 7r- ,7r- .
5“" lSHUI-JfiJi “111$“ .1.-+1)l l "1 1-1,11+1 1. HI”

x6[1rI, k+1’ 7r1+1, k+ll

Hence by Chebychev’s inequality and Lemma 3.1, V 6>0,

i

20
P1 1321' sa(1rjfi , 11 mm“, k) ( > 6/2]

S jg) P1 1 Sn(7rjI k+1 1 7rj+1I k+1l1> 6/21
< 46-2 7:222 . I '_. "1+1
_ n _ Irma... 111111 x

1

2k+1_1

l 2
jg) [F(7rj1k+l + ("1’ 713+1. k+l + £111) 13(7rj,1<+l+ £11r17rj+1,k+1 '1‘ fin,” /

S 4 6-2 T1122 ;7n17m‘( 1P( :1— _1)l1n+l
l

where the last inequality fOIIOWS from the Cauchy-Schwarz inequality applied to
the factor involving F and the fact that

jg) [F(7rj’ k'H + a, 7r1‘1'11k+l + a.) 511 V aeR_

Similarly, we also obtain
Pl 8x112 ll SHUT-jig I h. 7'3" X_+l 1c l1> 6/21 S46 ) T11~zi:;7ni711r(1p( 1 1‘) llm+l-
Hence from (3.17) we obtain that
(3.18) P[ sup 1511(Xll > 6)
x 61
S 4 (N '1' 116-2 71122 £72111 Inr 1P1 (1 1‘) HUI-H + PISXUP [HM (X)> 6/41-
1 1

We now analyze the first term in this upper-bound. Since for large 11, 7'n>1, and

by the Cauchy- Schwarz inequality, $71115 11 12/ (.3 6) yields that

IE +1< 2 + log.) [)1(d)/6] + c In 11,

fi

21

where c is a constant and In denotes natural logarithm. We shall next show that,
(3.19) an 1: (In 11) 7,2}; gym", |p(i-1‘) ("1+1 = 0(1), 0 < 1/111.
1

Consider first the case when (111 + 1)19 < 1.

1511 nl

an S( In 11) (nlm<ax 72. ) 11'1 T1122 ¥Ip(i-r) lm+l
1
z L(n) [n2 1 (“my [112 ‘ m9] .

Here, the last asymptotic equivalence follows from (A.1), (11.3)(1) and
(11.1). Assertion (3.19) for this case, now follows from (11.3)(ii). Assertion (3.19)

for other cases follows from (11.1) in a similar fashion.

The proof of (2.5) now follows from (3.14) (3.18) and (3.19). //

Remark 3.1. The main difference between the above chain and the Dehling-
Taqqu chain is in the. definition of 1; in (3.6). Note that when 7111 : n‘l/2 , the
above chain reduces to that of the Dehling-Taqqu. Also, (2.5) generalizes some of
the results of Dehling and Taqqu (1989) from the ordinary empiricals to the

Weighted empiricals. //
Proof of (2.6). Define,
l/"(X) 3: 1Un(x) ' Snfxll ' 1 U;(X) ' Siifxll
= r1: Z7..1[J111(X+£,,;)-J11.(X)l (mu-‘Hmtnti. x e 1.
I
Since by (2.5), sup [Sn(x)l : oI,(1), and also sugIlSflxH = oI,(1) when (2.5) is
x 61 x

applied with {111: 0 V i, 1 g i 5 11, it is enough to show that

(3'20) 5“!) ll’nfxll = 011(11'
x 61

i

22
Note that

VAX): = T" $71115111'l11101111x1H 111 (710/1113.

where un xis some number between x+ {III and x, 1 g i g 11.

Let wn: = lm<aiit< _11{ lntl}. Then for any k > 0,
sup{ lun(x)| : [x1 > k, x e I}
Ssupf 1Jin(x)1: N > k ‘ W11, X E 1} Th'Zl‘rniénil 1Hm(7li)1/m!-
1

By (A.6), sup{ lan(x)| : Ix) > k, er} ._. 0 as k _. 00. Hence, in view of (3.15)

withj = 1, to prove (3.20) it suffices to show that V 0 < k < oo,
(3.21) sup{ )1/I,(x)| : (x) g k, x e I} = oI,(1).
Fix a k. Viewing 11,, as a process 011 [—k k] ()1, it is enough to show that

(3.22) (a) 1/,,(x) : oI,(1) V x e I,
and
g c (x - y)? , for some c e (0, 00)

(b) Ell/11(xl ' ”11(3)”

For (a), note that
E[Vn(x)]2 = 71122 27111710 Jm(x, x+€III) .lm(x, x+£IIj) pm(i—j)/ ml
1 1

< Unix. x+t -)1'-’ (we 13.1 > n" 1.322: ,le(i-r)l"‘~0.
1

15 511'“ s 1 s n

Since Jm is continuous in a neighborhood of x and by (A.1), (11.1) the rest of the

term is bounded. For (1)), note that the mean value theorem entails that for x 5 y,

x1y 6 l'kv kl F11.

”11(3’) - V11(X) = Th]Z:7,,i{[-l1n(y+«£,,,)--lrn(X+€,,,)1 - 1-l1n(Y)--l111(X)1} (1110'le(771)
1

23
= Thl?7ni (y-x)[an(unixy) - Jin(uxylle(Ui)(m!l-11

where unixy E [x-H’ni, y+£ni], uxy E (x, y]. Therefore,
Ell/“(1‘) - Vn(}')l2 = (Y’X)2 11122 Z7ni7nj Anixy Anjxy Pm(i-j)/ (11102 : Cn(Y'x)2»
1 J

where, Anixy: Jin(unixy) - J;,,(uxy). Note that, lAnixyl 52 .1112 llJ,‘l’,’(x)l (00 V n,
i, x, y. Hence using (A.l) and (F.1) once more as in (a) we obtain that C": 0(1).
Now (3.22) follows from Billingsley (1968; Theorem 12.3). This completes the
proof of (2.6) also. //

Proof of (2.7) . This follows from (2.6) in conjunction with the following:
SZPI'TH‘ZvndFe+sin—Fem?mam)! = 0(1),
x 1 1

which can be proved by applying the mean value theorem on {F(x+£ni)}-F(x)}

and using the uniform continuity of f and (AA). //

Proof of Theorem 2.2. Proof of (2.8) follows directly from the Hermite
expansion at each fixed x and the continuity of .l,,,, which in turn follows from the
assumption of the continuity of the d.f. F. Proof of (2.9) follows from (2.8) in a
simple manner. Finally, assertion (2.10) follows along the same line. as in the proof

of (2.5) and (2.6) with suitable inmhfications. //

fi

CHAPTER 2
RANK ESTIMATION
2.1 Introduction.

The idea of estimating location parameter based on rank statistics finds its
root in the seminal work of Hodges and Lehmann (1963). Since then, a major
branch of nonparametric statistics deals with the rank-estimation (R-estimation)
of parameters by minimizing certain dispersions based on the ranks of
observation. Generally, these dispersions are expressed in terms of linear rank
statistics. Thus, the widespread applicability of linear rank statistics for a variety
of testing problems leads to its use in estimation in a natural way. The key tool

for studying the R-estimators is the AUL of linear rank statistics.

To explain R-estimation in a regression set up, let 1,!) be a nondecreasing
real-valued function 011 (0, 1) and {RiAv lgign, AGRP} denote the residual

ranks, i.e., RiA is the rank of Yni - xiiiA among Ynj - xflJ-A, 1 gj 5 n. Define
(1.1) S(A) 2: Xij(xm - x) WRiA /(n+l)) = [ 51(A), SP(A)]E
MA) := sum /<n+1) w“, - x:,,A).
Note that S is a linear rank statistic. .lureckova (1971) defined a class of
linear rank estimators of B, the regression parameter, by

. . n
(1.2) fl-IU i: argnllllA E Rp{j§1'S.l(A)l}

For square integrable score function 1/2, Jureckova (1971) obtained the AUL
of the linear rank statistics S(A) and the asymptotic normality of the

standardized fiJU by exploiting the contiguity of a sequence of densities of the

 

25
i.i.d. errors. Koul (1969, 1970) obtained the AUL of S for right continuous and

bounded 1,!) under weaker assumptions on the design matrix X and the error
density. He used weak convergence techniques and the inherent monotonicity of
the w.e.p. to come up with a better result for the bounded score functions. Koul
(1992, Theorem 3.2.4) also obtained the AUL of rank statistics under more general
heteroscedastic errors that allow the study of the robustness of linear rank

estimators under independent errors.

Jaeckel (1972) defined another class of R—estimators by

(1.3) [9.] :: argminA E It.“ 3(A),

where

5(A) ;= Zz/J(RiA/(11+l) (Yni - xffiA).
l
Jaeckel showed that if the score function 1/) satisfies
(L1) i1 I/i'(i/(n+1)) = 0.
I:

then 11) is nonnegative and convex on R". Moreover, using the AUL results of
Jureckova and the observation that almost everywhere differential of j is -S,

Jaeckel proved the asymptotic equivalence of ,6 J U and ,BJ.

Our aim in this chapter is to investigate the asymptotic behavior of
Jureckova-Jaeckel type estimators under the model (1.1.1) and (1.1.2). We
consider nondecreasing, right continuous bounded score function 1/) and follow the
weak convergence approach of Koul. The question of the AUL of rank statistics is
first reduced to that of the asymptotic uniform continuity of the w.e.p. of the
residuals. The results of Chapter 1 are then applied to yield the asymptotic

representations of the above estimators in the LRD set up.

26
2.2. AUL of linear rank statistics.

To proceed further, we introduce more notation. Let X denote the nxp
matrix whose jth column consists of nxl entries of xm- z: anij/n, 15 j Sp

and W := X - X , the centered design matrix. For any matrix D of order n xp, let
(2.1) Ad z: glam)”2 , 13d: = “(may/2,

as long as the definitions make sense. Define the following process based on

residual ranks and weights corresponding to centered design as

Z(u, A) 2: leni' x) 1(R1A 5 nu), 0 g u 51, A E R".
I

We are now ready to state the AUL theorem of the linear rank statistics S.

Recall conditions (A.5) and (A.6) from Theorem 1.2.1.

Theorem 2.1. In addition. to (1.1.1), (1.1.2), (A.5) and (A.6) assume that
the following hold:

(W.1) (WtW)l exists for all n 2 1).

(W. 2) n mfg“ )Mw‘W) =0(1).

xni

(L2) ll) 6 \II:= {g: [0, 1]—>R1, g is nondecreasing, right continuous, g(1) - g(0) = 1}.

Then V l) E (0, 00),

(2.2) sup “13$le [1 + A;,l s) - fi(11)]-sf(F'l(u))“:—- 0,,(1),
05u51,seN(b)
(23 B;,ls A‘ s f1v(F = 1,
) regigemb) ll [03+ w l+3 l‘W F)" Op”
where,

 

27
(2.4) $(u) := £(xni- i) [I(ei S F'l(u) - u], 0 5 u 51,
s == soc...- 2) was.» - i1. 17):: (i)¢(U)du- //

The following lemma gives an approximation to BQNIS in terms of the
Hermite polynomials that is useful in determining the limiting distributions of the
above estimators.

Lemma 2.1. Let {n}, i 2 1} be a stationary, mean zero, unit variance Gaussian
process with correlation p(k) :2 E(nln1+k), kg 1. Suppose 6i :2 C(ni), i2 1
and assume that (1.1.2), (W.1), (W2) and (L2) hold. Then,

(2.5) B133 = stud) + 0.11).
where,

l
quj) 2: E 1/I(F(€))Hq(1;):—(j)'.lq(F'l(u))di/v(u),q 31,
and for any nxp matrix D,

(2.6) 3,1 := 13;,1 2d,,,H,,,(1,,)/m!. //
l

The next corollary gives the asymptotic representations of the suitably
normalized R—estimators defined in (1.2) and (1.3). These representations are
obtained from (2.3) in a routine fashion as in the proof of Theorem 4.1 of
Jureckova (1971) and Theorem 3.3 of Jaeckel (1972). We omit the details for the

sake of brevity.

Corollary 2.1. Under the assmnptions of Theorem. 2.1 and condition (L.1),

(2.7) Aw ([9,. -19) = Aw (23,,“ - 13) + 0,111
= {Ifd1/'(F)}“B;v‘3+ 0.11).
(2-8) = {If dd(F)}'l swimw) + 0,,(1). //

 

28
Remark 2.1. In the i.i.d. errors case Koul (1992, Corollary 4.4.1) derived

the asymptotic representation of 31, under the conditions (W.1), (W*.2), (A.5),
(L.1) - (L.2) where (W*.2) is a slightly weaker condition than (W.2), namely,

(w*.2) 111;,(wtwrlwni = 0(1).

max

1 S i S 11
Using the AUL of S, Koul obtained that

(WW/213m) = {If darn“ (W‘wrl/‘ZS + 0.11).

. l _ .
which converges in distribution to N(0, {ff(l'1/1(F)}“) [[1/v(u) - Ml du Ip xp ).
0

Note that the limiting representations in the LRD case differ from those of
the i.i.d. errors case in tw0 fundamental ways. Firstly, they have different
normalizations. If, for example, li‘rantW/n exists and is positive definite then
(W"W)1/2 is of the order of 111/“2 whereas AW is of the order of nmg/Q/Lm/gh) .
Secondly, unlike the i.i.d. errors case, the limiting distribution may not be always

normal. The value of m is very crucial for determining the limiting distribution.

If, either G is strictly monotone and continuous or G is an odd function
with the property that {x e R: C(x) g 0} equals either (—oo, 0] or (0, 00] then m=1
(See Remark 1.2.2). In such cases the first. approximation of AWLdJ-fl) is exactly
N(0, 031‘“) where I‘,,:: 7,,‘2(Wt'W)—l/2W"RW(WtW)‘l/2. R the dispersion matrix
of (711, . .nn)t and a} :2 {ff di/2(F)}‘2 [El/’(F(F))I)]2. Yajima‘s (1991) results can be

used to calculate the limit of I‘nunder some additional conditions on the design. //

Remark 2.2. Conditions (W.1) and (W .2) are satisfied by many designs, in

particular, by polynomial designs with Xnij: ij, lsisn, 13151) and by

trigonometric designs with Xnij= cos(i;1j) or sin(i;1j), 111- # t‘k forj ¢ k. //

 

29
2.3. Proofs.

To begin with we introduce some more processes that will be useful in the

sequel. Accordingly, define

FJM A) =n‘2HY Ym- e,Asyt yEL

Tum, A) z: 2: (xni- yam/n,- xffiA 5 F-1(u)) 0 g u g 1, A 6 RP.
l

The basic idea of the proof of (2.2) can be sketched as follows. Note that

(3.1) z(u, e + 11;,1 s) = 2(x x)I[Rank one, - lei/1;} s)< nu]

l

xIll

:21): x", x)I[F,,S(€i - x-tAls )<u]
l

where Fns(-) is the empirical distribution function of {6i — xniA'w1 s, 1_<_i<n}.
Therefore B'w1%(u, )8 + A,”l s) can be approximated by the process Bhflu, 3)
(defined in (3.5) below) in the sense of (3.6). Now use (3.4) below along with the
AUL of the TH process to conclude the AUL of the 55 process. To that effect, the
following preparatory lemma states the AUL of the F 11 and TI] processes which are

interesting for their own sake.

Lemma. 3.1. Assume that (1.1.1), (1.1.2), (“7.1), (W2), (A.5) and (A.6)
hold. Then for every b e (0, oo),

(3.2)311py E 1, s e N(l)){lai1llF"(y’ e + 11;} s) - F,,(y, m] - ,,I/2 inn-WIS f(y)|} = 0,,(1),

em HB‘lT MUfl+A si-Tmtm-Meuow=am.
and
(3.4) FFiils (u) ' 1 2 611(1):

30

where
,_ -l/‘2 - -
on .— n rn—iO and op(1) denotes a sequence of stochastic processes

converging to zero uniformly in 0 g u _<_ 1, s E N(b), in probability. //

Proof. We first consider (3.3). ObserVe that for fixed 3 6 RP,

fc)II[(<F'l (u 11)+xIIIAW s]

xlli

B‘w1'rn1u, 19 + M} s) = r;.1 231W1Wi'1”1x’
l

is a vector of VI, processes with 7111:: jth coordinate of (WtW)'l/2(XIII- i) and (III:
xfIiAQ} s, 1 _<_ i g 11. Notice that (“7.1) entails (A.1) and this choice of {7111} satisfies
(A2). Note also that

xI1II(W1W)'1/2” 11s11

 

 

t
x.A S <T max
lgi_<_nl m w I nl<i_<_n

1n "'””1L"‘”11nii = 011).

       

_ 1/2 f. t
_11 111521511” xlll(W W)l

so that (A.3) holds VsEN(b). Finally, to verify (A.4), observe that, by the

Cauchy-Schwarz inequality applied to the. second step,

:111w1Wi-1/11x xix :..1W‘Wi'1”1sll

lll

1ww1 ”1s12111w1w ”1x1 xi"

5 11121.3( (x
l g 1 S n 1”

l/\

 

H 1.1W1Wi'1111

151311

"S" “”1 1:||<W"Wi‘1”11x..i- @111”

= inisiii ll xltll(wtw) 1/ N 11811 111/21)/2= 0(1)-

Therefore, by (1.2.7) and the fact that B;,}Z:(x xx) XIII A1 —- Ipxp, it
l

111

follows that for all s 6 RP,

supu e10. 1] 1113311111111, 11 + A;.1 81- T1.1y. 1311 - “11111111111 = 0.11)-

 

31

The uniform convergence over 8 e N(b) is achieved by exploiting the monotonicity
of the indicator function and the d.f. F along with the compactness of N(b) as in

Theorem 2.1 of Koul (1991). This completes the proof of (3.3). Assertion (3.2) can

be proved similarly by taking 711i: 11'1/ 2, V 1 g i g n and {III as before.

For 134). note that IFF;.1.1ui - u1511111111411) - F...F;.1.1uil + 1FnsF;.1.1ui - 111.
The first term is 6p(1) from (3.2) and the fact that Fns(y) = Fu(y, B + A}: s)

Vy 6 I. The second term is at most 1/ 11. Hence (3.4) follows. //

Proof of Theorem 2.1. We first prove (2.2). Define

(3.5) $(u, s) := 2(xni- i)I[ei 3 13111301) + xiiiAiii s], 0 g u _<_ 1, s 6 RP.
I

From representation (3.1) and by the fact that

(3.6) P[ei - xfIiAgvls = ej - xijAds , for some i, j; 13 i géj g n] = 0,

we obtain that w.p.1

 

 

1311.1... + .1 - .1., .11..) .111 (Wm-w

= 0(1).

V0311gl,s€R1).

Therefore, it suffices to prove that
13.7) 811%. s) - 22.11111 - f1F“(u)) = 6.11).
Now by (3.3), 133%, s)
= B13T..1FF;.1. (u). fl + 11;) s) = B:.1T..1FF;.1.1ui. m + s fF'1FF;.1.1ui + 6,11).

Therefore using B§1§5(u) = Bngn(u, 3), the uniform continuity of the function
foF‘1 and (3.4) it is clear that (3.7) will follow from the tightness of {B'JTn(u, ,6),

u E [0, 1]}. Therefore, it remains to prove that V a > 0, 3 6 > 0 3

(3.8) lim supIIP[ sup “B-“HTnhl, [3) - Tn(v, ,6” H > a ] < (Y.
)u-v)<6

 

32
But (1.2.4) applied p times, with the choice of {7111} and {5m} as in the proof of

(3.3), yields that

139) an E 10. 1] 1113-1111:, 11) - tum-1111113111 = 0.11),

where E "3,111 .-= 0(1).

Note that for u 5 v e [0, 1], the Cauchy-Schwarz inequality yields that

13.10) lJm(F’1(V)) - Jm(F"(u))1 s E1/11I1u<F1ei 3 vii‘1 E1”1H;1..1nis 1mi1v-uii1”1.

Hence (3.8) follows from (3.9) and (3.10). This proves (2.2) also. (2.3) follows from

(2.2) using an argument similar to the proof of K0111 (1992, Theorem 3.2.2). //

Proof of (2.5). Let 23(11) :2 2(XIII- it) [1(6i < F‘l(u) - 11], 0 _<_ u 31. Note
I
that because of the continuity of F,

(3.11) P[ 23(11) = %(u), for some 116(1), 1) l = 0

. 1
Since S = - f 53(11) (ll/(11), (2.5) follows from (3.11) and (3.9) and the. boundedness
0

of the function 1/). //

CHAPTER 3
MINIMUM DISTANCE ESTIMA TION
2.1 Introduction.

In minimum distance (m.d.) method of estimation, one estimates the
unknown parameter by a minimizer of some discrepancy measure between a
function of observations and that of a family of underlying distributions.
Wolfowitz (1957) discussed m. d. estimation as a general principle of estimation
that can be applied in many statistical problems and showed that under
identification and continuity assumptions, m.d. estimators (m.d.e.) are strongly
consistent. Research on m. d. estimation proliferated during the mid seventies and
eighties. Beran (1978) discussed asymptotic normality and robustness of the one
sample location estimators obtained by minimum Hellinger distance. Parr and
Schucanny (1980) and Millar (1981) discussed asymptotic normality and
robustness of a large class of m.d.e. that includes Cramer-Von Mises type
distances in the one sample location mo1ilel. Boos (1981) discussed minimum
Cramer-Von Mises type distance estimation in the. one and two sample models
and its application to the ‘goodness of fit’ tests. The above authors, in some way
or another, viewed m.d.e. as a functional defined on an appropriate subset of
univariate distribution functions, that satisfies Frechet differentiability condition
and proved the asymptotic normality of these estimators by techniques delineated

in Serfling (1980, Chapter 6).

Koul and Dewet (1983) and K0111 (1985a, 1985b) visioned appropriate
extensions of m.d.e. from the one and two sample location models to the multiple

linear regression model via weighted residual empirical processes. Their technique

33

34

of proving the asymptotic normality and qualitative rolmstness of the m.d.e. is
completely different from that of the predecessor’s work. It uses the asymptotic
uniform quadraticity of the Cramer-Von Mises type statistics based on weighted

empirical processes.

While most of the asymptotic literature on m.d.e. assume independent
observations, little seems to be known for the dependent observations. Motivated
by the important application of linear models with LRD errors, we study in this
chapter the large sample behavior of m.d.e. of the regression parameters under the
model (1.1.1). In Section 2 we derive the limiting representations of the
normalized m.d.e.’s. The following empirical processes with weights {wni:= )in - i,
1 g i _<_ n} and {x1

1i? 1 g i g n} are useful in this chapter.

(11)VW(A9 Y) :: aniI(Yni S xti'A + Y) :: Z: “nil“:i S XL (A ' fl) + Y)
I‘w(Aa y) I: ZIWniF(X)',i(A-fl) + 3')

WA, y) =-—- 2w,,i{1(eisx,‘,(A-m+ y) - F(x‘.<A - fl) + n}, Ae RP, yeR‘.

Ill

Similarly, define Vx, ”X and UK with weights {xni}.

For linear regression model, \Villiamson-Koul minimum distance estimator

of the regression parameter [9 is defined by

(1.2) BK: argminA 6 up MW(A),
where

Mw< A) z = 1.9"” (W"W)"/'-’v,..( A. y) Ill (1y.

Under the assumption that the error d.f. F is synnnetric around zero Koul

(1985b) defined, for each H, a minimum distance estimator of fl by

35
(1.3) fl; := al-glllillA E RPMX(A) ,

where

[H

Mx(A):= rn~21[)(X‘X)“/2{ 15’9““ - am 5 y) - 1m, + xtfiA < n}

 

[Zde

and H is a nondecreasing right continuous function from R1 to R1.

Note that [9K and [3; are the estimators 8 and [3+ respectively defined in
Koul (1985b) for the independent errors case. The motivation for considering
these estimators and its finite sample properties are discussed in Koul (1985b). In
particular, as noted by Fine (1966), for p = 1 with xni = 1, 1 g i _<_ ml, and xni
= 0, n1+l g i g n, 3K reduces to the two sample Hodges-Lehmann estimator of
the location parameter. Also, for Xni = 1, H(.\') = x [ H(x) = 1(0 3 x) l, )6; is the
Hodges—Lehman!) estimator of the one sample location parameter [ median

estimator ] (see Koul 1985b).

3.2. Asymptotic representation of minimum distance estimators.

In this section we first derive the asymptotic uniform quadraticity of
Mw(A). To that effect, apart from (“7.1) and (W2) in Section (2.2.2), we need
the following assumptions on the model (1.1.1) and (1.1.2):

(W.3) For every 8 6 RP, I" 13’“; pww + A;,} s) - s f(y)”‘2 dy = 0(1),
(D.1) The d.f. F of 6 has a continuous density f.
(13.2) [f2 dy < oo.
(D.3) IF (l-F) dy < oo.
Define,

 

(MA) == I HB'w‘vww, y) + BJW‘ X (M) fear-11y, A 6 RP.

36
Theorem. 2.1. In addition to (1.1.1), (1.1.2) assume that (W.1) -

(W.3), (D.1) - (D.3) hold. Then, for all b E (0, 00),

(2.1) E sup |Mw(fi+ A?) s) - wa+ A'w‘sn =00). //
sEN(b)

Proof. The technique of the proof is similar to that of Theorem 2.1 of Koul

(1985a). As there, it is enough to show that V b e (0, 00),

(2.2) E sup I "BJUMfl, y) + s {(y)|[-’ dy = 0(1),
86 N(b)
(2.3) sup JHB—vi pw(,3 + A'“! S) - 8 fly)“:2 dy = 0(1),
86 N(b)
(2.4) E sup IIIB'Wl use + A‘wl s. y) - Bw-luww, y>|F<1y = 0(1).
86 N(b)

Proof of (2.2). Using H a+bII 2 _<_ 2 (Hall? + II MI 2 ) for a, b e Rp and Fubini,

1.11.3. of (2.2) 32 [Elle-wl wa, ”If? (1y + 2 b2 If? (1y.
. P
New. IE || 13;: we, will dy = r..'-’,21IE[:aW..U {1ei 5y) - Fem? dy,
J: I

. - - t -1 2
where, for any matrix D, adnij 2: the Jth entry of the vector (D D) / dni'

Using the Hermite expansion of {I(cigy) - F(y), yEI} in (1.2.3) and the

properties stated in (1.2.1) the integrand of the jth summand is equal to

- 7 I' I
= Tn 2 El: gaw'fij OWNJ (1 gm (“gm qu) l/(l! Jq,(y)/q ‘ EHq(”1)HqI("f)

z: Maya/q!Jen-r).

: -2 Cl " 0' '
Tn ? Er: me WHI‘J (12m
2 m '_ . ‘2
< 2 Jg(y)/ql nlmsaiitsna wnij XXII) (1 1)l/ mu

37
(2.5) s F(Y)(1-F(Y))nlma2< ...(W‘WW MWL-"‘<n>>:>:1pm<i-r>l

$1511

(2-6) S K F(Yl(1' PM)

Here, (2.5) is obtained from X) J3(y)/ql = E[I(€;SY)'F(Y)]2 = F(Y)(1’F(Y))
q_m

while (2.6) from (F.1) and (W..2) Now (2.2) follows from the assumptions (D2)
and (D.3). The assertion (2.3) follows from (W.3) and a monotonicity argument of
Koul (1985a, Theorem 2.1). //

Proof of (2.4). For fixed 3 6 RP,
EIIIB'wl mm + As‘ s, 11)-131.! wa.y>lr+’dy
S2IEllel Sw(fl + Aw.l S5 3') ' Bid Sw(fls 3)”.1 (1y

mi x Aw 8 +3) ' Jm(Y)}(1n!)-1H111(7li)

   

+2

 

[2 dy

Ill

             

where, for A 6 RP, y E R],

Sw(A,Y)==ZVni{I(€i SXffim'flHW-F (x X,’,,(A fllfl’H 111( X.‘,,(A- -fl)+Y)(m' ) 1H111(71i)}
I

From (1.2.3), T1 is equal to

p 2
2 Ira-2 E[ raw",- :5“ Jay, x,.iA;J s + Y)(<1)Hq(n;)] dy,
j=l 1 q _ m

where for a function h : Til—+111, 11(x, y) stands for h(y) — 11(x), x g y.

Now using the properties in (1.2.1) as in the proof of (2.2) the jth integrand

of the above expression

Tn-Z Z: Zlawnija wnrj Iq '2 J( “(y xiliAw S+Y)'](I(Y9XIII'AW1 S+y)/q! l lpn](i_1~)|
Zm+l

38

-2 2 _ :-
0" 25 ¥'“wni1'“wmj 1,11%“ 310% KEN s + y)/q!]l/’

[>2 13W, xtrAQJ s + 11)/(111W lpm(i-r)l
<1> m+1

3,1th )1" n"‘"'"”’L-m<n)z: nee-m.

|/\

max F xt-Aw l
lgign (y, m s+y) 111m<axSn

|/\

1 -1
K fusaisn F(y’ xniAw S + y)

Using (1.2.1) once more, it is easy to see that the integrand in T2 is

bounded by K max 1,2,,(y, x’ A}, s + y) which by the Cauchy-Schwarz

1 <1 <n '”
inequality lS bounded by K max F( y, x A“, l s + v).
I <1 <11 “' V
_ tw -l '2
Note that lmSaX<n | xfliAiNlslgr,,l111sz11i<S"”xni(W W) / H ”SH
1 2 -l - 0 ’ 2
(2.7) = 11 / 1.1-151,3“)!11 xf]i(W‘W) /'-’ ”0(11 m / LW (11)) =o(1),

by(W.2) and the fact that for any slowly varying function V and V6>0, V(n)/n‘5 :

0(1) . Therefore, with an :: 1"ng l xltliAw1 s s,| Fubini’s theorem yields that,
I II
y+an
1 max |F(x .A W‘s + y) - F(y)| dy 3)) 1 {(1)11111151 = 2 an -._.— 0(1).
1 513 n "i y—an

Again the uniform convergence in (2.4) is achieved in a routine fashion, see, e.g.,

Koul ( 1985a). This completes the proof Theorem 2.1. //
The following corollary gives the asymptotic. representations of the suitably
normalized minimizer of MW(A).
Corollary 2.1. Under the assmnptions of Theorem 2.1,
(28) A113,, - 19) =- (If2 dx)‘l 13;: 1w, 1) dF(y) + ope),
(2.9) =- (If? dx)-lsw1Jm<y) dF<y) + ope),
= (If? (11)-1 swEFemmu) + 0,,(1). //

39

Proof. Proof of (2.8) is routine from the asymptotic uniform quadraticity
result (2.1) and hence omitted. For (2.9) note that
"1131} W, y) - stmon <1F(y)lr~’
s supl HB'W‘ Uta/1y) - stm(y)lF = op( 1),

y E
by (1.2.4) when applied p times,j th time with 711i = awnij’ 1 g j g p. //

Remark 2.1. In the independent errors case Koul (1985b) derived the
asymptotic representation of BK, which when specialized to i.i.d. errors case states

that under the conditions (W.1), (W*.2) (W.3) and (D.1)-(D.3)
(WW/WK - fl) = —(1f‘-’ dx)-‘<WLW)“/‘-’1UW<H y) are) + ope),

where (W*.2) is as in Remark 2.2.1. Therefore, remarks similar to that of Remark
2.2.1 is also applicable here. Also note that condition (W.3) is satisfied in the
presence of other conditions when for example, the density f is uniformly

continuous.

To cite an example of error distribution under which the limiting
distribution of 3K is nonnormal, consider C(x) = x2 - 1. In this case 111 = 2 (see
Dehling and Taqqu, 1989). Hence from the representation (2.9) it is clear that in
the case of the two sample (111, n._,) location model for the centered chisquare
random variables with design vector, X111 : 1, 1515 ml and xni : 0,
n1+1 5 i gnzz nl+112, “0 L’l(n)(/3K - /3), the normalized two sample Hodges-
Lehmann estimator of the location parameter converges in distribution to the
random variable [)1(1-A)]’1{Z2()1) — AZ2(1)}, where Azz lim(nl/n) is assumed to
exist, O<A<1 and {Z2()1):)\ 6 [0,1]} is the Rosenblatt process (see Dehling and

Taqqu, 1989). Clearly, the limiting distribution is nonnormal. //

40

Now we turn to the asymptotic representations of the m.d.e. fl; under the
symmetric error distribution. For the sake of clarity we assume that the
integrating measure H is symmetric around zero. The. techniques of proofs are

similar to those of Theorem 2.1 and Corollary 2.1 and are omitted.

To derive the asymptotic uniform quadraticity of the dispersion Mx(b)
and the limiting representation of H}: we assume the following conditions on the

design matrix and the underlying distribution:
(X.1) (XtX)'1 exists for all n 2 p.
(X.2) n {n<a)i(< nxi1i(XtX)-lxni : 0(1) .
(X3) For every seRP,IllB;31Jx</3+ Ax-‘s, 11)-1m. 11)] - 111(1)le dH<y) = 0(1).
(D*.1) The d.f. F of 6 has a continuous symmetric density f.
(D*.2) (i) 0 <If dH <00, (11) I) < )1“ (1H <00.

0 0

(D*.3) o <°)°(1-F) (1H <00
0

(N4) 8113), MW) dH<y) =1f<y> dHu).

Theorem 2.2. In addition to (1.1.1) and (1.1.2) assume that (X.1)-(X.3)
and (D*.1)—(D*.4) hold. Then

(2.10) A1091- fl) = - (2112 (1111113331{UaHvHUan-ynfiy) dH<y) + ope).
=- (211‘2 dH)'leI[-l111(y) +.1.-.,<—y)1f<y) dH(y) + ope).

= (21f2 dH)-‘s,. EP(()H1.1(1I) + 0,11),

41
where,

X
p(X) == po(x)-p0(-x), p0(x) ==_))Ode- //

Remark 2.1 From representation (2.10) it follows that in the case of one
sample location model for symmetry, the normalized Hodges-Lehmann estimator
nm0/2 L'm/2(n)(flil(' - [3) converges in distribution to the random variable
()1? dx)-1 EF(c)Hm(1)) zmu), where {2”,(1) ; 1610, 1]} is the mth Hermite
process as defined in DT. //

CHAPTER 4
REGRESSION QUAN TILES AND RELATED PROCESSES
4.1. Introduction

In their fundamental paper, Koenker and Basset (1978) (KB) introduced
regression quantiles as a natural extension of the notion of sample quantiles and
order statistics from the one sample location model to the multiple linear
regression model. Most of the asymptotic literature on these assume either
independent errors or weakly dependent errors as in Portnoy (1991). In this
chapter, we study the large sample behavior of the regression quantiles processes
under the dependent setup (1.1.1). We also investigates the asymptotic behavior
of the regression rank-score processes, L- and linear regression rank-score statistics
of Jureckova (1992a) and Gutenbrunner and Jureckova (1992) under the LRD
setup. In particular we obtain the AUL of the regression rank-score statistics

under the LRD errors.

For an a E (0, 1), KB defined an em regression quantile as any member

[911(0) of the set

(1.1) 13,,(0) :2 {be RP: 31: 110(Y11i - xt‘ b) = minimum},

111
1:1

where ha(u) := a uI(u>0) - (1 - (1) 111(115 0), u E R1, 0 _<_ u g 1. Note that, hl/2(u)

1' I u I /2 and so, 13,,(1/2) reduces to the well-known least absolute deviation
(LAD) estimator of ,8.
Theorem 3.1 of KB gives the following linear programming version of the

above minimization problem (1.1):

43
(1.2) minimize a 1:“, r+ + (1 - a) 1), r'

subject to Yn-Xb=r+-r' , (b, r+, r') eRpx 1x11",

where 1:, z: [1, ..... 1] 1 X" and Yn is the response vector. By the linear
programming theory, 13,,(0) is the convex hull of one or more basic solutions of
the form

(1.3) bh = 11,311,,

where h is a subset of {1,2...n} of size p and X11 ( Yb ) denotes the sub-design

matrix ( sub design vector ) with rows x)“, i e h ( co-ordinates Yni, i e h ).

In general, one can choose 3“.) from (1.1) in such a way that it is a
stochastic process called regression quantile process that has sample path in [D(0,
1)]p. There will also be ‘break-points’ 0 = (10 < a, < < OJ“: 1, such that
3“.) is constant over each interval (oi, “1+1)’ 0 g i g .ln-l. See Gutenbrunner and

Jureckova (1992) (GJ) and references therein for more. on this.

The corresponding dual program, 111entioned in the appendix of KB is the

following:
(1.4) 111axi11iize. Y),a with respect to a
(1.5) subject to Xfla = (1-(1') Xhlna a6 [0, 1]".

GJ investigated the statistical properties of the optimal solution of (1.4)
when the errors are independent. Note that a maximizer of ( 1.4) also maximizes
al'(Yn - Xt) subject to (1.5), for any t E RP. Now choose t = 3,,(0), where for some
p-dimensional subset hn(cr) of {1,....n} an optimal solution for (1.2) is given by
311(0) = X31191) Yhn(0‘)' Then one particular solution of (1.4) corresponding to

this choice of 3,,(0) can be given as follows:

44

For i not in hn(a), let

A

(1.6) 5111(0) .-= 1, Y", - xgis,,(a)>0,

A

01 Y ' ' xt ,6,,(a)<0,

111 111

II

and for i in hn(oz), 51,],(0) is the solution of the p linear equations:

1.7 =1- " .-" .IY.-".‘,,- >0.
( ) jEE}(O)xnjanJ(a) ( 05:511an jglxn‘l ( 11) anfl(n) )

GJ call 5,,(0) the regression rank-scores for each a e (0, 1). When p =1 and

Xni '2 1, 1 Si 5 11, GJ observe that

(1.8) 3111(0) = 1, (1 < (Rm - 1)/n
2: R111 - 1101, (R111 — 1)/n 5 a 3 RM /11
= 0, Rni /n < a,

where Rni := Rank(Yni). Hence in the one sample location model, the processes
{sni(.), 1 Si 3 11} reduces to the familiar rank process (See Hajek and Sidak, 1967,
Section V35). GJ observed that 5,,(.) satisfying (1.6) and (1.7) can be chosen in
such a way that it has piecewise linear paths in [C(O, 1)]n and
5,,(0) = 1n = ln- 21,,(1). See GJ also for a discussion on the generalization of the
duality of order statistics and rank process from the one sample location model to
the linear regression model by the RQ (z: regression quantiles) and RR (:2

regression rank-scores) processes.

Using these processes one can construct various statistics such as L-
statistics and RR statistics that are useful to make inference about the regression

parameter 1?. In the context of i.i.d. errors, different types of L-estimators were

45
proposed by Ruppert and Carrol (1980), Koenker and Portnoy (1987), Portnoy

and Koenker (1989) and GJ. Among them, Koenker and Portnoy (1987) and GJ
considered smoothed L-estimators that are asymptotically equivalent to the well
known M-estimators but have the added advantage of being invariant with respect
to scale and reparametrization of the design. In this paper we shall consider these
type of L-estimators and will observe that under appropriate conditions, the above

asymptotic equivalence continues to hold even in the LRD setup.

As pointed out by Jureckova (1992a), one of the major advantages of using
RR statistics based 011 the residual is that the corresponding estimators of the
some of the components of 5, when others are treated as nuisance parameters do
not require the estimation of the nuisance parameters. The basic result needed to
study these estimators is the AUL of the RR statistics based 011 residuals as given
in Jureckova (1992a) for the i.i.d. errors. The corresponding result under (1.1.1)
and (1.1.2) is given in section 4. Section 2 obtains the joint asymptotic
distribution of the finite number of suitably normalized RQ ’s and asymptotic
representations of RQ and RR processes. Section 3 applies this results to yield
the asymptotic behavior of L- and RR statistics. The proofs heavily depend upon

the uniform closeness result of Chapter 1.

4.2. Theorems and Proofs.

To find out the asymptotic distribution of RQ, we first define the following

minimum-distance type estimator of 6 by

 

2
9

(2.1) ,amd(01) Z: argminA "(X50442 ‘.I(A, 01)

where,

46
€(A, a) := 2x11i{I(Y11i' x11iA _<_ 0)- a}, 0 5 a 51, A6 RP.
I

By the continuity of the d.f. F, SKA, a) is an almost everywhere

differential of the function 2.3%de - xt-A) with respect to A. Therefore,
I

DI

intuitively, minimizer of Zha(Yni - xi1iA) and f9md(a) should be asymptotically
I

equivalent. The following lemma is enunciated towards this direction. It also gives

the joint asymptotic representation of the finite. number of RQ. To state it, we

need to introduce

13(01):: [9 + F 1(.1( (1)01, 01 2: (1,0 ”0)", and (1(0) := f(F'l(a)), 030' g 1.

Also, recall the definition of Sx from (3.2.8), conditions (X.1), (X.2) from Theorem
3.2.2 and conditions (A.7), (A8) from Theorem 1.2.2. Moreover, the following
design condition is assumed:

(X.0) The first column of the design matrix X consists of one only.

Lemma 2.1. Assume that (1.1.1) (1.1.2) and (X0) - (x2) hold. Then
(i) sup, 6 [0, 1] "Bruno. 0) - J...(F"(a))sxll = 0.11)
(ii) If, in addition, (A.7) hold.- then )1 1.) e (1), 00) 11.11.1111 a e (0, 1),
sung/(1.) “131mm )+ A3.l s a) - 21mm), «)1- s<1<a (11)): o, (1),
(iii) If, in addition, (A.7) and (A.8) 11,0111 111.1311. )1 a e (0, 1/2) 11.11.11 v 1) e (0, 00),
sup

ae1a1-1)a1se111. llB'11rWHA‘s 0)- H<H<a).o))-sq(a)ll=op(1).

(iv) Under the conditions of (ii), for every (1 E (0, 1),

47
(a) Axmmdw) - fi(a)) = - c1110) Bra/3(a), a) + ope)

(b) Anise) - 19mm» =- ope).
(c) Consequently, for any 0 < (11 < (1.3, ...., < (1k < 1,

(2.2)

where, for any two matrices A and B, A e) B stands for its Kronecker product.

//

Proof. (i) follows from (1.2.4) when applied p times, j"h time to 7111 2' jth
coordinate of Dilxni, 1 g i g n, 1 gj g p.

(ii) Note that

Bglmaa) + A339», a) - wan), 0)) - s ,(..)

= 7;} gngxmnu, g F‘1((1)+ xf‘liA;,ls) - 1(6i g F‘l(n~))} - 3 (1(0)
1

Now (ii) and (iii) follow from (1.2.9) and (1.2.10) of Theoren'1 1.2.2 respectively, as
in the proof of (2.3.3).

(iv)(a) Note that v aE(0, 1)., v oeRP, omen/1(a) + 111,119, a) is an
increasing function of r > 0. This fact together with an argument like the one

given in the proof of Theorem 2.1 below implies (iv)(a) in a routine fashion.

(b) First we show that for every 0 < a < 1,

(2‘3) “Bxlgifihiah 0)” 2 011(1):

48

From Theorem 3.3 of KB, we have the following algebraic identity:
(granny...- xgifiaa) s 0)-a) - 2: Jenna... xrfima) 3 0101111,)“, = was),
1 i E hn(a) "
where, hn(a) is as in (1.6) and each element of the pxl vector wn(a) belongs to
the interval [a -1, 01]. Hence
Bkl‘fl'(Bn(a), 0) : Bid . Z xniil ‘ a) "I“ Biclxi, (a) W11(0)-
1 E hn(a) 11
Therefore, (2.2) follows from (2.3) and (X2) by noting that the right hand side is
t -l _

bounded by 2 p lmsaiicS n ”xnin I) — 0(1).

Now the claim in (b) follows along the same line as in the proof of (2.5) below.

The claim about (c) follows from (b) with the help of Cramer-Wold device. //

Remark 2.1. An interesting special case of (2.2) is when k :1 and (11: 1/2.
Here regression quantile reduces to the celebrated LAD (least absolute deviation)
estimator. As an example of errors when the limiting distribution of the LAD
estimator is nonnormal, consider the model (1.1.1) when the error r.v. is a chi-
square centered at its median, i.e., C(x) : x2 - v, where v is the median of x? r.v.
Then, as shown in DT, 111 equals two. Hence in the one sample location model
with chi-square errors centered at its median, it follows that the asymptotic
distribution of the LAD estimator is the same as that of u.{nl ' 6L(n)}'l 2(1)? - 1)
which converges weakly to the r.v. 112.2(1) where {Z2(a), C16 [0, ll} is the

Rosenblatt process as in example 2 of DT. //

Remark 2.2. A very intriguing pl‘ienomenon is observed regarding the
asymptotic efficiency of different estimators when the errors are exactly Gaussian,
i.e., C(x) = x. In this case, in equals one. and all the estimators discussed in this

thesis are asymptotically normally distributed. lVIoreover, (2.2) implies that the

49

asymptotic dispersion of the regression quantiles ([9,,(01); a e (0, 1)} is independent
of a. In addition, it is same as the asymptotic dispersion of the least squares
estimator, M-estimators and that of Koul’s 111. d.e. flit, irrespective of the scores —
functions (in the case of M—estimators) and integrating measure H (in the case of
m.d.e.). Similarly, in this situation, R-estimators and Koul’s m.d.e. [9K continue
to have the same asymptotic dispersion irrespective of the score function C. This is

in complete contrast with the i.i.d. errors case. Finding an estimator with better

asymptotic efficiency in the case of Gaussian errors is still an open problem. //

Remark 2.3. Recall that in the i.i.d. errors case, under (X0), (X.1), the

assumption that lm<ax x:i(XtX)’lxni = 0(1), and (A.7) one obtains that V a e (0, 1),

_1Sn

A

(2.4) Dx(fin(0) - Mal) => Nl01 Ipxp 0 (1 - 0’) (1'2(0)l W-

This essentially follows from the work of Koul (1992, section 5.4) by viewing
fin(a) as an M-estimator corresponding to the convex score function 110. Under
stronger conditions on the design matrix it also follows from KB. Hence, remark

similar to that of Remark 2.2.1 is also applicable here by comparing (2.2) and (2.4).

In the following theorem, we depict the asymptotic representation of the
regression quantiles process uniformly over the compact subset of (0, l). The basic
idea for its proof can be found in Jureckova (1971) and Koul (1985). Here, for an
Rp valued stochastic process {Xn(a), a E [0, 1]}, “anla := sup{ "Xn(a)" ; a g a S La}
and we say that Xn = 03(1) ( 0;“,(1) ) if H X” IL, = Op(1) (o,,(1)) for every a E (0, 1/2].

Theorem 2.1. Assume that (1.1.1), (1.1.2), (X.0) - (X2) and (A.7) - (A.8)
hold. Then

(2.5) Ax<3....1(a)-A(a)) + (Ma) Bin/1(a), 0) = 0;;(1).

50
(2.6) Ax11‘3..(a)- ammo) = 03(1). //
Proof. The proof will be given in several steps. Let D,,(a) denote the set of
minimizers of "D;l g (A, a)". Note that H D,1 ‘5' (A, a)” can take at most 2"
possible values and the set Dn(a) is nonempty for each a in (0, 1 / 2]. Now define
Ana) by AxlAaa) - 19(0)} =-—= - <1"(a) Bra/3(a), a). T11 to1rove (2.5), it

is enough to show that for every a in (0, 1 / 2],

(2.7) sup, 6 ,asupA E W, ll AAA-[1(a)] - Adina) - [3(a)] H = ope)

Step 1. "133} '5'(A,,(a), A)" = 0;;(1).

Proof. Follows from the above Lemma (2.1)(iii) by observing that
AX1A..<a)-H(a)) = 0

Step 2. 1111“ R, ”13;,1 am, a)" = 03(1).

Proof. Immediate from Step 1.

Step 3. For a b>0, define

Ob(a):- — {AeDn(a );x||A( A [i((1))”<b}.and

(),(a 21;.) — {AeD,,(a 1;,("11 A an 1)” > 1.)
Then v M > 0,

         

))(|> 11 0,,(n e) #oVo'Ela] =0(1).

   

(2.8) P[ snpa E I a'supA E 0b(

II(

Proof NotethatA(AA,, (1 = A,[A-(as )-q]-( 1(a))[n;,121(s(a),a)].

Therefore, the left hand side of (2.8) is less than
P[supa E IasupA E 011(0) ll Ax[A-fi(a)] - (1,-1(a)) [B;1€r(/3(a),a)]))”> M,
SUPGEIHHPAEO((1) (|(1'1((1)3(IB ‘.I( A, a) l” < M/2, 0b(0 (1') #4) V0613]

+P[sup(316IaISUPAEO()(1llq.1(a)13xlzrl(A 0) )|| > M/2 0,,(0 a) ¢¢Vaelal

51
s Ptsupa 61am”, 6 ohm Hq“(a)B;.‘[‘I(A, a) - Wu), an - AAA-Ma» M > M/2,

Ofia)¢¢Vaeg]
+ P[supa E [asupA 6 Ohm) ||(1-1(a)B;,'cy(A, a) H _>_ M/2, ohm) ¢ ¢ Va 6 Ia].

Now, the first and second terms are 0(1) by Lemma (2.1)(iii) and Step 2 respectively.

Step 4. Given 6, M >0, 3 6>0 and "0 such that V n 2 110,

(2.9) P[infa E Iainfusuz a ”Wm/3(a) + A338, 0)" > M] > 1 - a.

Proof. The polar representation of vectors, the Cauchy-Schwarz inequality
and the fact that V 06(0, 1) and V 96R", HtB;1[€I'(fl(n) + rAQO, a) is a

monotone increasing function of r > 0, yields that for any 6 > 0,
. -1 -l
1anle 6 "13,( mam) + AK 8, a)“

3 infr2 6 arm : 1 otBflsmm) + Age r, a) 3 inf othflerwm) + A29 6, a).

l|0||=l

Therefore, using Lemma 2.1(iii), for all s1.1fficiently large n, the left hand side of

(2.9) is not less than

P[infa otngflma) + A330 6, a) > M]

e lawfnoll = 1

2 P[infa E lainfflon : 1 [at-Bgonma), a) - 6 (101)] > M + 1] - 6/2

2P[ 6 inf (1(a) - inf a'n-‘wam, a) > M + 1 -5/2.
(1 61a x

a e lamrnon = 1

Now, using infa E [a q(a) > 0 and infCt 6 lainfl O‘B;19'(fl(d), a) _-_-. Op(1), we

ION-:1

can choose 6 sufficiently large so that the above probability is not less than 1- e/ 2.

52

Step 5. Given 6, M >0, 3 no such that V n 2110,
(2.10) P[supa E [asupA 6 011(0) “mm-{3(a)} - AX[A,,(a) - ,B(a)]” > M] < 6.

Proof. By Step 4, choose 6 and 11 large such that P[ 66(0) 7t (25 for some a
in Ia] < 6/3 . Then the left hand side of (2.10) is less than

P[supa E IasupA 6 W0) ”Ax[A-fi(a)] - AX[A,,(a) - pm] ll >M, 0,,(a) ¢ ¢ Va e la]

+ P[()6(a) 75 43 for some a in la]

 

S P[Supa E [aSUPA 6 06(0) [AAA-[“0” ' AxlAn(a) ’ fl(n)l ” >1“, 06(0) ¢ 45 V0 61a]

+P[supa E [asupA 6 55(0) “AX[A-fl(a)]—AX[A“(G)-[i(a)] H > M, 05(0) ¢ (:5ch e lam/3.

The first probability is less than 6/ 3 by Step 3. The second probability is less than
6/3 by the choice of 6. This completes the proof of (2.5). Assertion (2.6) can be

a:

following the same lines by observing that “Dying/a), a!) H = 0,,(1). //

We now turn to the RR processes. To define these, let {Cni’ 1 _<_ i g n} be a
triangular array of p x1 vectors and let Cn x p be the matrix with rows cai,
1 g i g n. Define sequences of weighted RR processes by

03(0) 3: Zc.,;{ft..i(a)- (1' (1’)}, 0 S a S 1,
and an approximating sequence.l of weighted empirical process
Uf',(a) := ZCniUifi > F'l(n)] - (1 - 0)}, 03051.
For convenience we now recall the following algebraic identity from (5.15) of GJ.
(2.10) 5M0) - (1 - a) = 1(6, > F'l(o)] - (1 - a)
-{I[e, 5 Ma) + Xf,i(/9..(a) - fl(a)) - He, 3 F"(a)1}

+ 5mm) IlYni = xii/Bum”, 13 i 5 n, 0 g a g 1.

53
This identity is useful in approximating Bf, by U3. The following theorem gives

the desired asymptotic representation of LIE.
Theorem 2.2. In addition to (1.1.1), (1.1.2), (X.O) - (X.2) (A.7) - (A.8)

assume that (C.1) - (C2) hold where,

(C.1) (CtC)'l exists for all n 2p. (C2) 11 max c-'1(CtC =0 1 .
l < i < n 6'“

Then the regression rank-scores process admits the following representation:

(2.11) 3.1mm - U§<a>1 = Be‘C‘XAglAa/‘mm - 2(a)) q(a) + 0;;(1),

and
(2.12) Be‘foa) = - Set-new» + 0;;(1).
Consequently,
(2.13) Erma) = - (S. - Bg‘C"XA;.’SXJ tum-1(a)) + 03(1). //
Remark 2.4. Let g :: BglC‘XAQ. Then from its definition 9 = Dglc‘xng.
Hence by the Cauchy-Schwarz inequality 9 is bounded. //
Proof of the Theorem. Let En( Ax[fl,,(n -fl(o)]. From (2.10) we obtain that

Bette) = Be‘foa) - gEm) q<a>
-(E;‘:c..,{1[e, s E 1m )+x.,,A;‘E.a( > He, s E-‘<a>1}-9En(a)q(a)1

+ BE] 2: )Clllalll(a)1[\7ni—_ xni('Bn(a 0)]

iEhna

=Bg‘U;(a)-9E,.(a) 1(a)- R1(a)+E)(a ) (say).

By Theorem 2.1, (2.6) and Remark 2.4, R1 2 03(1). By (C.2), supa E [0, 1] “19(0)”

= 0(1) almost surely. Hence (2.11) follows. The relation (2.12) follows as in the

proof of Lemma 2.1(i).

54
Remark 2.5. As in Remark 2.1, the nature of the approximating process in

(2.13) is quite different from that in the i.i.d. errors case. The leading r.v.
(214) Z := -[S. - 1330"“:le J...<F"(a)>

in the right hand side of (2.13) is a product of a random quantity, independent of
a, and a nonrandom continuous function of a, which also depends on the Hermite
rank m. If m =1, then Z is a multivariate normal r.v. with mean 0 and

dispersion matrix proportional to
(2.15) Bglcta - X(X‘X)‘1X) R (I — X(X‘X)'1X)CB;1,

where R is the dispersion matrix of (771,712. . .1'/,,)’. For 111 other than one the
limiting distribution may not be normal. Also note that unlike the i.i.d. errors

case the limiting distribution may not be distrilmtion free. //

4.3. L—estimators and regression rank scores statistics.

In this section we derive the asymptotic distribution of smoothed L-
estimators based on RQ processes. For a finite. signed measure 1/ on (O, 1) With

compact support, an L-estimator of ,6 is defined by

(3.1) Tn: [9,,(a) (1,/(a).

c%-—

Note that for u with dI/(a) = I(aga g 1-a)da, Tn reduces to an analog of the

trimmed mean. The following theorem is an immediate consequence of Theorem 2.1.

Theorem 3.1. Under the assumptions of Theorem. 2.1, with u :2 1/(0, 1),

(3'2) AxlTn ' fill ' e1 iF-I(G)d"(“)i : ’SxiJndF-lini) (14(0) (ll/((7) + 0})(1)' //
0

55
Remark 3.1. Consider the linear model (1.1.1) where now {6i} are i.i.d. P.

Let 1,!) be an absolutely continuous function from R1 to R1 such that [wdF =0, 0
<I¢2dF<oo and its almost everywhere derivative 112’ satisfies 0 < Iw’dF,
f (II/)2dF <00. Then from GJ, it can be deduced that if c has a continuous
positive density f, then the normalized M-estimator corresponding to 1/2 and an L-
estimator corresponding to the measure 1/ given by the relation dI/(a) =
t/J’(F'l(a))/Ew'(e)da coincide asymptotically. The same phenomenon happen in

the LRD setup with the same choice of the measure V.

To explain this further, suppose that the 111 in (1.2.2) is also equal to the
Hermite rank ml of 2/2(G(7/)), i.e., m = inf{k31: Ey”(G(1}))Hk(1])7£O} and w is
constant outside compact. Koul (1992) showed that under the assumptions that the
function z—oEld/(e - Z)-1/”(6)| is continuous at zero, (1.1.1), (1.1.2) and (X.0) — (X2),
(33) AM- fl) = [Ew‘v’eH‘Sx are) + ope),
where [9 is the M-estimator of ,8 corresponding to w and JmW’) :2

Edi(C(n))Hn,(1)). Note that with (ll/(n) = [Ee‘5’(c)]‘l2/"(F'l(a)) do,

Jm(F"(a)) q“(a>du<a)

C'qi-fi

1
= [Ex/“(en‘lEHmen(new) s Elm» tl*’(F"(0)) (14(0) .10 = [Er/“(6)1“and).
Hence from (3.2), we readily obtain that

Ax(Tn - fl - c1[ET/”(6)1-1Elfll‘llfll ) = 'lE’r/'I(‘c)l-l Jm(?/’) Sx + 0p“)- //

Next we turn to the. linear regression rank-scores statistics as defined in

GJ. They are of the form VIC, = chibni, where {hm} are the scores generated by
1

. l
(3.4) bni: -£l)(0) (121mm), 1 g i _<_ n,

56

for a function b from (0, 1) to R1 which is of bounded variation and constant
outside a compact sub-interval of (0, 1). From (1.8) it is easy to see that, in the

location model, bni reduces to the familiar one sample score

Ri/n
= n I b(a) do
(Ri - l)/n

A

ui

Theorem 3.2. Under the assumptions of Theorem 2.2,
(3.5) B'C1(Vf,- chib) = - (s, - g 3913 + 0,,(1),
I
where

(3.6) 13:2 b(o) do and b 2:: -E{l)(F(G(7/)))Hm(7])}. //

Og-fih

. 1
Proof. Integration by parts yields that hm: b(0) + (a (o) (lb(o'). Hence,
0

Ill

1 . .
BglgUf, db(a) = Bgl gems”, - 13;.l gm b(0) + (1 - a)db(o)]
l 1

CE—

: BEIWii ' ZcuiB)'
1

Therefore (3.5) follows from (2.13), since. by Fubini’s Theorem

(l)~1,..(F"(a)) dun) = - Bumemanate-2)}. //

Remark 3.2. Observe that the random coefficient in the leading term on
the right hand side of (3.5) is the same as in (2.14). In general, 1) depends on the
unknown G. This in turn implies that unlike the i.i.d. errors case, the sequence of

r.v.’s {HERVE - chibfl is not asymptotically distribution free. (a.d.f.).
1

However, if G is strictly increasing with d.f. F, then G 2 FIG) and in this
case m =1, b = - E b(<I)(7]))7]. Hence in this case the r.v.’s are a.d.f. In order for
this result to be useful for testing about the slope parameters fl it is necessary to

estimate 0 that appears in Bc and R in (2.15). Let f)" be an estimator of 0 such

57
that (3,, - 0)loge n = 013(1). The estimator of 9 given by Yajima (1988) is known
to satisfy this condition. Let BC and R denote BC and R respectively after 0 is
replaced by 9,, in these entities. Then, the sequence of statistics R‘I/2 13;1 (VIC, -
Ecnib] converges in distribution to a N[0, I], x I,] r.v. and this can be used to test
i

different hypotheses concerning ,6.

4.4. Asymptotic uniform linearity of linear regression rank-scores statistics.

This section obtains the AUL of the RR processes and statistics based on
residuals. These results are similar to those obtained by .lureckova (1992a) for the
i.i.d. errors case. They are useful for testing sub-hypotheses and the estimation of
some slope parameters when others are treated as nuisance parameters. See

Jureckova (1992b) for other applications of the AUL results.

Accordingly, let {r 1 g i g n} be a triangular array of p x1 vectors and let

ni’

Rn x p be the matrix with rows If“, 1 g i g n satisfying the following two conditions:

(R.1) (R‘R)‘1 exists for .11 .121). (R2) max (

lgngn

 

9.15.9” = 0(1).

lll

Let Ynit :2 Yui - rt'.A;lt, 1 g i g n, t E RD and Ynt := [Ynlt , Yn‘zt ,.., Ynndt.

111
Let We). on Y... >. inure. Y... ). mo. Y... ) denote €r(fl(a). at time). no)
etc. respectively, when {Yni} are replaced by {Ynit} in their definitions. The
following lemma is similar in spirit to Lemma 2.1 and Theorem 2.1. It gives the
asymptotic representation of the regression quantiles processes based on the
residuals {Ynit} and the proof is similar to that of Lemma 2.1. Here 6,,(1) (6p(1))
denotes a sequence of stochastic processes that converge to zero (bounded) in
probability, uniformly over a _<_ a g 1 - a. lltllg L, V £1.6(0, 1/2], L e (0, 00). Also

racall the notation 03(1) from Theorem 2.1.

s
, (1.1.2), (x.0) - (x2), (A.7), (A.8), (11.1)

(J!

Lemma 4.1 Assume that (1.1.1

V

and (R2) hold. Then

(4.1) IlBglmmo). o. Y...) - and). o))- E .‘X‘RAJ t ,(.. (o)1— — o (1).

(4.2) sup{ 113113190.“ )+A,,l s, a, Yn )- (16(0). mY -crs<1( a)”}=0p(1)

where the above supremum is taken over a g a g 1 - a, HS“ 5 K, ”til 3 L.

(4.3) Axmmao. Y...) - (3(a)) = - n"(o){S.J...(F"(o)) + EQX‘RAflt q(o))+o,.(1),
(4.4) A. (dunno. Y...) - Add. Y...)) = 5,,(1),

(45) Annie. Y...) - (3(a)) = 0.11). //

The following theorem gives the main result of this section.

Theorem 4.1. Assume that (1.1.1), (1.1.2), (X0) - (X2), (A.7), (A.8),
(RJ), (R.2) and (C.I) - (C3) hold, where

(0.3) dx =
Then
(4.6) 330510, Y...) = 13313.net) + B;‘C"RA;‘t oto) + 6,.(1).

Moreover, if the score function I) is of bounded variation and constant

outside a compact subinternal of(0, 1), then V 0 < L < 00,

(4.7) suthlIS L B1Vf,(Y -V,C,]- BEICtRA}1 t ith) db(a) = 0,)(1). //

 

 

 

 

59
Proof. Let E n(oz, t):— —— Ax [,Bn( (0, Ym)- )]. From (1. 6), for 1 sign,

ani(a’ Ynt) : I[Ynit > Xiii 311((1’ YIN-n + 311““) Ynt) IlYnit: xltll 1811(0 Ynt)l
: 1 - Hg g F'1 (a ) + x,,iAx1E11(Oa t) + rfliA} lt]
+ad11i(a Ynt)IlY11it=x1t1i'Bn(a Ynt)l

Hence. 13.111131... Y...) - 13.105101

ni X

=-B‘.‘Zc..{1[e.s F“(a) + xf..A;.‘E..(a. 1) + rhiA‘Jtl-Iie. s W 1) + x A ‘E 16“ )1}
l

+ Bel :c'a‘ (a, Ynt) IIYnit: xlll fl|l(n Yul” Bt‘l Xenia iii ((1') IlYni : xltll 3M0”

111 am

(4.8) = - Rl(a, t) + 112((1, t) - 113(0), (say).
By (C2), R2(a, t) = 6])(1) and 112(0 61—) — 01 *,(1). To handle R1(a, t), let

T(a, s, t) 3=B,C..i{1l‘i <F l()( “)+X].iAx S+ri1iA r 112,] s,tERp.

l

~th

Applying Theorem 1.2.2(iii) 1) times, jt'll time with 7111 = J component of

D”1 c 1<j<p, and 5

C ni’

teRp,

— xniAxl s + rniArl t to conclude that V a6 (0, 1/2], 3,

[ll

T(a, s, t) - T(a, 0, 0)— BC1 zcuix' 11iA ~C1B chir’ 11'1A ’Cl—t - op(l).

In view of (C3) and an argument similar to the proof of (2.3.2) yields that
(4.9) ...,.(T(.., s. t) - T1a.o.0) - Dz‘C‘RA.‘ 111111)} = 0.11).
where now the snpremnm is taken over a g (1 g 1 - a. ||s|| s K, Ht" 3 L. Hence, (4.5)
and (4.9) yields that
(4.10) 33 geninq g F‘1(a) + xffiA 1E”(a t) + rfliA t] T(,.. 0,0)

- B;10‘RA;11 (1(a): 5,,(1).

60

By a similar type of argument,

Agent.) - T(.., 0, 0) = 0;;(1).

Ill

(4.11) 13-32%in _<_ F-1(..) + x).
l

Combining (4.8), (4.10) and (4.11) to get (4.6). Assertion (4.7) follows from

(4.6) by using integration by parts and the assumption that 26111: 0.
l

61
BIBLIOGRAPHY

Boos, D. (1981). Minimum distance estimators for location and goodness of fit. JASA
76, 663-670.
Beran, R.J. (1977). Minimum Hellinger distance estimates for parametric models. Ann.
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