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presented by
Sunil Kumar Dhar

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MINIMUM DISTANCE
ESTIMATION IN AN ADDITIVE EFFECTS
OUTLIERS MODEL

by

Sunil Kumar Dhar

A DISSERTATION

Submitted to
Michi an State University
in partial ful lllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Statistics and Probability
1988

5/7742Z3/

ABSTRACT
MINIMUM DISTANCE ESTIMATION
IN AN ADDITIV E EFFECTS OUTLIERS MODEL
BY
SUNIL KUMAR DHAR

Consider the additive effects outliers (A.O.) model where one observes
Yj,n = Xj + an, 0 5 j _<_ n, with Xj = pxj_1 + (j, j = 0, i1, i2,..., |p| < 1. The
sequence of r.v.s {Xj, j 5 n} is independent of {vj,n’ 0 S j S n} and an, 0 S j S n,
are i.i.d. with d.f. (1—7n)I[x 2 0] + 7nLn(x), x E R, 0 S 7n S 1, where the d.f.s Ln’
n 2 0 are not necessarily known and cj's are i.i.d.. This thesis discusses the class
of minimum distance estimators of p defined by Koul (1986, Ann. Stat. L1,
1194—1213) under the above A.O. model. These estimators are shown to be
asymptotically normally distributed and their influence functions are also
computed. .

The second part of the thesis presents the asymptotic behavior of another
class of minimum distance estimators defined by Heathcote and Welsh
(1983, J. Appl. Prob. fl, 737—753) under the above model. This class of estimators
is obtained by minimizing the negative of log modulus square of the empirical
Y

j—1,n’
function of t, for each value of the e.c.f.. Uniform consistency and uniform strong

characteristic function (e.c.f.) of the residuals Y. 1 5 j S n, as a

— t
,n
consistency of these estimators are proven, uniformity being taken over all possible
estimators defined above. The weak convergence of these estimators to a Gaussian

process is also established under the above A.O. model.

In both the problems it is observed that the asymptotic biases vanish if
Jfi 7n = 0(1) or if Jfi 7n = 0(1) and Zn 4 0 in probability, where Zn is a r.v.
with the d.f. In The asymptotic biases are non—vanishing when (5 7n -+ 7 and

Zn -o Z in probability, where Z is some r.v. and 0 < 7 5 1.

To my parents and my wife

ACKNOWLEDGEMENTS

I wish to express my heart felt thanks to Professor Hira Lal Koul for being
my guide during my entire stay here. I sincerely thank him for suggesting the
problem, inspiring and guiding me during the preparation of this dissertation.

My sincere thanks to Professor James Harman for his constant help and.
encouragement. His constructive criticism was a great help in improving the
results of my thesis.

I am thankful to Professors James Harman, Habib Salehi and Joseph
Gardiner for serving on my committee. To professor Habib Salehi I extend my
special thanks for always taking interest in my well being.

I wish to express my gratitude to all my teachers for everything they taught
me.

Finally, thanks to Ms. Cathy Sparks for typing the manuscript.

CHAPTER

TABLE OF CONTENTS

1 Koul's estimators in an additive effects

1.0.
1.1.

1.2.

1.3.

outliers model.

Introduction and summary.

Assumptions and existence.

A

Asymptotic approximation of ph(H)
when centered.

Asymptotichnormality and influence
function of ph(H).

2 Heathcote and Welsh's estimators in an

2.0.
2.1.
2.2.

2.3.

additive effects outliers model.
Introduction and summary.
Definition of a class of estimators.
IJniform (strong) consistency of

Weak convergence of the process

BIBLIOGRAPHY

vi

PAGE

C}!

50

5O

59

CHAPTER 1.
0. Introduction and Summary.
Let F and Ln’ n 2 0, be symmetric distribution functions (d.f.s)
on the real line IR, symmetric about 0. Throughout this thesis F is
assumed to have a density f 2 0. Let {7n’ n 2 0} be a sequence of

numbers in [0,1] converging to 0 as n -» oo. Define

(0.1) fln(x):= (1-7n) I[x 2 0] + 7nLn(x), x 6 IR,

where I[A] denotes the indicator function of the set A. Let

Cj’ j = 0, :1, i2,..., be independent and identically distributed (i.i.d.) F

random variables (r.v.s), with E53 < 00. Let vjn, o g j g n, be i.i.d.
fin r.v.s.

We consider the model in which one observes, at stage n, r.v.s Yj n’

0 g j 5 n, satisfying

(0.2) Y. = x. + v.

Jan J Jan, j = 0, 1, "-a I],

with {Xj} obeying the autoregressive model of order one (AR(1)), viz.

(0.3) X. = pX.

< 1 '= a1 a2,...,
J J lpl ’ J 0, i

-1 + (j)
where {Xj} is stationary. Moreover, {Xj’ j 5 n} is assumed to be
independent of {vj n’ 0 S j g n}, n 2 0. This chapter studies the
problem of estimating p.

Denby and Martin (1979) called the model in (0.2) and (0.3) the

additive effects outliers (A.O.) model. All the above assumptions on

{Yj’ 0 S j 5 n}, {Xj}’ {Vj,n’ 0 5 j 5 n} and {6].} will be referred to as the
model assumptions. The assumptions on {vj,n’ 0 5 j 5 n} reflect the
situation in which the outliers are isolated in nature. Isolated outliers are
defined by Martin and Yohai (1986) as the outliers any pair of which are
separated in time by a nonoutlier. Martin and Yohai (1986, page 796,
Theorem 5.2 and Comment 5.1) also made the assumption of independence
of the process {Xj’ j S n} and {vj,n’ j = O, 1, 2, ..., n}, n 2 0.

In practice, an apprOpriate model for time series data with outliers
may be difficult to Specify. Fox (1972) and Martin and Zeh (1977) point
out the importance of finding the difference between various types of
outliers in order to effectively deal with them. The two types of outliers
in time series analysis that have received considerable attention are AC.
and innovations outliers (1.0.). In the 1.0. model one observes Xj of
(0.3), and that large data points are consistent with the future and perhaps
the past values. On the other hand, in the AD. model outliers are
generally not consistent with the past or future values of the unobservable
process Xj‘ The additive effects outliers may occur due to measurement
errors like key punch errors (Denby and Martin, 1979) or round off errors
in which case LI1 is taken to be uniform d.f. on the interval [—.5, .5]
(Machak and Rose, 1984).

Denby and Martin (1979) studied the least squares estimator,
M-estimators and a class of generalized M—estimators (GM—estimators) of p
under the above models; they took F and LI1 to be 40,03) and
1(0,a2), respectively. Under their A.O. model all of these estimators have

non—vanishing asymptotic biases with a possible reduction in biases for

GM—estimators.

This chapter of the thesis studies the behavior of the class of
minimum distance estimators of Koul (1986)(KL) defined under the AC.
model (0.2) and (0.3). To define this class of estimators, let h be any
Bore] measurable function from R to R, H be any non—decreasing

function on R, define

—1 2 n
(0.4) Sh(x,t) = n / jglhwfl,n){1[vj,Il g x + th_1,n] —

— I[— Yj,n< x — th_1,n]}, x e [R
and

(0.5) M(t) = J 8121(x,t) dH(x), t e IR.

Denote ph(H) to be a measurable minimizer of M, if it exists. Then
ph(H) satisfies
(0.6) inf M(t) = M[ph(H)].

KL studied this class of estimators under the 1.0. model. Among,
other things, he proved that the estimator with h(x) E x has the
smallest asymptotic variance within a certain class of estimators Lhm).

Section 1 gives the assumptions that facilitate the small and large
sample study of ph(H). It also contains a proof of the existence of
ph(H). Section 2 discusses the asymptotic distribution of the pr0posed
class of estimators. Theorem 2.1 uniformly approximates M(t), t 6 IR, by
a quadratic function of t, uniformity taken over small closed
neighborhoods of the true parameter p. Its proof uses the technique
presented by KL (proof of Theorem 3.1) and Koul and DeWet (1983, proof
of Theorem 5.1). The techniques of Koul and DeWet (1983, Corollary 5.1)

and Koul (1985, Lemma 3.1, Theorem 3.1) are used to obtain an
asymptotic approximation for (E [ph(H) - p] in Theorem 2.2.

Throughout this thesis, the asymptotic bias of ph(H) is defined as
the mean of the asymptotic distribution of Jfi [ph(H) — p]. Section 3
contains the study of the asymptotic normality of a suitably standardized
ph(H) and its asymptotic bias. Section 3 thus begins by reproducing‘in
Theorem 3.1, the Central Limit Theorem (C.L.T.) for [—mixing set of
processes proved by Withers (1981, Theorem 2.1, and 1983). Lemma 3.2
gives a general method to verify that the set of processes involved in the
approximation of Jfi [ph(H) - p], is a—mixing. Definitions of [-mixing
and a—mixing sets of processes are as in Withers (1981) and have been
restated in this section for the sake of completeness. Using Ibragimov and
Linnik (1975, Theorem 17.2.2), the Lemma 3.3 gives a general method to
compute the asymptotic variance. Theorem 3.1, Lemma 3.2 and Lemma
3.3 are used to prove, in Theorem 3.4, the asymptotic normality of
Jfi[ph(H) — p], when apprOpriately centered. At this point, Remark 3.0
disscusses conditions under which the model assumption of symmetry of 'F
or Ln's about 0, could be dr0pped in the study of the asymptotic behavior
of ph(H). Theorem 3.5 contains conditions under which Lhm) has
vanishing and nonvanishing asymptotic bias and Remark 3.1 states the
corresponding asymptotic normality results.

As in KL, Remark 3.2 notes that the Optimal estimator which
minimizes the asymptotic variance is the one with h(x) 5 x. For this
estimator we see that p and the asymptotic bias of than have the
same sign.

Remark 3.3 gives a smaller set of sufficent conditions which imply the

assumptions of Section 1. Remarks 3.4 and 3.5 discuss these assumptions

when H(x) E x or H is bounded, respectively.

Remark 3.6 points out that all the assumptions of Section 1 are
satisfied by h(x) 5 x, H given by dH = {F(1 — F)}"1dF and F
either equal to the d.f. of a double exponential or A002).

Finally, using Martin and Yohai (1986, Definition 4.2), under some
regularity conditions, the influence function of ph(H) is computed in'
Remark 3.7. The influence function turns out to be pr0portional to the
asymptotic bias of 511m).

Before proceeding further, observe that the process {Xj} is stationary
ergodic and Xj—l is independent of ‘j’ j 2 1. From the assumptions on
an's and Xj's it can be seen for each n, that the process {(Xj’ vjm),
0 S j 5 n} is stationary ergodic and hence so is {Yj,n’ 0 g j S 11}.

These observations will be used in the sequel repeatedly.

Notation. Throughout this thesis, by op(1) (Op(1)) is meant a sequence
of r.v.s that converges to zero in probability (is tight or bounded in

probability). Also, let ZIl be a r.v. with d.f. Ln, n 2 0.

1. Assumptions and existence.
This section contains a list of assumptions that will be used
subsequently. Using some of these assumptions it also contains a proof of

the existence of Lhm) of (0.6). We begin by stating the

Assumptions.
. 2 _
A1. n'yn - 0(1), where 7n 6 [0,1].

A2: H is a non-decreasing continuous function such that

limes

|H(X) - H(y)| = IH(-X) - H(-y)| V x, y E R-

H generates a unique Lebesgue—Stieltjes messure, hence H will

also be used to represent a measure in the sequel repeatedly.

10'

(a) 0 < 123x?J < a. (b) For some 5 > 0, 0 < E|h(X0)|2+6 < a
2 2
EXOh (X0) < 00.

For some 6 > 0, sup E ] |h(x0+z)|2+5dLn(z) < oo .
n

xh(x)20 Vx or xh(x)$0 Vx.
0<]fde<e, where k=1,2.

x— x 00 . II— f2 x— x 00 .
(a) Strip ] Ef( zn) dH( ) < (b) nm ] E ( zn) dH( ) <

3 CO > 0 such that ]|f(x—u) — f(x)| dx 5 com, v u e IR.

- (a) £1151 E|X0|h2(X0)[f(x+sX0) — rm] dH(x) = 0,
(b) lira; ] sxgh2(x0)[i(x+sx0) — f(x)]2 dH(x) = 0.
8—)

In all the assumptions to follow 0 5 CI1 E [R are such that Cn -i 0.

All:

c
(a) HE 533;] ‘1 ]EU |X0|h2(X0)f(x+sX0—z) dLn(z)]dH(x) ds < a.
n —c

II

12'

13'

14'

15'

16'

17

Note.

7

(b) 1175 X13331 E[x§h2(x0)] f2(x+sX0—z) dLn(z)] dH(x)ds < a.

II

- 33p ] EU |X0+zl |h(X0+z)| Ef(x+pz-jZn) dLn(z)] dH(x) < 00,

holds with (a) j = 0 and (b) j = 1.

'I'Ei 222 z f2x -' z x 00,
It] ] EU (x0+ ) h (x0+ )E ( +pz 1an dLn( )] dH( ) <

holds with (a) j = 0 and (b) j = 1.

C
° 1 11E th2Xz fszz—'dL ~
r—Irlmgcgj ] [[I 0+| (0+){](+p+s[0+itu) non}

n
-dLn(z)] dH(x) ds < 00,

holds with (a) j = 0 and (b) j = 1.

C
- H71; 733—] ‘1] EU (x0+z)2h2(x0+z)] f2(x+pz+s[X0+z]—ju) dLn(u)]°

II n
_Cn
.dLn(z)] dH(x) ds < 00,
holds with (a) j = 0 and (b) j = 1.

- 1m ] Eh2(Y0 n)Gn(x+pv0 n)[1 — Gn(x+pv0 n)1 dH(x) < a.
n , 3 ,

TEE] EUh2(X0+z)[E{F(x+pz—jZn)—F(x—pz—jZn)}]2dLn(z)]dH(x) <. 00,

holds with (a) j = 0 and (b) j = 1.

In Remarks 3.2 — 3.6 various sets of sufficient conditions that imply

the above assumptions are given.

Existence.

Lemma 1.1. Assume that A2 and A6 hold. Then
either (i) H(IR) = 00 or (ii) H(R) < to and h(O) =
implies the existence of ph(H).
m. The proof will be given only for the case xh(x) 2 0 V x E IR;
the proof in the case xh(x) S 0 V x 6 IR is exactly the same, with h
replaced by —h.

Define

c(x)°= n—1/2h(0) I my. = o]{1[v. < x] — 1[— Y. < x]} x e [R
' j=1 j—1,n j,n- j,n ’ ’

(1.1)
n

d:= 11-1/2 2 |h(Y
i=

IY_ #0, b:= max Y. .

j— —1,n
Observe that c(x) = 0 for |x| > b and hence c is H—integrable.

Now rewrite

[1
(1'2) Sh(x’t) = n 1/2 j21h(Yj-1,n)I[Yj-l,n* 0]{I[Yj,n 5 th—l,n+ XI ’

— I[— Yj,n< — th—1,n+ x]} + c(x), x, t E IR.

The first term on the r.h.s. of (1.2) is bounded by (1. Hence

(1.3) c(x) — d S Sh(x,t) S d + c(x), t, x E IR.
Further xh(x) 2 0 implies
(1.4) Sh(x,t) ——i c(x) i d as t ——+ i 00.

Moreover, under A2, H is continuous and by calculations similar to

those in KL (equation (2.1) - (2.3))

9

—1
(1‘5) M(t) = n E: f h(Yi-l,n)h(Yj-l,n)“H(Yim _ tYi—lm) -

Hence M is continuous on R. Now consider

QM} H(IR) = so. If (1 = 0 then from (1.3) M E J c2dH. Hence a
trivial measurable minimizer exists. Now let (1 > 0. Since c and c2
are H-integrable,

] (c(x) a (1)2 dH(x) = a.
Hence from (1.4) and the Fatou Lemma,

M(t)—ioo as t—i too.
This and the continuity of M ensure the existence of a measurable
minimizer of M.
Case (ii). H(IR) < 00. From (1.1), h(0) = 0 implies that c E 0. Thus
(1.3), (1.4) and the Dominated Convergence Theorem (D.C.T.) give

M(t) ——. d2H(IR) as t -—» a a.

This with (1.3) and the continuity of M ensure the existence of a

measurable minimizer of M. o

2. Asymptotic approximation of ph(H) when centered.
For stating the main results of this section we need some more
notation. Let Gn denote the d.f. of v. n + 6-, j S 11. Since v. and

J J Jan
6 j are independent,

(2.1) Gn(x) = (1—7n)F(x) + 7nEF(x—Zn), x 6 IR.

A density of GH is

10

(2.2) gn(x) = (1-7n)f(x) + 7n Ef(x—-Zn), x 6 IR.

Define

2
(2.3) cm) = ][sh(x,p) + 111/20 - p){a,(x) + anus] we), t e r,

where an(x) = EY0h(Y0)gn(x + pvo) and an(x) = an(—x), x 6 IR.

We shall first uniformly approximate M by Q, uniformity taken
over small closed neighborhoods of p. Using this approximation we obtain
the asymptotic approximation of ph(H) in terms of the minimizer of Q.

From here on we shall suppress n in the r.v.s v a and Yj n,

j,n’ n
etc., for the sake of convenience.

Theorem 2.1. Let all the model assumptions (0.1) — (0.3) hold. Further

let A1 - A4, A7, A8 and A1 - A17 hold. Then for any 0 < b < 00

0

E mm Imn-mm =MU
nl/zlt-pISb

Proof. The techniques used in here are as in KL. Define, V x, t 6 IR,

—1/2 n -1/2

W = o o— o o < o —

(x,t) {n jglh(YJ_l)I[vJ ”VJ-1+6] _ x+n tYJ_1]} p(x,t)
(2.4) with

n—l
p(x,t) = 11—1/2 2: h(Yj_1)Gn(x+n'1/2tvj_l+pv

i==0 1‘1)

11

h

Note that the jt summand in W(x,t) is conditionally centered, given

(vj_1, Yj—1)' From (0.4) and (2.4) we get

(2.5) Sh(x,n_l/2t+p) = W(x,t) + W(—x,t) + p(x,t) + p(—x,t) —

I1
- 11-1/2; h(Y

J=1 H)’

From (0.5) and (2.5),

M(n‘1/2t+p) = ”W(x,t) — W(x,0) + W(—x,t) — W(-x,0) +
+ Sh(x,p) + tWK) + aI-X)l + u(-Xit) - u(-x,0) -

2
— ta(—x) + p(x,t) — p(x,0) — ta(x)] dH(x).

From the above representation of M(n_l/2t+p), using (2.3), the Hélder

inequality, the Transformation Theorem for integrals and A2,
|M(n—1/2t+p) - Q(n_1/2t+p)|

(2.6) 58|W(t) — W(0)II21 + 8|u(t) - x40) -ta|12{ +

1/2
(|W(t) — W(onfi) +

1/2
+ (no) — 11(0) - talfi) ],

1/2
+ 4081,02) + tla + a llfi) [

where W(t), Sh(p) and p(t) are functions W(x,t), Sh(x,t) and p(x,t)
with their integrating variables suppressed and | I; denotes the square

of the L2(H)-norm. From (2.6) it suffices to prove:

12

(i) E sup |u(t) - u(0) - talfi = 0(1).
ItISb
(ii) mg IW(t) - W(MIE, = 0(1).

2
(111) 11m E sup ISh(p) + t[a + 2: “H < 00.
n ItISb

Proof of (i). Define
h+(x) = h(x)I[xh(x) 2 0] and h—(x) = h(x) — h+(x) v x 6 IR.

Replacing h with hit in each of the functions a and It gives new

functions, say a* and pi.
)2 2+ 2b2, a and b in IR,

From the inequality (a+b S 2a

W0) — f0) - aims}?

at i tn—l a; 2
(2.7) s 2] [u (tx) — u (0,X) - 51,30 th (Y,)g,(x+pvj)] dH<x> +

n—1 2
+ 2t2] E j :0 tht(Yj)gn(X+pvJ-) - ai(x)] dH(x)

= I(t) + 2t2

11, Itl s b,
where I(t) and 11 represent the first and the second integral on the

r.h.s. of (2.7), respectively. From (2.2) and (2.4) we can rewrite

_1/2n—1 a; n-l/2t
I(t) = Hn 1.20th (rpm—711)]O [f(x+st+pvj)—f(x+pvj)] ds +

13

+ 7 nJ22]1/2t[f(x+st+pvj-Z) - f(X‘I'IWJ ‘ZII d3 dL112(1(Z)}]H(X I

(2.8) g 4(1 — 7n)2] n“1/2bnn2; v2h2(v)

[f(x+sY-+pv.) — f(x+pv.)]2ds dH(x) +
- 1 1 1

n—l
+ 472] n‘1/2b 2 Y2h2(Yj-)
n
j— =0
n—l/Zb

.] ] [f(x+sY.+pv.—z) — f(x+pv.-z)]2dLn(z) ds dH(x).
—n—1/2b .I J .I

Inequality (2.8) follows from the Cauchy Schwartz inequality and the
moment inequality. Use (2.8), the Fubini Theorem, the stationarity of
{(vj, Yj)’ 0 5 j 5 n}, (0.1) and (0.2) to get

—1/2

Iililllgblfi) g 4n1/2(1-7n3)l)]nn_1/21)b] iix2h2(x0 )[f(x+sX0 )—f(x )1 2de( ) ds +

+ 4111/27 (1- -7n ) 2b]::: /:] EU (X0+z) )2h2(x0+z)

[f (x+s[X0+z]+pz) - f(x+pz)]2 dLn(z)] dH(x) ds +
(2.9)
11—1/2b
+ 4111/272 n(1- 7n ) bE‘l/2b ] [Bxgh2(x0)] [f(x+sX0—z) —

— f(x-z)]2dLn(z)] dH(x) ds +

14

-1/2

+ 4.1/2 57. j b ] EU (x,+z)2t2(x0+a.

-1/ b
H [I(X‘I'SIXO‘I'ZHPZ—U) - f(x+pz—u)]2dLn(u)] dLn(z)] dH(x) ds.

A1 and the continuity preperty A10(b) show that the first term on
the r.h.s. of (2.9) converges to zero. The remaining terms of (2.9) go to
zero by A1, A4, A8(b), A11’ A13 and A15. Thus

(2.10) E suprKt) — -o(1).

Now consider

EII = ] EEl-n— 1'2; Y.h (Y. j).gn(x+pv) — a *2d(x)] H(x)

(2.11) 2(—17n)2]E[—n§1Y.h((Y.)i(x+pvj) —

2
— EY0h*(Y0)i(x+pv0)] dH(x) +

+ 272]131]-—n 7'20 th (Yj)] f(x+pvj—z) dLn(z) _

i 2
— EYOh 0(0)] f(x+pv0—z) dLn(z)] dH(x).

The inequality (2.11) follows from (2.2) and the inequality

(a+b)2 s 2a2+ 2b2, a and b in IR. From Al, A4, A b), A b) and

8( 13(
the stationarity of (vj, Yj)’ 0 S j S n—l, the 2nd term in (2.11) goes to

zero as n -1 00 and theJ first term can be written as

l5

2n‘1(1-7n)2 ] Var{Y0h*(Y0)f(x+pv0)} dH(x) +

(2.12)

-2 2 n—l . :l:
+ 4n (1—7n) 2 (11-7)] Cov{Y0h (Y0)f(x+pv0),
i=1

, th*(Yj)i(x+pvj)} dH(x).
The first integral in (2.12) can be written as

(1—7n)Ex3h2i(x0)] i2(x) dH(x) +

+ 7n] EU (X0+Z)2{h*(X0+Z)}2f2(x+pz) dLn(z)] dH(x) _
— (1-‘rn)2[153X0h"‘(xo)]2 ] f2(x) dH(x) _

— 27n(1—7n)Ex0h*(x0)] 1002]] (x0+z)h*(x0+z)i(x+pz) dLn(z)]-
~dH(X) —

_ 7121 HE] (X0+z)h*(X0+z)f(x+pz) dLn(z)] 2dH(x),

which in turn converges to Var[XOhi(X0)]] f2dH. This follows from A1,
A4, A7, A13(a), the moment inequality and the Hdlder inequality. Hence
the first term in (2.12) converges to zero. The second term in (2.12) can

be written as

n—l
4n-2(1—7n)4.21 (n—j)Cov{x0hi(x0), xjh*(xj)}] i2dH +
J:

16

+ 4n—27n(1—7n)3 DE: (n—j) Cov[X0 h'h(X0 )f(x),

,] (xj +z)h* (x. +z)f(x+pz) dL n(2)] dH(x )+

(2.13)

+ 4 2 311—1 *
n 7,0- 7,)j§ (n1) Cov[x,h (xj)f(x).
,] (x 0+z)h*(x0+z)i(x+pz) dLn(z)] dH(x) +

+ 4n 2117 2(1- 7n )2 jEI l(n—j) CovU (X0+z)h (X0+z)f(x+pz) dL n(z ),
, ] (Xj+z)h (Xj+z)f(x+pz) dLn(z)] dH(x).

By assumptions A1, A 4, A7, and A13(a) and the Hiilder inequality, the

second, third and the fourth terms in (2.13) go to zero. Since,

(2.14) Var{n 121] X."'h (Xj )}=
l:

nIOVar[X h (X0)] + 2n_2 jEl (n—j) Cov{X0 h (X

to prove that the first term in (2.13) goes to zero, from A4, A7 and (2.14)

it suffices to prove that

n

(2.15) Var{n"1 2 xjh*(xj)} -» 0 as n e 00.
i=1

But from A4 and the Stationary Ergodic Theorem

11
(2.16) {12 x.h*(x.) .7 nxohfixo) a.s..
j=1 J J

17

Also, from A 4, the sequence {th:(Xj)} is uniformly integrable of order

2, hence so is the sequence {n12 X h (Xj )}, which follows clearly from
j=1-‘
Chung (1974, exercise 9, p. 100). Thus, from (2.16), (2.15) follows, which

in turn gives

(2.17) E11 4 0 as n a 00.
Thus (2.10) and (2.17) applied to (2.7) prove (i).

Proof of (ii). Replace the h in w by hit and call the new r.v. wi.

Fix t in [—b,b]. Using the Fubini Theorem and the fact that the

jth summand in W* is conditionally centered, given (vj_1, Yj_1),

Elwm) - when?I

(2.18) _ --J1J'in21ijE[{h (Y _1)}2E[{I[Vj+£j g x+n'1/2th_1+pvj_1] —

—1 2
— I[vj+cj S x+pvj_1] — Gn(x+n / tY._1+pv.

J H) +

+G nj_(x+pv 2v(j_1,Y )2. )H dH(x),

       

which can be dominated by
2 E{hi(Y )}2IG (x+n—1/2tY + v ) — G (x+ v )| dH(x)
0 n 0 P 0 n p 0 '

Using (2.1), the Fubini Theorem and representation of F in terms of its

density, this term can be dominated by

18

j E|X0|h2(X0)f(x+Xos) dH(x) ds +

J EU IX0+ZIh2(X0+z)f(x+pz+s[x0+Z]),

. dLn(z)] dH(x) ds +
(2.19)

J EU |X0|h2(X0)f(x—z+Xos) dLn(Z)] ,

- dH(x) (is +

In” |X0+z|h2(X0+z)J f(x—u+[X0+z]s+pz) .

- dLn(u) dLn(z)] dH(x) ds.

From Al, A3(a), A4 and A10(a) the first term in (2.19) converges to zero.
That the remaining terms also converge to zero, follows from A1, A11(a)

and A14, giving
(2.20) E|w*(t) — w‘=(0)|12I .7 o, t 6 IR.

Thus to complete the proof of (ii), use the monotone structure of

Wi and pi, the compactness of [-b,b], HE ElalfI < co and (i), just
n

as in Koul and Dewet (1983, p. 929, Theorem 5.1, proof of (ii)). The

details are similar, hence deleted.

19

Proof of (iii). From (2.5) taking t = 0, we get

(2.21) S(x,p) = W(x,0) + W(—x,0) +

n
+ n 1/2j21h(Yj-l){Gn(x+pvj—l) — Gn(x—pvj_1)}.

We shall now proceed to prove that

(2.22) l'iTn- E|S(p)|I21 < a.
n

Using the fact that the summands in W are conditionally centered, the

stationarity of the process (vj’Yj)’ 0 S j S n—1 and the Fubini Theorem,

E J W(0)2 dH

1+6. 5 x] — an_(x+pv12)} dH(x )

I .—
{ [vJ (W J

(2.23) =1 En 1 21h2(Y.
J:
= J 13‘,112(Y0){I[vl-pv0+c1 S x] - Gn(x+pv0)}2 dH(x).

The lim sup of the r.h.s. of (2.23) is finite by A , A (b) and A
1 3 16

Next, using the stationarity of (vj’ Yj), 0 5 j g n—1,

n-1
EJ {fl/21.20 h(Yj)[Gn(x+pvj) — once-7779]}2 dH(x)

(2.24) = j Eh2(Y0)[Gn(x+pv0) — Gn(x—pv0)]2 dH(x) +

20

—1
+ 2:14:31 (n-l') E[h(Yo)[Gn(X+P‘/o) - G,(x—pvo)lh(Yj)o
-[Gn(x+pvj) — Gn(x—pvj)]] dH(x).

The lim sup of the first term on the r.h.s. of (2.24) is finite by (0.1),
A1 and A17. The expression inside the sum in the second term on the

r.h.s. of (2.24) can be written as

(2.25) (n—j)7121J EU h(X0+z)[Gn(x+pz) — Gnu-772)] dLn(z)-
.1 h(Xj+z)[Gn(x+pz) — end—772)] dLn(z)] dH(x),

which follows from the independence of {Xj’ j S n} and

{vj, 0 S j 5 n} and the latter being i.i.d. fin. Thus applying the
Cauchy—Schwartz and the moment inequalities to the integrand in (2.25)
and using the stationarity of {Xj}’ the second term on the r.h.s. of (2.24)

can be dominated by

(2.26) (n—1)712J EU h2(X0+z)[Gn(x+pz) — Gn(x—pz)]2dLn(z)]dH(x).

From A1 and A17, the lim sup of (2.26) is finite. Hence (2.22) holds. The

proof of (iii) now follows from rm E|a|12I < 00, which in turn follows
n

from (2.2), A4, A7, A8(b) and A13. This also completes the proof of the

theorem. D

Note. In Theorem 2.1, the proof of (2.15) gives an alternate way to prove

KL's equation (14), p. 1211.

21

In order to prove the next theorem, analogous to the definition of
[311(11), define Lh(H) with M replaced by Q. Using the definition of
511m), (0.4), (2.3) and the fact that Sh(-,p) is even,

311(9) [§+a] dH

 

Mam — p) = —j [Mala

(2.27)

Sh(p)a

= — 21—7 dH.
|3+§|H

Theorem 2.2. In addition to the assumptions of Theorem 2.1, let us
assume that A6 holds and also for each 11 let ph(H) be a measurable

minimizer of (0.5); then

(228) n1/2Ip,(H) - pl = n1/21p,(H) - pl + 0,0).

Proof. The line of proof is as follows:
(i) For any 17 > 0 and 0 < z < 00 there exists a N and b,

0 < b < 00, depending on n and z such that

P( inf M(n—1/2t+p) 2 z) 2 1— 17 ‘v’ n 2 N.
|t|>b

(n) 7/fi 173,01) — pl = 0,0) and
«a lp,(H) - pl = 0,0).

(iii) Mlphlnll = lehmll + 0,0) and
Mlph(H)l = leh(H)l + 0,0)-

22

Proof of (i) follows exactly as in Koul and Dewet (1983, Corollary
5.1) or Koul (1985, Lemma 3.1).

Proof of (ii) follows from (i), (2.22) and the reasoning given in Koul
(1985, Theorem 3.1).

Proof of (iii) follows from (ii) and Theorem 2.1. From (iii) and

(2.27) we get

(2.29) nlp,(Hl - 73,01)? |a+2l§ = 0,0)

From (2.2), (2.3) and the symmetry of the function gn w.r.t. to the
y—axis, we get

|a+a_|fI = 4(1-7n)2[EX0h(X0)]2 j g3 dH +

2 2
(2.30) + 7n) [EJ (X0+z)h(X0+z)[gn(x+pz)+gn(—x+pz)] dLn(z)] dH(x) +

+ 47n(1—7n)EX0h(X0) J gn(x)EU (X0+z)h(X0+z) .
- le,(x+pz)+e,(—x+pzll dL,(zl] dH(x).

From Al, A2, A4, A7, A8(b), A12, the moment and the Holder

nd rd

inequalities, the 2 and 3 terms on the r.h.s. of (2.30) go to zero and

the 1St term converges to
_ 2 2
(2.31) 2q _ 4[EX0h(X0)] Jr dH.

From A3, A6 and A7, the r.h.s. of (2.31) is strictly greater than zero.

Hence from (2.29) the proof is complete. I:

23

Note. In order to study the limiting distribution of n1/2[ph(H) — p] we
need to apprOpriately center (2.28). In view of the Theorem 2.2, for fixed
h and H, the asymptotic behavior of ph(H) will not be affected if for each
n 2 0, ph(H) is replaced by any convex combination of the measurable

minimizers of (0.5). Further note that the proofs of Theorems 2.1 and 2.2

only need Eh2(X0) < co instead of A3(b).

3. Asymptotic Normality and Influence function of ph(H).

In this section we apply Withers (1981), (1983) C.L.T. for Z—mixing
sequence of arrays. We also discuss the sufficient conditions under which
the asymptotic bias of ph(H) is vanishing or nonvanishing. Finally we
compute the influence function and show that it is directly proportional to
the asymptotic bias of Lhm). To study the limiting distribution of
JH [ph(H) — p] when centered, from (2.27), (2.30), (2.31) and Theorem

2.2, we need only to study

II
(3.1) — (In-”21,31 h(Yj_1)pn(Yj—ij_rl

when centered, where q > 0 is as in (2.31) and

(3.2) (pact) = J a dH — I a dH.
Let

p, = Eh<Y0)p,(Y,-pY0) and
(3.3)

{Jan = h(Yj-1)¢n(Yj—ij_1) ‘ #nr 1 S j S n.

24

From (0.1), (0.2) and (3.2),

Y ‘Y
p,(y) = (1—7,)Ex0h(x0)[[ g, dH - j gn an] +

-oo

3'
(3.4) + 7n” EJ (X0+z)h(X0+z)gn(x+pz) dLn(z) dH(x) —

_ J—y EJ (x0+z)h(x0+z)s,(X+pZ) dL,(Z) dH(X)]-

-00

From (3.4), A2, A3(b), A4, A5, A7, A8(a), A12 and 7n 6 [0,1], note that
{1.11, 1 g j g n are real valued r.v.s and ”n < 00 for all n 2 0.
For the sake of completeness we shall reproduce the following

definitions and theorem from Withers (1981 and 1983) and use them to

n
prove the asymptotic normality of 11—1/2 23 {j n'
i=1 ,

Consider a series of random processes

(3.5) €= {t,.n21l. where £,= {tj,,.d,sisN,l,

which are not necessarily real (although possibly complex), defined on sOme
probability space (SLAP), with (in, N11 integers such that
-ooSdn<NnSoo and Nn-dndooasn4oo. Forany(not

necessarily real) r.v.s 0, d) we set

JIM) = the a-algebra generated by 0,
(3.6)
04%) = suplPUnJ) - I’(1)I’(J)|.

25

where sup is over all I 6 1(0), J 6 94¢). Define

011(k) = dnslgugl’én- ka({€dn,n’""€j,n}’ {£j+k,n""’€Nn,n})
(3.7)
a(k) = max an(k), 0 S k < 00.

{n: kSNn- dn}

The set of processes 5 is said to be (it—mixing if a(k) _, 0 as

k -+ 00. Set

[H(k,u) = max suplCov[exp{iu g: 65 n},
dns j SNn- k p=dn

for u real, 0 5 k 5 (ln — ND, 11 z 1, where sup is over {dj 2 0 or 1}

and the covariance of complex r.v.s means that

Cov(0,¢) = E0 $ — E0 E6.
Now set

(3.8) €(k,u) = max (H(k,u), 0 S k < oo, u real.
{nszNn- dn}

The set of processes 5 is said to be t—mixing if for all real 11,

l(k,u) —7 0 as k --1 oo. From Ibragimov and Linnik (1971, p. 307), as
pointed out by Withers (1981), [H(k,u) S 16 an(k), 0 5 k < oo, V u E [R,
thus

(3.9) l(k,u) S 16 a(k), 0 _<_ k < 00, 11 real.

26

Henceforth assume dn = 1 and N n: n. Define

_ . < < < —

2 . .
an = Var Sn and C110) = sup|Cov(§d,n, €m,n)l’ 0 < j < n,

where sup is over {d, m:|d - ml 2 j},

(3.10) 80) = max Eng).
{n:j<n}

Theorem 3.1. (Withers, 1981, Theorem 2.1, 1983). Assume that the
following hold: For some 17 > 0 and n1 2 0, 5 satisfies the moment

inequality

1+17/2+n
(3.11) sup Elsn(a,h)|2+’7 = 0(b 1
a,n

) as b-loo.

g is Z—mixing and for all real u
(3.12) l(k,u) = o(k—5) as k -+ on, where 6 = 2171/17.

(3.13) 0121-400 as 11—400, Limofi/n>0
n

and

m
(3.14) 2 c(j) < .0.
i=0
Then
031(sn— ESn) —7 40,1) in distribution.

27

Lemma 3.2. Let 0 n’ w be Borel measurable functions from IR2 to IR and

£1.11: 9n(Yj-1,Yn’l'n

{Xj’ j = 0, $1, $2,..} is a stationary process that is strongly aX—mixing

)1<J<Il n>1 Let an=w(Xj, vjn)’ where

(Ibragimov and Linnik, 1971, Definition 17.2.1) and the sequence of

independent r.v.s 0 5 j g n} is independent of {Xj’ 0 S j S n},

{V vj,n’
n 2 0. Then 5, as in (3.5) with (1n = 1 and N11 = n, is strongly

org—mixing with
(3.15) 05 _<_ ax.

Proof. In (3.6) let 0 = (€1,n”’°’€j,n)’ ¢= (5 €j+k n’” ”énn)’

I = [(€I,n""’€j,n) .6 B1] and J = [(§j+k,n,...,§n,n) e 82],.

where 131 e 3 (1111), the Borel a—field, and 132 e 3(an_J_k+1),

l g j g n, 2+j S j+k S 11. Let us suppress the n in vj n and define
7,171 .: 111“ -7 le
[X0,X 11'" 'ij " [011W 0"", 9nX(XJ-_17Xj)]7

, j+1 j+1
TY’j'J'l. [R -1 IR

[x0,x1,...,xj] -1 [111(x0,v0),...,w(xj,vj)]

and T* vby 13* andjby

Y ,n-j—k+2 by replacing in T

yd+1’

*

n—j—k+1, where v: (v v0, v,1...,vj) and v = (vj+k—1"”’vn)' Then,

|P(10J) - P(1)1’(J)| =

28

= |P[{w(x,.v0),...,p(xj.vjll e 7111,03,), _1
{‘“(XHk-l’Vj+k-1)"""”(Xn"’n)} e 6...,_k+7<87>] —

- r[{.(x,,v,1,....p<x,.v,->l e 21331)]- _1
'P[MXHk-livj+k_1)p---p“’(xn1"n)} E ¢1.n-l'-k+1(B2)] I

—1 —1
(3.16) g E[|P[(X0,...,Xj) 6 TN. +1¢1,J.(B1), (xj+k_1,...,xn) e

,..-1 —1 _
E Ty ,n—j-k+2¢1,n—j-k+1(B2)]

—1 —1
— P[(x0,...,xj) e TY, j +161,1.(131)] P[(Xj+k_1,...,Xn) e

(v0"”’vj’vj+k—1"’"Vn)]‘

 

 

,..-1 -1
E Ty ,n—j—k+2¢1,n—j-k+l(B2) ]

Inequality (3.16) holds because for all k 2 2, (V0""’Vj’vj +k—1""’Vn)
is a sequence of independent r.v.s and independent of (X0,...,Xn). From
the definition of strongly cit—mixing sequence of stationary r.v.s, and the
stationarity of {Xj}’ we see that the r.h.s. of (3.16) can be bounded by.
ax. Taking sup over all B1 and B2 in .2 (W) and .2 (an—j—kH)

and then taking max twice as in (3.7), we get that (3.15) holds. 13

Note. The proof of the Lemma 3.2 goes through even when w and 0n are

 

replaced by can and 01711 for each j and n.

29

Lemma 3.3. Define 5]. n as in Lemma 3.2 satisfying all the conditions
there. In addtion, let {vj n’ 0 5 j 5 n} be identically
(3.17) distributed fln , with 7116 [0,1], 7n = 0(1) and ax

satisfying, for any n > 0 E lozx(j)'1 < oo.
j-l
Further let 0 and h, be real valued Borel measurable
(3.18) functions with 0 defined on R2 and h on R, such that
011$(x,y) Ch(x) and 0 Il(x,y) -1 0(x,y) for each x, y 6 IR,

0<CER. Letforsome 6>0,

(3.19) E|h(o(x0))|2+5 < a and sup EJ |h(w(X0,z))|2+6dLn(z) < to,
II

where w(-) =w(-,0); then

(3.20) 11—10121: ’1Var 2 g.

j=112n

3

where

(3.21) r2=Var[0(w(X0),w(X1)]+2j§100v[0{w(X0),w(X1)},0{w(Xj),w(Xj+1)}].

Proof. From the definition of Y's and the conditions satisfied by Xj's

JH
and v. 's, {Yj 11’ 0 S j 5 n} is stationary, hence we can write

2n—l
(3.22) {10:11 =Var(51n)+— j2 (n —j)Cov[0n (Y 1n),6n(vj,n,vj+1’11)].

0, n’Y
From (0.1), the definition of Yj n's and the conditions satisfied by Xj's

and vjn's,

30

(3.23) Var(t1,,) = (1-7,)2E0,2,{w(xol.w(x1)} +
+ 7,(1-7,)Ej 0,2,{w(Xo).w(X1.z)} dL,(z) +
+ 7n EJ 0§{w(X0,z),w(X1,vl’n)} dLn(z) — (251,192,

which in turn converges to the first term on the r.h.s. of (3.21). The
above convergence follows from 711 = 0(1), (3.18), (3.19) and the DOT.
From (0.1), the definition of Yj n's and the conditions satisfied by Xj's

and vj n's, for j 2 2, we get

Cov[0n(Y0,n,Y1,n)a (Yj,n’Yj+1,n)1

= (1—7n)4Cov[0,{114X0)1“’(X1)}’ 0,1{111(Xj).w(Xj +1)}] +

+ ,n(1—7n)3Cov[0n{tp(xo),w(xl)}r J0,{w(XJ-72)7W(Xj+1)}‘1Ln(z)] +

+ 7n(1-7,)3COVU ”n{w(xopz)7“(X1)}dLn(z)l”n{“’(xj)""(xi+1)}1 +

+ 7n(1—7,)3CovHonmxohuixliz” dLn(Z)’0n1“’(Xj)’w(XJ+1)}1 +

+ 7,(1-1,)3COV[”n{“’(xo)’“1xl)1’1 11n{t(xj),cp(xj +1,z)}dL,(z)] +

+ 7121(1_7n)200v[0n{,,(x0),u(xl)}, fl 0111‘“x j’z)’w(xj+1’u)} .
~dLn(z) dL,(u)] +

(3.24) + 7,2,(1-7,)2CovU0,{u(X0).w(X1,z)} dL,(z).
7 I 0n{"-’(Xjaz)r“'(xj+1)} dLn(Z)] +

31

7121(1—7n)200v ”J 0n{w(X0,z),w(Xl,u)} dLn(z) dLn(u),
, o,{p(le.p(xj+,)}] +

7,2,(1—7,)2Cov[[ o,{p(x,,z).p(x,)} dL,(z).
, j o,{p(le.w(xj+,.zll dL,(zl] +

7,2,(1-7,)200v[[ 0,{w(X0.z).w(X1)} dL,(z).
.jo,{w(xj,z).p(xj+,ll dL,(zl] +

7121(1‘7DJZCOVU 0n{w(X0),w(Xl,z)} dLn(z),
, j 0n{w(Xj),w(Xj+1,z)} dLn(z)] +

73(1-7,)Cov[[ 0,{w(xo.z),w(X,)} dL,(z).
. fl 0,1ptxj.z).p(xj+,,u)l dL,(z) dL,<u)] +

73(1—7,)Cov[[j 0,{w(x0,z).w(x1.u)l dL,(z) dL,(u),
, j 0,1ptxj.z),p(xj+,ll dL,(z)] +

130-1..)va onlutxtlpdxrpll dL,<z),
’ J] 0n{w(xjiz)1“’(xj+lru)} dLn(Z) dLn(u)] +

73(1—7n)COVUJ 0n{w(X0,z),w(X1,u)} dLn(z) dLn(u),
,j0,{p(xj).p(xj+,,zll dL,(z)] +

73COV ”I 0n{w(X0,z),w(Xl,u)} dLn(z) dLn(u),
,jj o,{w(xj.zl.p(xj+,,ull dL,(z) (11,011].

32

From computations similar to those in (3.24), 711 = 0(1), (3.18), (3.19)
and applying the D.C.T., show that for j = l

(3.25) Cov[0n(Y0,Y1), 61109.5(j +1)1 ..
Cov{0{w(X0).w(X1)}. 0{w(Xj).w(Xj+1)}l-

Similarly, from (3.24), (3.25) holds for j 2 2. In general, any covariance
on the r.h.s. of (3.24) can be represented by

COVI¢n2(X01X1)7 ¢n3(xjixj+1)]-
From (3.18) and (3.19) we see for each n 2 l

2 6
(3.26) E|¢n2(X0,X1)| + 5 c1 < co and

2 6
E|¢n3(X0,Xl)| + 3 c2 < 00,0 < c c2 e IR.

1,

Applying Ibragimov and Linnik (1971, Theorem 17.22) to the sequence {Xj}
With 13 = 11 T = J " 11 J .). 27 E = ¢n2(X01X1)1 17 = ¢n3(Xj’Xj+1)’ and
from (3.26),

(3.27) leovlp,,(x,,x,). p,,(xj.xj+,lll so ax(1—1)‘5/(2+1),

where 0 < C 6 IR depends only on cl, c2 and 6. Thus from the
conditions satisfied by ax in (3.17), (3.25), (3.27) and the DOT. for
counting measure, we see that the second term on the r.h.s. of (3.22)

converges to the 211d term on the r.h.s. of (3.21). Hence (3.22) and the

convergence of (3.23) to the apprOpriate limit imply that (3.20) holds. :1

33

Theorem 3.4. (i) Under (3.17) — (3.19) and the assumption 72 > 0

n
111/2 21% } -7 ADJ?) in distribution.
j- J,Il j,n

(ii) Let Al — A17 and all the model assumptions (0.1) to
(0.3) hold. Then

(3.28) n1/2[ph(H )— p + Iraq—1] -+ 40,0121) in distribution,

where

(3.29) 6,21 = q—2[EXoh(X0)]2Eh2(X0)E¢2(cl),

y
(3.30) ,p(y)=j th—I de VyeIR,

q is as in (2.31) and ”n as in (3.3).
Proof. We shall prove (i) and then prove (ii) using a special case of (i).
In view of Theorem 3.1 we shall first show that (3.11) holds for 5 as in

(3.5) and (3.17). Let 17 = 6 as in (3.19) and 111 = 1+6/2 then

E3n(ab)2+6

 

 

(3.31) ,2?”

IA

E|0 H(YYOD,

IA

01(1-7,lElh{p(x0)112+1 + 7,13Ilhlptxopll12+15 dL,(zll.

34

The above inequality follows from the definition of Yj n's and the

conditions satisfied by Xj's and vj n's, (0.1), (3.18) and the Jensen
inequality . From (3.19), (3.11) is satisfied. From the Lemma 3.2 and
(3.9), (3.12) is satisfied. From Lemma 3.3 and r2 > 0, (3.13) is satisfied.

Since {Yj’ 0 S j 5 n} is stationary, cn in (3.10) can be written as

Cn(j) = SUPICOVICIJV €|d_ml+l,n“1

where sup is taken over {d, m: |d—m| 2 j}. From this, (3.10), the
same argument as in (3.26) to (3.27) and (3.19), we get

(3.32) «20) s 80ax(i—1)‘5/(2+ ‘1), Vi _>. 2.

From the conditions satisfied by ax in (3.17), (3.19) and (3.32), (3.14) is
satisfied and hence the C.L.T. holds for 5.

proof of (ii). We shall first show that the C.L.T. holds for 5 as in (3.3)
and (3.5). Thus take in (i), 0n(x,y) = h(x)¢n(y—px), x, y 6 IR, Yj’ Xj and

vj n as in the model assumptions (0.1) - (0.3) with w(x,y) = x+y.

(D
Note X. can be as. represented as 2 pkc. k' Using A3(a),
J k=0 J"

A9 and Pham and Tran (1985, Theorem 2.1) with 5 = 2, A(k) = pk,
we get that {Xj} is strongly (ix—mixing with ax(n) g Cplplzn/3 V n 2 1.
Also note from A1, A7, A8(a) and (2.2)

x x
(3.33) J gn dH -+ I f dH uniformly in x.

-00 -oo

35

Thus the r.h.s. of (3.4) converges to EX0h(XO)¢(y), uniformly in y, by
(3.33), A1, A3 and A12. By A7, 16 is bounded and hence tbn is
uniformly bounded. Since F is symmetric about 0 and 11) is an odd,
E¢(£1) = 0. Thus letting 0(x,y) = h(x)i/)(y—px)EX0h(X0), x, y 6 IR in

Lemma 3.3 we see 72 = afiqz. From (3.29), (3.30), A3 and A7,

0121Q2 > 0. We now see that all the conditions of (i) are satisfied. Hence
C.L.T. holds for 5 as in (3.3), (3.5). Thus from (2.27), (2.30), (2.31) and

Theorem 2.2, (ii) holds. 1:1

Remark 3.0. One of the assumptions under which we have studied the
asymptotic behavior of ph(H) as an estimator of p is F and Ln’s are I
symmetric about 0. One could generalize the results by making an
attempt to discard this assumption. A careful study of the results shows
that due to this change, the arguments involved in the proof of (2.22) fail.
The proof of the Theorem 2.2 can be easily modified without this
assumption of symmetry or any additional assumptions. In view of the
Theorem 3.1, the Lemmas 3.2 - 3.3 and the Theorem 3.3(i) we see no
addtional modifications are needed to incorporate this change. Thus it only
remains to modify the arguments from (2.21) - (2.26). Note in (2.21) and
(2.24) we need to replace Gn(x-pv0) by 1 — Gn(—x+pv0), since GI1 is

not symmetric. Thus in view of (2.24) we will now need

rim I Eh2(Y0)[Gn(x+pv0) + Gn(—x+pv0) — 1]2 dH(x) < o. .

36

Qase (i). Let F be symmetric about 0 but Ln's need not be symmetric
about 0. In this case (2.25) can be replaced by

(n—1)7;";[(1-7,)2Eh(x0)h(xj)j [E{F(x-Z,)+F(—x-Z,)-1}l2dH(x) +

+ (1—7n)j [E{F(x—Zn)+F(-—x—Zn)-l}]- _
.E[h(x0)j Eh(Xj+z)[Gn(x+pz)+Gn(—x+pz)—1] dLn(z)] dH(x) +

+ (1—7,)j IE{F(x-Z,)+F(-x-Z,)-1}l-
-E[h(Xj)J Eh(X0+z)[Gn(x+pz)+Gn(—x+pz)—l] dLn(z)]dH(x) +

+ J EU h(X0+z)[Gn(x+pz)+Gn(—x+pz)-l] dLn(z).
J h(Xj+z)[Gn(x+pz)+Gn(—x+pz)—l] dLn(z)] dH(x)].

Thus in view of the arguments involved in (2.26) and under proper
assumptions Theorem 2.1 holds.
Qase (ii). Both Ln's and F need not be syrmnetric about zero. In this

case we will need to assume for each n 2 1,
Eh(Y0)[Gn(x+pv0)+Gn(-x+pv0)—1] E 0,

after that we can use the same technique as presented in Lemma 3.3 and

some limit theorem to prove (2.22).

Theorem 3.5. Let all the model assumptions (0.1) — (0.3), A1, A2, A4,

A5, A7, A8(b), A13 and A16 — A17 hold; then

37

(a) If h is a continuous function on R and Zn -1 Z in distribution

and 75711-170 then fipn-ip,where 0<7CEIR and
(3.34) ,7 = 7c[EX0h(XO)]J f(x)EJh(X0+z){F(x—pz)—F(x+pz)}dLn(z) dH(x).
(b) If either Z n -1 0 in distribution or ,5 7n -1 0 then Jr? ”11 -) 0.
13201. From (2.27) and (3.1) — (3.3),

.63 ”n = E J Sh(p)a dH.

For large enough n we shall justify the interchange of expectation and

integral above using the Fubini Theorem. Consider

(3.35) I ElSh(p)| |a| dH 3 U 2312107) dH]1/2U a2 dH]1/2.

Inequalitity (3.35) follows from the Hdlder and the moment inequalities.
That the lim sup of the r.h.s. of (3.35) is finite follows from (2.22) and
the same reasoning as in (2.30) and (2.31). Thus, by the stationarity of
the process (Vj—l’ Yj—l)’ the independence of (vj__1, Yj-l) and

vj + cj, 1 S j S n, and (0.1), for large enough 11, we get

(3.36) ,5 ”n = Jfi'ynj a(x)EJ h(X0+z)[G(x—pz) — G(x+pz)] dLn(z) dH(z).

Using (2.1) and (2.2), (3.36) can be rewritten as

38

75 7,0—7,)3Ex,h(xo)j f(x)Ej h(x0+z)1F(x—pz) - le+pz>l~
- dLn(z) dH(x) +

+ .5 7§(1-7,)21Exoh(x0)lj EF(x—Z,)EI .
- h(X0+z)[F(x—pz) — F(x+pz)] dLn(z) dH(z) +
(3.37)
+ Jfi 7121(1-7n)J E (X0+z)h(X0+z)gn(x+pz) dLn(z)] -
E h(X0+z)[F(x—pz) — F(x+pz)] dLn(z)] dH(x) +

+ .(n 731 I a(x)EJ h(X0+z)E[F(x—pz-Zn) — F(x+pz—Zn)]dLn(z)dH(x).

The fourth term in (3.37) converges to zero by A1, A17(b), the same
reasoning as in (2.30), (2.31) and the Hfilder inequality. The third term in
(3.37) converges to zero by A1, A13, A17(a), the Hiilder and the moment
inequalities. That the second term in (3.37) converges to zero follows from
A1, A4, A8(b), A17(a), the Hiilder and the moment inequalities.

For each x 6 IR,

EJ h(X0+z)[F(x-pz) — F(x+pz)] dLn(z)
(3.38)
7 RIM X0+z)[F( (x-pz) - F(x+pz)l dL<z )

where L is d.f. of the r.v. Z; we get (3.38) from A5, Zn —+ Z in
distribution and the D.C.T..

The proof of (a) and (b) now follows from the convergence of the
first term in (3.37) to the appropriate quantity which itself follows from
A1, A4, A5, A7, (3.38) and the D.C.T..

39

Remark 3.1. From Theorems 3.4 and 3.5 (a) we see
vfi [ph(H) — p] -1 I (—p q—l, 0121) in distribution
and from Theorems 3.4 and 3.5 (b) we see

,In [ph(H) — p] -7 1(0, 0%) in distribution.

Remark 3.2. KL has shown that the function h(x) or x, x 6 IR minimizes
0121. Let px(H) be the estimator corresponding to h(x) a x, and
measure H. Then, the asymptotic bias of 3x01) under Theorem 3.5 (a)

looks like

—1
[212ng Pm] 7c} f(x)EZ[F(x+pZ) — F(x—pZ)] dH(x),
which has the same sign as the sign of p.

Remark 3.3. Consider

S1:

11 6 IR, is bounded and continuous at 0.

2
The function defined by q1(u) = J [f(x+u) — f(x)] dH(x),

S2: The function defined by q2(u) = J f(x+u) dH(x), u 6 IR, is

bounded and continuous at 0.

2 2
s3: sgp EJ (|X0|+|z|) h (X0+z) dLn(z) < a.

S .

2
4. 11—15 EZn < 00.

Il

Then under 81’ 82, S3 and S4, we can show that the assumptions in

Section 1 reduce to Al — A7, A9 and A16‘

40

Proof. A8(b) follows from 81’ A7, the Fubini Theorem and the inequality
(3.39) I f2(x+u) dH(x) 5 2I [f(x+u) — 1(x)]2 dH(x) + 2 I f2 dH.

A8(a) follows from 32 and the Fubini Theorem. A10 follows from the
Fubini Theorem, A3(b), A 4 and the Bounded Convergence Theorem. A12
follows from the Fubini Theorem, 32, S3 and the moment inequality. A13
follows from the Fubini Theorem, S1, S3, A7 and (3.39). A11(a) and A14
follow from the Fubini Theorem, boundedness in S2, S3, A3(b), A 4, A5
and S 4. A11(b) and A15 follow from the Fubini Theorem, boundedness in
81’ S3, A4, S4, A5, A7 and (3.39). Further, A17(b) can be bounded by

I EI h2(X0+z)IEI Iplzli(x+n—zn) duII2 dLn(z) dH(x),

-pIZI
which in turn is

pIZI
(3.40) 5 2p I EI h2(X0+z)|z|E I f2(x+u—Zn) du dL(z) dH(x),

-pIZI

using the moment inequality. That the lim sup of the r.h.s. of (3.40) is
finite follows from the Fubini Theorem, S1, S3, A7 and (3.39). A17(a)

follows similarly as above.

Rema_rk 3.4. In the case when H(x) a x, all the assumptions of

Section 1 reduce to A1, S3, S4, A3 — A7 and A9.

Proof. Note that A2 and 82 are trivially satisfied. In view of A7 and the

41

translation invariance of the Lebesgue measure, to prove S1 we need only
prove that I f(x)f(x+u) dH(x) is bounded as a function of u and is
continuous at 0. Boundedness follows easily by the Héilder inequality and
the translation invariance of the Lebesgue measure. From A7, f e L2(H),
hence from Rudin (1974, Theorem 3.14), we have for any 7) > 0 3 a

continuous function 65” vanishing outside a compact set such that

(3.41) lg)” — le < 7).

Consider

II f(x)f(x+s) dH(x) — I f(x)f(x+t) dH(x)

 

3 II f(x)If(x+s) — p,(x+s)] dH(x)I +

(3.42)
+ II f(x)[¢n(x+s) — ¢n(x+t)I dH(x)| +

 

+ II f(x)[¢n(x+t) — f(x+t)I dH(x)

The continuity of I f(x)f(x+u) dH(x) as a function of u now follows
from (3.41), (3.42), A7, uniform continuity of the function a)", the H61der
inequality and the translation invariance of the Lebesgue measure.

We shall now prove that A16 holds. Note A16 holds if AL). holds,

where
*

A16: (a) I F(X)[1 — F(x)] dH(x) < e .

(b) 1175 I EF(x—-jZn){1 — EF(x—-Zn)} dH(x) < a. , j = 0, 1.

42

(c) 131m IEU h2(xosz)F(x+pz)[i—EF(x+pz—zn)] dLn(z)I dH(x) < to.

(d) 11m I EUh2(X0+z)EF(x+pz—jZn)[1—EF(x+pz—jZn)]dLn(z)I dH(x)
< 00 , j = 0, 1.

Since F is continuous and Elcll < 00, we have

oo 0
(3.43) I [1 — F(x)] dx < co and I F(x) dx < a.
0 - 00
Thus
00 0
I F(X)[1 — F(x)] dx 5 I (1 — F(x)] dx + I F(x) dx < a.
0 -00

Using the Fubini Theorem, the translation invariance of the Lebesgue
measure and A5, we see that A:6(d) with j = 0 follows from A:6(a) and
also to prove A:6(b) with j = 1 is the same as to prove A:6(d) with

j = 1. A:6(b) with j = 1 can be rewritten as

IIIF( x[)1 — F( (x—z+u)] dx dLn(z) dLn(u)

(3.)44 =IIIF(x)[1 — F( (x—z+u)] dx dL n(z) dL n(u )+

+ IIIO FIX111‘FIX“+“)I dx dL,(z) dL,(u).

The second term in (3.44) is bounded by the 2nd term in (3.43). Now

43

consider the first term in (3.44), which can be written as

III F(x)[l — F(x—z+u)]I[u 2 z] dx dLn(z) dLn(u) +
(3.45)

+ IIIO F(x)[1 — F(x—z+u)]I[u < z] dx dLn(z) dL,(u)-

The first term in (3.45) is bounded by the first term in (3.43). Rewrite
the second term in (3.45), using change of variable and splitting the range

of integration, as

III:—Z F(x+z—u [)1 — F(x)]I[u < 2] dx dL n(z) dL 11‘“ )+
(3.46)

+ II F(x+z—u)[l - F(x)]I[u < z] dx dLn(z) dLn(U).

The first term in (3.46) is bounded by 2E|Zn|. Hence from S 4 we get
that the lim sup of the first term in (3.46) is finite. The second term in
(3.46) is bounded by the first term in (3.43); consequently A:6(b) and
A:6(d) with j = 1 hold. By the Fubini Theorem, the symmetry of Ln's
about 0 and the translation invariance of the Lebesgue measure, A:6(c) can

be written as

131m E{I h2(X0+z) dLn(z)} I F(x)[1 — EF(x—Zn)] dH(x),

:1:
which is the same as proving A16(b) with j = 0, in view of A5.

44

Proceeding exactly as in (3.44) — (3.46) and using (3.43), S 4 and A5,
we get that A:6(c) holds.

Remark 3.5. In case H generates a finite measure, all the assumptions
of Section 1 reduce to Al - A7, A9, A10(b), A11(b), A15 and, A8(b) and
A13 with lim sup replaced by sup.

PM. From A5 and the fact that H generates a finite measure, we see
A16 and A17 are easily satisfied. Using the H6lder inequality, A5 and
A15(b), we see A14(b) holds. Using the Hdlder inequality, A5 and A15(a),
we see A14(a) holds. Using the Héilder inequality, A3(b) and A11(b), we
see A11(a) holds. Using the moment inequality and A13 with lim sup
replaced by sup, we see A12 holds. Using A3(b), A10(b) and the Héilder
inequality, we see A10(a) holds. Using A8(b) with lim sup replaced by

sup and the moment inequality we see A8(a) holds.
Remark 3.6. For H given by

dH = 17%? where f(x) = 2_1exp{—|x|}, x 6 IR,

[Note here we could take f to be the density function of a 40,02” we
shall show that in view of Remark 3.3, the assumptions in Section 1 reduce

to A1, 713(5), 33, A4 — A6, and

S5:

56: 131m nI h2(X0+z)exp{|pz|} dLn(z).

liTn—EeprZ|<oo,
n I]

45

Prmf. Note
F(x) = 1 — 2_lexp{-x} if x 2 0
= 2-lexp{x} if x < 0.

Let x, u E IR, and note

 

exp{-k|x[-j[x+u[} _ x _ -—x —1
(3.47) F(x)[l_F(x)] Sexp{-(k1)| 012 exp{ I |}] ,

where k, j = 1, 2. Since the r.h.s. of (3.47) is integrable over the real
line, applying the D.C.T. to the functions in (3.47), we get 81’ S2 and A7

hold. Let x, u E R. From the inequality

e-IX-UI e-IXI

2 ’2

 

52—11111 [expl-lx-ull + exp{-IXI}I
and the translation invariance of the Lebesgue measure, we see
I )f(x-u) -f(x)| 4x: In).

* * *
Since A16 implies A16’ we shall verify Al6 holds. A2 and A16(a) are _
trivially satisfied from definition of H. A3(a) follows trivially from the

m .
a.s. representation of XO as E p16_j. S5 implies S4 holds. Using A5,
i=0
:1:
S5 and S6’ lengthy calculations show that A16(b) — (d) hold.

If in addition to the restrictions of this remark we assume h(x) a x,

then the assumptions in Section 1 reduce to A1 and S5. For these

46

assumptions to be satisfied we could choose Zn to be 40,02) and
7n = 11-1/2 for all n.

Remflk 3.7 (Influencg flinctign). We shall define a functional T on a

subset, say P0, of the set of all stationary ergodic measures on
(Rw’m, @, where R7°°’°° is a collection of all sequences of the type

y = (..., —1’ yO’ yl’ y2,...) and .2 is the Borel o—field on R7°°’°°.

Proceeding as in Martin and Yohai (1986, equations 3.1 — 3.3), the
functional T, p 6 P0, is defined as T(p) satisfying the equation

2
(3.48) %,I [Illa/011117, s x+ty0l — Il—yl < x—tyol} (1747)] dH(x) = o,

where P0 is the set of all stationary ergodic measures on (Rw’m, .2),
such that the integral in (3.48) exists and is differentiable. Further, T(p)
minimizes the double integral in (3.48) as a function of t, and of all the
minimizers it is defined as the one with the smallest magnitude. Under
the assumptions A2, A6 and following the same argument as in Section 1,
we see that (3.48) has at least one solution which minimizes the double
integral in (3.48), as a function of t. For the sake of completeness, let us
repeat the definition of the time—series influence functional as in Martin
and Yohai (1986, Definition 4.2). For T as in equation (3.48) and u;
the measure generated by the process (0.2) where 711 = 7 and

Ln = L V n, the influence functional (IF) of T is defined as

7 __ 0
(3.49) IF = lim m7) TMY)

7-10 7

47

provided the limit exists.
In (3.48) taking [1 = p}, then computing the innermost integral and
differentiating within the integral sign w.r.t. t under some regularity

conditions, we get T7 E T013) satisfies

(3.50) I [(1—7)Eh(x0){e(x+1T7-plxo) - G(x—[T1—plX0)} +

+ 7EI h(X0+z){G(x+[T7-p](X0+z)+pz) - G(x—[T7-p](X0+z)-pz)} dL(z)I-
°I(1-7)EX0h(X0){s(X+IT7-pIX0) + g(x-IT7-plX0)} + 7EIh(X0+Z)(X0+Z)'
°{g(x+IT7—pl(X0+z)+pz)+g(x-IT7-pl(X0+z)+pz)}dL(z)I dH(x) = 0.

Note in (3.50) G and g both depend on 7. Setting 7 = 0 in (3.50)
we see T0 E T013) satisfies

(3.51) I Eh(X0){F(x+[T°-p]X0) — F(x—lTO-plxoll-
oEX0h(X0){f(x+[T0-p]X0) + f(x—[TO—p]X0} dH(x) = 0.

Assuming the derivative of F exists and equals f we see from A6 that
T0 = p is the only solution to (3.51) which makes the double integral in
(3.48) with p = #3 minimum. Thus differentiating the l.h.s. of (3.50)

7
under the integral sign w.r.t. 7, then setting 7 = 0, solving for g—1 7:0,

we get

IF =q‘1Ex0h(x0) I f(x)EI h(X0+z){F(x+pz)-F(x—pz)} dL(z)dH(x).

Note that IF = — 721(asymptotic bias of Jil— [ph(H) — p]), where the

bias is computed under the assumptions of Theorem 3.5 (a).

CHAPTER 2
0. Introduction and Summary.
Heathcote and Welsh (1983)(H—W) proposed a class of minimum

distance estimators pn(s) of the vector p in an autoregressive model of

order k, defined so as to minimize

-2 -1 11 ' 2
Mn(t,s) = — 3 log (n—k) 2 exp{is(X. — X._1t)}
' j=k+1 1 '1 ‘

as a function of t, where X34 = (Xj-l’xj-2"”’Xj-k)'

Here we shall study the behavior of this class of estimators of
p t (-1,1), when k = 1, under the AD. model (1.0.2) - (1.0.3). The
model assumptions of this chapter exclude the assumption F, Ln's
symmetric about 0 and E63 < 66, but assume the first moment of (0
exists and E60 = 0.

Section 1 contains the definition of a class of minimum distance
estimators pn(s), s E of, of p where of is a compact set of the type
[—b,-a] U [a,b], 0 < a < b.

The main result in Section 2 is stated under Theorem 2.5, wherein it
is proved that the sup norm distance between pn(s) and p converges to
zero in probability or almost surely, the sup being taken over of. This
convergence in probability (almost surely) is referred to as uniform (strong)
consistency of the estimators. The idea of the proof of Theorem 2.5 is'
taken from Csiirgéi (1983). Lemmas 2.1 to 2.4 are used to prove this

2

result. Lemma 2.1 uniformly approximates exp {- s Mn(t,s)}, under the

model (1.0.2) - (1.0.3), by exp {- 32Mn(t,s)} with Yj n replaced by

48

49

Xj’ where uniformity is taken over all s E of and t in some compact
set K containing the true value p in its nonempty interior. Remark
2.1 uses Lemma 2.1 to give sufficient conditions under which p is the
unique global minimum of the limit of Mn(t,s) for each s E or: Lemma
2.2 contains a standard result. For the sake of completeness, Lemma 2.3
gives a Glivenko—Cantelli type result for empirical d.f. of stationary ergodic
random vectors from Gaenssler and Stute (1976). Lemma 2.4 finds the
uniform limit of exp{— s2Mn(t,s)} under the AC. model for

(3,13) 6 of x K.

Section 3 contains the discussion on the weak convergence of the
process {45 [pn(s) — p], s e of} and its asymptotic bias. This section
begins by obtaining an approximation for Jfi [pn(s) - p] using the Taylor
series expansion. Theorem 3.1 gives the uniform limit of the coefficient. of
,5 [pn(s) — p] in the above expansion. Using this theorem, the techniques
as in H—W and the results from Billingsley (1968), Theorem 3.2 proves
the weak convergence of the process {45 [pn(s) — p], s E of}, when
appropriately centered, to a Gaussian process in C(eY)—space. The finite
dimensional distribution convergence of the above process is proved by
using the C.L.T. for l—rnixing set of processes of Withers (1981 and 1983)
and Billingsley (1968). Remark 3.1 observes that the asymptotic bias of
the process pn(s) converges uniformly to zero if either Jfi 7n = 0(1) or
75 7n = 0(1) and Zn = op(1) and that it mnverges uniformly to a
function if Zn -1 Z in probability and JR 711 -+ 7, where Zn is the r.v.
associated with Ln and Z is a r.v.. The corresponding weak
convergence results for the process {J17 [pn(s) — p], s E of} under the

above observations are also stated in this remark.

50

1. Definition of a class of estimators.
Define
07:: [—b, —a] U [a,b], 0 < a < b,

—2 -1 11 . 2 .
(1.1) Mn(t,s) = — s log| n jglexp{1s[Yj,n— Yj—1,nt1}1 if s E o)"

)2 if s=0.

fill-1

11
E Y. — Y. t
i=1 ( 3’11 I'll“
Let K be a compact set containing the true parameter p in its
nonempty interior. Then pn(s) for each s E eYU {0} denotes a
measurable minimizer of Mn(-,s) when restricted to K. For Mn as in

(1.1), p11 can be uniquely defined to be sample continuous on of

Further pn(s) satisfies

(1.2) Ire? Mn(t,s) = Mn(pn(s),s).

Note that pn(0) is the least squares estimator which has been studied in
detail by Denby and Martin (1979) under the AD. model; hence we shall

not allow 8 to be zero.

2. Uniform (strong) consistency of [111(3), 8 E at.
In this section we prove uniform (strong) consistency of the estimators

A

pn(s), s E of The idea of the proof for this result is taken from
Cséirgéi (1983) and H—W.

51

Lemma, 2.1. For each n, let Yj n = Xjn + vjn’ j = 0,1,...,n be r.v.s.

II
Let pr =n‘12|v. |A1. Let K c112 besuch that,
n j=1 ,n n

c = sup {Isl}, c = sup {lstl} and c +c 5c<oo.
1’11 (t,s)eKn 21“ (t,s)eK 1’“ 21“

Then

11

I1

(2'1) (t 33153 111—1731 [exp{sz‘ “14.1111 ‘ exp{isIXip‘ “14.11”“
1 I1 _

S [(c+2) V 4Itin.

Proof. The Triangle inequality and |eit — eisl 5 It — s] A 2,

t, s E R, show that the l.h.s. of (2.1) can be bounded by

_1 11
(2.2) n j21[{cl,n1vj,nl + c2,n1Vj—l,n1} A 2].

The expression inside the sum in (2.2) is bounded by
1cl,n1Vj,n| A 2] + [C2,nlvj—1,n1 A 2].

The r.h.s. of (2.2) is now dominated by

[(c1 n v 2) + (c2 n v 2)]nn.

, 3

From which the lemma follows. 0

52

Notg. Under condition:

(a) an -7 0 in probability
(2.3) or

00
(b) 2 END < oo,
n=0

l.h.s. of (2.1) converges to zero in probability or as, the latter follows
from the Markov inquality and the Borel—Cantelli Lemma applied to an.
In the case when vj n are an distributed for any d.f. Ln in (1.0.1),

_ -1 ’ _
Erin — n (n+1)7nE{|Zn| A l}, and thus 7n — 0(1) or 2 7n < 00
become sufficient conditions in (a) or (b). Further note from here on we
shall supress the n in the random variables (r.v.s) Yj n and Vj n'
Remark 2.1. We shall now present the sufficient conditions for the unique

true value p to be the global minimum of the limit of Mn(t,s) for

each 3692’. Fix 6>0 and sea’. From (1.1) weget

n
(2.4) M,(pi«5.s) — M,(p.s) = - s 2IopIIn 1215110le,- -(p1=6)Y,-_ll}|2/
J:
—1 11 2
/In 2 exp{is[Y. — pY._1]}| I.
j=1 J J
The r.h.s. of (2.4) can be written as

_ s_2logIIn-lj§1[exp{is[Yj- (pt6)Yj_1] — exp{is[Xj- (p:t6)Xj_1]} +

53

(2.5) + exp{islxj- (M)j_1l}II/Inj_l§[-exr>{iSIYj(NJ--11}-

2
- exp{islx, - px,_,l) + explislxj - px,_,l)I I.

From Lemma 2.1 and the Stationary Ergodic Theorem (S.E.T.), (2.5)

converges in probability (a.s.) to

-2 2 2
2. —
( 6) s 1°g{l¢x1—(p46)x0(311 /|¢,l(s)l 1,

where 43X denotes the characteristic function of a r.v. X. Under the

assumptions I())f (s)| > 0 and |¢X (s)| < 1, (2.6) > 0 V s e of
1 0

Thus for sufficiently large n, with large probability (with probability 1), a
minimum is achieved at the true value p. If the distribution of 61 is

infinitely divisible then [966 (s)| > 0 V s 6 cf is satisfied. Also, if the
1
distribution of ‘1 is not lattice type then |¢X (3)] < 1 V s e of
0

follows from (6X (3) = ¢X (ps)¢€ (s) and Chung (1974, Theorem 6.4.7).
1 0 1

Lemma 2.2. If ¢n(t) are characteristic functions on Rk such that

¢n(t) " 41(1) for each t 6 RR, then for any compact subset K of R1"

~ ~

sup 171,0) - 12(1)) .. o.
tEK

~

Proof. The proof follows from Ash (1974, p. 333, Theorem 3.2.9). 0

54

Lemma 2.3. Let X1, X2,... be a sequence of strictly stationary and

ergodic random vectors taking values in Rk. Then

(2.7) PUT-IE sup IFn(xl,...,xk)-F(x1,...,xk)| = 0) = 1,
-oo<x1,X2 , ...,Xk<oo

where Fn(x1,x2,...,xk) is the joint empirical distribution function based 011
X1,...,Xn and F(x1,...,xk) is the joint distribution of the random vector
X1.

Proof. See Stute and Schumann (1980) and Gaenssler and Stute (1976). :1

 

To state the next lemma, let

(2-8) Dn(t,s) = % g eXPI13IYj_th_1)} - ¢£IS)¢X0(S[p—t1) 7

1:1
(t,S) E K x of

Lemma 2.4. Let {Xj} be as in (1.0.3) with cj's i.i.d.. Let vj n's be _as
in Lemma 2.1. Then

(a) under (a) of (2.3)

(2.9) sup Dn(t,s) -+ 0 in probability
(t,s)EKxoY
and

(b) under (b) of (2.3), (2.9) holds with convergence in probability

replaced by as. convergence.

55

Prggf. We shall give the proof of (b). The proof of (a) follows similarly.

From the triangle inequality we get

(2.10) Dn(t,s) S In‘1.nlexp{is(Yj—jY_lt)}— 12 1exp{is(Xj—jX_1t)}I
J= j=1
+ In“1 El exp{is(XjXJ-_1t)} — ¢€1(s) ¢X0(s[p—t])I.

That the first term on the r.h.s. of (2.10) goes to zero as. follows from
condition (b) of (2.3). Since {(Xj_1,cj)} is a stationary ergodic sequence,

Lemma 2.3 yields

 

 

1= H sup 21,-Ile <x x,_ <x,l -F (.,)rx 6,) —» 0)
x1,x2€|Rn 1:1 0
n
(2.11) 5 P( sup 2 exp{is lje+is2X j— _1}- ¢61(31)¢X0(S2) -7 0),

 

 

j=l
(8182)€K1 xK2

where K1 and K2 are any compact subsets of R. The inequality (2.11)
follows from the Continuity Theorem and Lemma 2.2. In particular, taking

K = of and K2 = {s(p — t): s E of, t E K} in (2.11), we get

1

(2.12) P( sup In.1 I213 exp{is(X.—X._1t)} — (I) (3)61)X (s[p—t]) -7 O): 1.
j=1 J J ‘1 0
(t,s)6Kxo>’

Now the lemma follows from (2.10) and (2.12). :1

56
Next, set
(243) M(t,s) = - s'zlosl¢,1(8)¢XO(SIp-tl)l2. 0.8) e K x pp:

Note that

(2.14) M(t,s) 2 M(p,s) V s and hence inf M(t,s) = M(p,s) V s.
tEK

Theorem 2.5. Let Xj's and vj n's be as in Lemma 2.4 with

(2.15) |¢61(s)| > 0 and 0 < |¢X0(s[p-t])| < 1, s E of and

t E K — {p}.
Then the following statements hold:

(a) Under (a) of (2.3),

(2.16) sup lpn(s) - pl -) 0 in probability.
set»!

(b) Under (b) of (2.3), (2.16) holds with probability convergence

replaced by almost sure convergence.

Proof. The proof of (b) is as in Csiirgii (1983, p. 345, Theorem 4.1). We
shall give the proof of (a). From (1.1) and (2.13),

lM,(t.s)-M(t.s)l s Colos[{D,(t.s)/l¢,1(s)l2l¢xo(s[/rtl)|2} + 1].

where C0 = sup{s_2: s E 9?}. Since

inf 1p, (9121px (plptlll2 > o.
(s,t)6e7kK 0

57

Lemma 2.4 and the above inequality imply that

(2.17) sup |Mn(t,s) — M(t,s)l = op(1).
(t,s)Ede’

From (1.2), (2.14) and (2.17)

(.213) sugIMn( (p(s s)—M(p,s)I = EJI’IEIIESU M (t s) 1211; M(t,s)
= op(1).

For 6 > 0, let K(6) = K — {tzlt-pl < 6}. Then, (2.17) also implies

(2.19) suEVIinf M (t s) — inf M(t,s) = op(l).
teK( 6) teK( 6)

Suppose that

(2.20) sup | p (s ) — pl does not converge to zero in probability.
sea

Then 3 770, 771 > 0 and a sequence of integers nk I 00 such that
(2.21) P(supl/3 (s) —pl > ,) > 7.
set»! 11k 1 0

Let 1) = 3_linf |M(p,s) — inf M(t,s)l. From assumption (2.15), 17 > 0.
SEof teK(nl)

58

From (2.18), (2.19) and (2.21), 3 k0(n,no) such that V k > k0(1),770)

1? < P[supIM (x3 (as) — M(p,s)l < n. suplé (s)—p| > n ]+
0 3695’ "k “k sear I1k 1

+ 770/2:

< P[supIM (£2 (8),s) — M(p,s)l < n. sum}; (s)—p| > n ,
3697 ”k “k 8693’ ”k 1

, supl inf Mn (t,s) - inf M(t,s)l < n] +
scthKMl) k t€K(nl)

+ P[sule ((1 (SL8) - M(p,8)l < n, supln (S) - pl > 17,
sea nk nk SEofi’ nk 1‘

, supl inf Mn (t,s) - inf M(t,s)l _>_ 77] + 770/2
scofi’tEKMl) k tEK(171)

(2.22) < P[supIM (i2 (8),s) — M(p,s)l < n. suplé (s) -pl > n,
369! DR 11k 3697 HR 1

, supl inf Mn (t,s) — inf M(t,s)l < 17] + 3170/4.
scofteKMl) k tEK(nl)

From the definition of 1), the first term on the r.h.s. of (2.22) is zero,
which leads to a contradiction. Therefore (2.20) must be false and hence

the result. CI

Note. The proof of Theorem 2.5 does not use the existence of f nor -
does it use any of the moments of 60 or Zn' Note that under the
assumptions c-‘s i.i.d. and |p| < 1, {Xj} of (1.0.3) is invertible and

J
strictly stationary ergodic.

59

3. Weak convergence of the prams Jr? (1311(3) - p), s E d’.

In this section we prove the weak convergence of Jfi [511(3) — p] as
a C(m valued random element. The idea of the proof for this result is
taken from H-W. The C.L.T. given by Withers (1981 and 1983) has been
used to prove its finite dimensional distribution convergence. We also
discuss the behavior of its asymptotic bias.

Recall from (1.1) that to minimize Mn(t,s) w.r.t. t is equivalent
to maximizing U121(t,s) + V121(t,s), where for (t,s) E K x or;

_ -1 “
Un(t,s) — n .2 cos(s[Yj — th_1]),

i=1
-1 n
(3.1) V (t,s) = n E sin(s[Y. — tY. ])
n j=1 J J-l
and
U11 5 VII 5 0, otherwise.
_ —16 2 2
LGt Inn : 2 3E (UH + VII). Then
1 n .
(3.2) mn(t,s) = — I13 .Ele—l{Unsm(S[Yj-th—1]) — Vncos(s[Yj—th_1])},
(t,s) E K x of:
= 0, otherwise.

By the Taylor series expansion

(3.3) mn(én(s),s) = mums) + % mn(t,s)I,=;(,)[bn(s) — p],

where

(3.4) was) -p| s line) -p|, s e or.

60

Theorem 3.1 below shows that % mn(t,s) t=p(s)’ uniformly in s,

converges in probability to a negative number. Hence
(3.5) 31612, lmn(pn(8),s)l = 0p(1)-

Thus from (3.3) and (3.5) we see that in order to prove the weak
convergence of ‘5 (pn(s) — p), it suffices to study the weak convergence of
the process mn(p,s), s E of. Before stating the next theorem, note that
(13,8) +

(3.6) %mn(t,s) = s_2{Un(t,s) Un(t,s) + Vn(t,s) i2 V

II

33%
S

2
+ [313 Un(t,s)] }, t e K, s e of:

_J
[\D

+ [% Vn(t,s)

503$ Un(t,s) = — 311-1 j: Yj_lsin(s[Yj — th_1]),
% Vn(t,s) = sn’ljg1 Yj_1cos(s[Yj — th_1]),
(3.7) $2 Un(t,s) = — 32n"1j§1 Y?_lcos(s[Yj — th_1]),
g2? Vn(t,s) = _ 3211—11:l Y§_lsin(s[Yj - th_,]),

for all (t,s) E K x of From here on it will be understood that sup is

taken over all s E of, unless Specified otherwise.

Theorem 3.1. In addition to the assumptions of Theorem 2.5(a) and all

the model assumptions (1.0.1) — (1.0.3), assume 1m EZI21 < co .
n

61

Then

2 2
(3.8) 32p 35 mn<t,s)1,=,(s) + |¢El(s)l Ex0 = opu),

with r = pn or pn.

Proof. Throughout this proof we shall need sup of each of the following

random functions

2 2
(3.9) IUnl, Ivnl, la Unl’ lg vnl, lie Unl, |th an

taken over all (t,s) e K x of; to be bounded in probability, which is
evident from (3.1), (3.7), E63 < co, the S.E.T. and

II
(3.10) E n‘1 2 v?

. 1 J = 7nEZ121-9 0’
J:

-1

which in turn follows from the rim 132;?l < 00.
II

We shall now prove (3.8) with r = En; the proof for r = pn is
exactly the same. From the triangle inequality the expression inside the

sup on the l.h.s. of (3.8) can be bounded by

‘3; mn(t,s)I,=-,;(s) - % mncsnggs)! +

(3.11) + '3: mn(t,s)|§=§(s) — 3: mn(t,s)| t=,,| +
= Y=X

2
+ |g€ mn(t,s)|t=p + l¢fl(s)| Exgl

62

Here, and in what follows, means that in the given

Y=X

expression replace Yj by Xj and t by p etc. The first term in

(3.11) can be dominated by

(30 lg; Un(t,s)|t_p() -;2 —2-Un(’=ts)|t =p(s)|+

 

+ |U,,(z(s),s)- Una hymn =XH62 U< ,,(,ts)I,___ ,(,)|+
Y:

(3.12) + If? Vn(ta3) It— _p(3 ’: V ”(ts |t= =p(s)|+

+ |vn(23(s),s)—v,, (2(8),)Iys =x§2l| vnmts I, _7(S)|
Y: X

+ H6; Un(t,8)lt=_.;(,,)]2 - [3E Un(t’8)|\t(:§(s)]2l +

2 2
+ [[3, Va“ News] ‘ [3; Vn(t’3)';:§(s)l l]-

nd th th

That the sup norms of the 2 ,4 , 5 and 6th terms in (3.12) go to
zero in probability follows from (3.1), (3.7), (3.9), (3.10), the Lipschitz

property of the sine and cosine functions, the S.E.T. and

(3.13) suplfi(s)l = 0p(1),
8695’

63

which in turn follows from Theorems 2.5(a) and (3.4). As the sine and.
cosine share similar pr0perties, it now remains only to show that the sup

rd

norm of the 3 term in (3.12) goes to zero in probability in order to

prove that the sup norm of (3.12) goes to zero in probability. Accordingly

I; Vnms 't=25(s) 22 V n‘tsHt: ___p(s)|

(3.14) < s2 n__11l 2n sin(s[Y -p(S)Yj_1])[Y2 j_ _l— xJ2_+1]|

+ s2n1j|§1xJ2_1{sin(s[Yj-z(s)vj_l]) — sin(s[xj-',3(s)xj_1])} .

Let C1 = sup{32} and 02 = sup{|s|3}; then the r.h.s. of (3.14) can be

dominated by

11

-1 -1 2
C1[2nZl|Xj_vlj_1|+n Ev +

 

i= i=1 j 1
(3.15)

—1H 2
+ 02n12 x2_1|v.| + |p(s)|nj21Xj_1|vj_1|].

j—l

 

This follows from (1.0.2), the fact that the sine function is bounded by 1
and Lipschitz of order 1, with constant 1. That the sup norm of the
expression (3.15) goes to zero in probability follows from (1.0.1), (3.10),

(3.13), Beg

< 00, Tim EZIZl < co, the Markov inequality applied to each of
n

the averages in (3.15) and the independence of {Xj’ j 5 n} and {Vj’

0 _<_ j S n}. This completes the proof that the sup norm of the first term

of (3.11) is op(1). We shall now show that the sup norm of the second

64

term in (3.11) is op(1). It can be dominated by

c [62 U(ts)| — U (p(s)s)| -a2 U t,s)| _
0 ’ t=ps n ’ n( t—p
E2 Y=X” Y: X as Y=X
-Un(p,s)| I +
Y=X

+oola2 v,,,=(ts)lt p(S)V(p(s),Ys)I x__—
Y=X

Y=X —

—62 _ancsnw v.,/ash |+

From (3.4), (3.7), (3.9), the Lipschitz pr0perty of the sine and cosine
functions, Theorem 2.5 (a) and the S.E.T., the sup norm of the third and
terms in (3.16) goes to 0 in probability. Since the sine and cosine
functions satisfy similar pr0perties, to prove that the sup norm of the
expression in (3.16) converges to zero in probability we only need now

nd

prove that the sup norm of the 2 term in (3.16) converges to zero in

probability. It can be dominated by

C a2 , —( - ‘92
oligarvnm It???” as V (“S Ii:§]v (p0 ”Y=X! +
(3.17)
+ Col [Vn(fi(3)as)| = _ Vn(p’s)|Y_x] .3272 Vn(t,S)| t: '

65

From (3.1), (3.4), (3.9), the Lipschitz prOperty of the sine function and

nd term in (3.17) goes

Theorem 2.5 (a), we get that the sup norm of the 2
*
to zero in probability. Let s = sup{|s|}. The sup norm of the first

term in (3.17) can be dominated by

n
2
(3-18)112 Xj [_{s IXJ-_1|surw|pn(8) - pl} A 2];
J=
this follows from (3.1), (3.4) and the sine function being bounded by 1 and
the inequality |sin(s) - sin(t)| 5 Is — t| A 2, t, s 6 IR. It remains to
prove that (3.18) goes to zero in probability. Let c > 0 be arbitrary, then

V n > n0(m)

(3.19) P(n12 x12 _1’“[{s |xj_ 1|sup|pns() — pl} A 2] > c)
j=1

s Pn< lg x2. [{s llelsuplp (s) pun] >c,

J
—1 —-1
,suplpn(S)-p| Sm )+m
8697
_ * _ _
g c lsxgus |X0|m1} A 21+ m1.

This follows from (2.16), the Markov inequality and the stationarity of '
{Xj}’ In (3.19), taking limit as n -+ co and then m -+ 00, E63 < co and
the DOT. give (3.18) converges to zero in probability.

H—W proved in their Theorem 2 that the sup norm of the third term

11 (3.11) goes to zero a.s.. This completes the proof of (3.8) as well. u

66

Next, set u(s) = Ecos[sel] and v(s) = Esin[scl], s 6 IR. Also, by

the random elements X11 and Y11 satisfying Xn(s) = Yn(s) + 5p(1) we

h 11 — = .
s a mean 31612, IXn(s) Yn(s)| op(1)

Theorem 3.2. Let the assumptions of Theorem 3.1 hold. Also let
(a) I |f(x-—u)-f(x)| dx < cm, u e s, for some 0 < c 6 IR,

(b) Var{cos(scl)] > 0, Var{sin(scl)] > 0,
Var[u(s)sin(scl) — v(s)cos(scl)] > 0, s E of and

(c) sup E|Zn|2+a < 00,0 < E|c1|2+a< 00, a > 0,
I]

hold. Then

(3.20) Ill/212nm — p + un(°)]

converges weakly in C(of) to a Gaussian process with mean 0 and

covariance

(Exfirluse): |¢(t)ll'2(st)‘1h(tss),

where
2h(t,s) = u(s—t)[u(s)u(t) + v(s)v(t)] +
+ u(s+t)[v(s)v(t) — u(s)u(t)] + v(s—t)[v(s)u(t) — u(s)v(t)] —
— v(s+t)[u(s)v(t) + v(s)u(t)]
and

(3.21) pn(s) = {1% mn(t,s)|t=5(s)] [Ecos[s(cl+vl—pv0)] .

- Evosin[s(cl+v1—pv0)] - Esin[s(cl+v1-pv0)]Evocos[s(£1+v1-pv0)]].

67

Proof. From (3.1), (3.3) and (3.5) - (3.7) we get

was (s )- ms 3; ms(tss)'t=3(s>

n
__n—'1/2j 2 lYj_1{Un(p,s)sin(s[Yj—ij_1]) — Vn(p,s)cos(s[Yj—ij_1])}

+ op(1)

= _n-I/ZE y. [{Ecos[s(c +v -pv )]}sin[s(c-+V--PV- )1
i=1 1—1 1 1 o J J J-1

— {Esin[s(el+v1—pv0)]}cos[s(cj+vj—pvj_1)]] —

II
(3.22) — n“1 2

j= 1 Yj_1{sin[s(cJ c.+v j.-pv J_1)—] Esin[s(cl+v1—pv0)]} -

vs mums) — Ecos[(cl+v1—pv0)l} +

n1j§1 Yj _j1{cos[s(c +vj —pv J_1)]— Ecos[s(cl+v1—pv0)]} .
- Jfi {v.,(ps) — Esints<e1+v1-pv0>1} —

II
— {n-ljlej-l} [Esin[s(61+vl—pv0)] -
~Jfi{Un(p,s) — Ecos[s(cl+v1—pv0)]} —
— Ecos[s(el+vl-pv0)]./s{vn(p,s) — Esin[s(cl+v1-pv0)]}] + 5p(1).

We shall now proceed to prove that the sup norm of the second, third and
fourth terms in (3.22) converge to zero in probability. To achieve this we
shall prove
(3.23) (5 {Un(p,s>— E cos[s( e1,+v1—pv0)1}

Jfi {Vn(p,s) — E sin[s(£1+vl-pv0)]}, s E of;

68

converges weakly in C(oJO to a Gaussian process,

In

(3.24i) sup|n_1J H{sin[s(c j+vj -pvj_l)]— Esin[s(cl+v1—pv0)]}| = op(1)
and I
(3. 24ii) suplnlJ n Yj_j1{cos[s(£ +vJ-pvj_1)]— Ecos[s(61+v1—0pv )]}|= 5p1( ).

In view of (3.22) — (3.24), to study the weak convergence of
n1/2[pn— p + 1111)] it suffices to study the weak convergence of the first

term in (3.22) when centered.

Proof of (3.23). Denote
{M(S) = c08[S(ej+vj-pvj_1)] — E cos[(£1+v1—pv0)] V s 6 9y:

Since Yj — ij_1 and Yk - ka_1 are independent for all | j—k| 2 2 .we
see a(k)= 0 Vk>2. Also, 3691’, because 7n —-10

(3.25) {10121 = 193612 n(s) + 2(n—1)n—1E§1,n(s)§2,n(s) —+ Var{cos(sco)],

which is positive because of the assumption (b), 0121 as in Theorem 1.3.1.
The remaining conditions of Theorem 1.3.1 are trivially satisfied.

Hence from Billingsley (1968, p. 49) the finite dimensional

(3.26) distributions of n __111/2 2 {jn (s) converge to that of a Gaussian
J: 1 ’11

process. Also, for any 3, t e of

1/211 _ —1/2“ 2
(327) El11 1215M“) n 1315,4111 —

69

vm[€1,n(3)-€1,n(t)] + (%)COV[§1,D(S)-§l,n(t), €2’n(S)—€2,n(t)]-

From the Cauchy—Schwartz inequality and the Lipschitz property of the
cosine function the r.h.s. of (3.27) is dominated by Clt—sl2, where
0 < C 6 IR.

Thus from Billingsley (1968, Theorem 12.3) the process

11 .
(3.28) n—l/2 2 {j n(s) is tight, and its weak convergence to a Gaussian
i=1 ’

limit in C(ofl—space follows from Billingsley (1968, Theorem 8.1).

The weak convergence of the second process in (3.23) to a Gaussian limit

in C(ofl—space follows similarly.

Proof of (3.24). The l.h.s. of (3.24i) without the sup can be dominated by

_1 n ..1 n . .
2n jgllvj—ll + In j21Xj_1{srn[s(cj+vj-pvj_1)] — s1n[scj]}| +
(3.29)

.1”
+|n .2 X

_1 n
sin[se-]| + |n 2 X. 1|.
1:1 J ' 1 J“

j—l

That the first term in (3.29) goes to zero in probability follows from the
assumption (c) and 7n = 0(1). From the Lipschitz property of the sine
function, the Cauchy—Schwartz inequality, the S.E.T., assumption (c) and
7n = 0(1), sup norm of the second term in (3.29) converges to zero in
probability. That the sup norm of the third term in (3.29) converges to
zero as. follows from H—W (Lemma 3.1). The last term in (3.29)

converges as. to O by the S.E.T.. The proof of (3.24ii) is similar.

70

It remains to study the weak convergence of the first term in (3.22)

when centered. To that effect let
£j,n(s) = Yj_1[{E cos[s(cl+v1—pv0)]}sin[s(cj+vj—pvj_1)] —
_ {E sin[s(cl+v1—pv0)]}cos[s(£j+vj-pvj_l)]] - 3%mn(t,s)|t:5(s)pn(s).

We shall first prove the finite dimensional distributions convergence of

—1/2 D .
n 2 5. n(-) usrng Theorem 1.3.4. Take
i=1 ’

0n(x,y) = x[{Ecos[s(cl+v1-pv0)]}sin[s(y—px)] — {Esin[s(cl+v1-pv0)]}-
°COS[S(y-p><)l] ,
0(XsY) = X{U(S)8in[S(X-Py)l - V(S)008lS(X-Py)]}s x, y 6 IR,

h:-=. . . .
x, X], Y], vJ

(D
x, y 6 IR, in Theorem 1.3.4. Since Xj = 2 pkcj
k=0

assumptions (a) and (c) and Pham and Tran (1985,Theorem 2.1) with

as in the model assumptions with w(x,y) = x+y,

—k a.s., using

6 = 2, A(k) = pk, gives {Xj} to be strongly a—mixing with

(3.30) a(k) 5 Cplplzk/3, V k 2 2, for some Cp > 0.

By assumption (b) and (c), 12 = sxg Var[u(s)sin(seo) — v(s)cos(sco)] > 0
for each s e of Thus all the conditions of Theorem 1.3.4 are satisfied.
Hence the C.L.T. holds for E as defined above, for each s E of Now

using the argument as in (3.26) we get the required finite dimensional

71

distributions convergence. Since for all i and j with |i-j| 2 2, cj, vj

and Vj—l are independent of {(vk,Yk), k = i-ls i},

(3.31) Cov[£in(t), fjn(s)] = 0 V s, t E of

Using (3.31), the same argument as in (3.27) and (3.28), we get

11
11—1/2 2 {j n(-) converges weakly in C(efl—space to a Gaussian process
i=1 ’

with mean 0 and covariance Exgh(t,s). Thus from (3.3) — (3.5), (3.8),
Billingsley (1968, Theorem 4.1) and (3.22), we get (3.20). 0

Remark 3.1. From (3.8), the assumption Jfi 7n = 0(1) and simple
computations using (1.0.1), we can see that Jfi ”n in Theorem 3.2 can be

replaced by

(3.32) Vn(s) = — s-l|¢61(s)|_2[EXg]_1[Ecos[s(cl+v1-pv0)]~

~Evosin[s(cl+v1—pv0] — E sin[s(cl+v1—pv0)]Ev0cos[s(£1+vl—pv0)]].

Note that Vn(S) represents the asymptotic bias of n1/2(pn(s) — p).

Consider the following assumptions:
o1og=mo
(b) J13 7n = 0(1) and Zn -1 0 in probability.

(c) Jfi 7n -1 7 and Zn -1 Z in probability.

72

Using (1.0.1) and the continuity of the sine and cosine functions, one

concludes that under (a) or (b), sup an(3)| -—1 0 and hence Jfi ”n in
see’

Theorem 3.2 can be replaced by 0. Using the Lipschitz property of the
sine and the cosine functions, the condition (c) implies that
sup lun(s) - p(s)| -1 0, where
86c»!
_ -l -2 2—1 .
(3.32) u(S) - -s melon [EXOI {7E} zsmlscl—pzn dL(z)}[u(s) +

+ 27 EJ cos[s(61+2—1(1—p)z)]cos[s2_l(1+p)z] dL(z)} —

 

- {7E} zcos[s(cl—pz)] dL(z)} [v(s) + 27EJ sin[s(cl+2_1(1—p)z)-

~ cos[s2-1(1+p)z] dL(z)}].

Consequently Jfi 1111(3) in Theorem 3.2 can be replaced by u(s).

Note. If f is a double exponential or 40,02) density, ZI1 is

such that l'iTn' EIZn|3 < co and 7n = 0(1), then simple calculations
11

show that all the conditions of Theorem 3.2 are satisfied. on

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