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This is to certify that the

dissertation entitled

On Sequential Procedures Based on the Minimum
Distance Estimator and Robustness

presented by

Yingqi Shi

has been accepted towards fulfillment
of the requirements for

Ph.D. degreein §La§l§tl§§

 

 

Date August 7, 1989

MS U is art Affirmative Acrt'On/Eq ual Opportunity Institution 0- 12771

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DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU to An Afflrmdlvo ActionlEquol Opportunity Institution
ammo-m

ON SEQUENTIAL PROCEDURES BASED ON THE MINIMUM
DISTANCE ESTIMATOR AND ROBUSTNESS

By

Yingqi Shi

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
1989

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0n Sequential Procedures Based on the Minimum

Distance Estimator and Robustness

Yingqi Shi

ABSTRACT

Problems concerned with the sensitivity of fixed-width interval estimation
procedures under contamination have recently received attention. Jureckova
and Visek (1984) showed that the Chow-Robbins procedure is not robust in
the presence of an single outlier. Geertsema (1987) considered the normal
distribution case and modified the Chow—Robbins procedure so that it is
robust under some specific types of contamination. In this thesis, for the
scale—model, a sequential fixed—width interval estimation procedure based on
the minimum distance estimator is pr0posed. We study the asymptotic
behavior of our procedure under the gross error model. It is shown that our
procedure is robust. Asymptotic efficiency, asymptotic consistency and the
rate of convergence of the coverage probability are obtained. Asymptotic

normality of the stepping time is also proved.

To my parents and my wife

iii

ACKNOWLEDGEMENT

I wish to express my sincere thanks to Professor Joseph Gardiner for his
guidance and encouragement during the preparation of this dissertation. I would
like to thank Professor H. L. Koul for many valuable suggestions that has lead to
the improvement of Chapter 2. I would like to thank Professors R. Erickson and
C. Weil for serving on my Committee. I would also like to express my appreciation
to Professor R.V. Ramamoorthi for his inspiring suggestions and many discussions.
Finally, I wish to thank Cathy Sparks and Loretta Ferguson for their superb typing

of the manuscript.

iv

TABLE OF CONTENTS

Chapter Page
1 Introduction and Summary 1
2. Minimum distance estimation and expansion 6

2.1 Minimum distance estimator 6
2.2 Expansion 7
2.3 Lemmas and Proofs 9
3. Sequential minimum distance estimation
procedure 21
3.1 Preliminaries 21
3.2 Sequential fixed—width interval
estimation procedure 22
3.3 Optimality of the sequential procedure 23
3.4 Proofs 25
4. Behavior of the sequential procedure in the
presence of contamination 35
4.1 Introduction 35
4.2 Main results 36
4.3 Proofs 39
4.4 Simulation Study 48
References 57

CHAPTER 1: INTRODUCTION AND SUMMARY

Sequential fixed width interval estimation has been extensively studied since
the pioneer paper of Chow and Robbins (1965). See, for example, the book by Sen
(1981) and the monograph of Woodroofe (1982).

The basic problem of fixed width interval estimation is to construct a
confidence interval for an unknown parameter 0 of prescribed width 2d (d > 0)
and prescribed coverage probability 1—2a, where 0 < a < 1/2. The best fixed
sample size procedure, say nd, possessing the desired pr0perty generally depends on
the underlying unknown parameter(s). Therefore, the sample size cannot be
Specified in advance to solve the problem. Dantzig (1940) proved that there is no
fixed sample size procedure that can produce a fixed width interval estimate for the
location 0 with unknown variance 02 in the normal distribution case. Zacks
(1971), and recently Takada (1986), showed that there is no fixed sample size
procedure for constructing fixed width interval estimates for a scale parameter. In
order to remedy this dependence on unknown parameters it is customary to define a
sequential sampling rule often called a stopping time. Essentially this rule, say Nd’
samples observations sequentially, updates at each stage a suitable estimator of the
unknown parameter involved in the fixed sample size procedure and stops sampling
as soon as the number of observations exceeds the estimated best fixed sample size
nd. Thus one is led to solve the fixed width interval estimation problem by using a

sequential procedure. The performance of the sequential procedure is usually

measuredby:
(i) lim P 061 ]
dao 9‘ Nd
and
N
.. . d
(11) 11m E —,
#0 0 nd

where IN is the fixed width interval estimator with stOpping time N d' When

d

(llim P ”[0 E IN ] = 1—2a, for all 0 we say the procedure is asymptotically
-+0 d
N

consistent. When lim E 0 31—d- = l we say the procedure is asymptotically efficient.
d-+ 0

The problem of fixed width interval estimation of the common mean u of
independent and identically distributed (i.i.d.) observations, with unknown variance
02, was considered first by Chow and Robbins (1965). They use the sample mean
as an estimator of p and the sample variance as an estimator of 02. They showed
that in the limit as (1 tends to zero the coverage probability is 1—2a and the
expectation of the ratio of the stOpping time to the best fixed sample size converges
to one. A closely related problem to fixed width interval estimation is bounded
length interval estimation. Here an interval estimate of width at most 2d is desired.
In a nonparametric situation, Sen and Ghosh (1971) considered a general class of
linear rank statistics and used them to obtain bounded length sequential confidence
intervals for the regression parameter. Jureckova and Sen (1981) proposed bounded
width sequential confidence interval based on M—estimators. Procedures based on
U—statistics and L—estimators are discussed in Sen (1981).

Csenki (1980) studied the convergence rate of coverage probability in the
case of location parameter with unknown variance 02. The main tool he used is the
fundamental result due to Landers and Rogge (1976), a Berry—Essen type theorem
for random summands of i.i.d. random variables. Csenki's result was extended to
procedures based on U—statistics by Ghosh and Dasgupta (1980), MukhOpadyay
(1981) and Mukhopadyay and Vik (1985). Geertsema (1985) obtained a
convergence rate on the coverage probability for a procedure based on the sign test
for estimation of the center of symmetry of a (symmetric) distribution.

Problems concerning the sensitivity of the fixed width interval estimation

procedures to contamination have just started to receive attention. Jureckova and

Visek (1984) studied the sensitivity of Chow—Robbins procedure to single outlier
contamination. Their results show that both asymptotic coverage probability and
the asymptotic mean of the stOpping time are affected by the presence of an outlier.
Geertsema (1987) considered the cases of single outlier contamination and a
particular gross error model. He compared Chow—Robbins' procedure with a
modified Chow—Robbins' procedure which uses, instead of the sample variance, a
fractile—difference estimator as the variance estimator. This modified procedure
behaves better for average st0pping time under single outlier contamination models
and for coverage probability under symmetric contamination.

In this thesis, we consider the problem of fixed width interval estimation of a
scale parameter. Let {Xn’ n 2 1 } be a sequence of independent and identically
distributed random variables having distribution belonging to a family
{F0(o), 0 e 9}, where F0(x) = F0(0x) and F0 is a fixed distribution. We are
interested in constructing a confidence interval for 0 with prescribed width 2d and
coverage probability 1-20, where a E (0, %) is prespecified. In addition to having
a procedure which is asymptotically consistent and efficient, we are also interested
in having a robust procedure in a sense which will be clear later. For this reason,
we pr0pose a sequential procedure based on minimum distance estimation (MDE).
We shall show that, actually, the robustness prOperty of MDE passes into the
sequential procedure.

We now give a brief review of the literature on MDE. Wolfowitz (1957) first
published several fundamental papers which initiated the minimum distance
method. He proved a consistency result and gave a number of intriguing examples.
Since then many researchers have studied minimum distance estimation. Blackman
(1955) proved the asymptotic normality of the LZ—distance MDE for a location
parameter. Bolthausen (1977) extended Blackman's result to a general parameter
model. Pollard (1980) studied the minimum distance method for testing problems

as well as point estimation problems. Boos (1981) considered weighted
Cramer—Von Mises distance for a location model and obtained the Optimal weight
in the sense that the corresponding MDE is asymptotically efficient. Koul and

De Wet (1983) studied MDE in a linear regression setting. Millar (1984) presented
a rather general approach to a broad class of MDE problems.

In the amt of robustness, Parr and Schucany (1980) studied the robustness
pr0perty of several minimum distance estimators for location and location—scale
models. They concluded, from both theoretical and Monte—Carlo results, that MDE
is competitive and seems easier to apply to general estimation problems than
maximum likelihood estimators. Parr and De Wet (1981) studied the influence
curve MDE. Millar (1981) gave, based on a reasonable mathematical framework, a
general theoretical justification for the robustness of MDE. Recently Donoho and
Liu (1988) showed that the minimum distance functional defined by a certain
distance )1 is automatically robust over contamination neighborhoods defined by p
and the minimum distance functional has superiority among Fisher—consistent
functionals with regard to sensitivity and breakdown point.

We conclude this introduction by giving a general overview of this thesis.
Section 2.1 of Chapter 2 describes our underlying model and our minimum distance
estimators. In Section 2.2 and 2.3 we establish a representation of the minimum
distance estimator in terms of a sum of i.i.d r.v's and a remainder term which
vanishes at a certain rate.

In Chapter 3, Section 3.1 we introduce some necessary prerequisites
regarding the sequential procedure. Section 3.2 outlines our sequential procedure
while Sections 3.3 and 3.4 establish the asymptotic efficiency, asymptotic
consistency a the rate of convergence of coverage probability together with a result

on the distribution of the stOpping time.

Chapter 4 begins with some preliminaries leading to the asymptotic behavior
of our sequential procedure under the gross error model in Section 4.2 and 4.3. As
an example a comparison with Chow—Robbins' procedure is made for the

exponential distribution. Section 4.4 lists results of a simulation study.

CHAPTER 2: MINIMUM DISTANCE ESTIMATOR AND EXPANSION

§ 2.1 Minimum Distance Estimator.

Let X1, X2,...,Xn be independent observations with distribution function
F ”(x) = F0(0x), where 0 E lt+ is an unknown parameter. The empirical
distribution based on the sample )(1,...,Xn is denoted by Fn' Let h be a
non—negative measurable function defined on (-oo, co). We are interested in the
following Cramer—Von Mises type of minimum distance estimator of 0.

Definition 2.1.1. Let
1).,(0) = l(Fn(t) - Footnzohzwndt.

an is said to be a minimum distance estimator (MDE) of 0 if an satisfies:
Dn(bn) = i221? an).

The following result gives a sufficient condition that ensures the above
minimum distance estimator is well defined. The proof is similar to Beran (1978)
and can be found in Boos (1981).

Lemma 2.1.2. Suppose h is bounded and F0 satisfies j F0 (1 -— F0)dt < 00. Then
0(Fn) = {0 e lt+z an) = 31161}pr Dn(u)}
is non-empty and compact.

Since 0(Fn) is compact, one can uniquely define tin, for example, as
(sup 0 (F11) + inf 0(Fn))/2. There are some connections between this type of MDE
and other statistics. Let the weight function h be fan, where f0 is the density
of F0. Then Dn(0) becomes the Cramer-Von Mises distance between FH and
F0(0t). It is known that the distance function has the following representation

(Anderson and Darling (1952))

DM = iii, (1:0("X(i))’giflil)2+ 1%.

where X“) S ....g X01) are the order statistics. This representation allows an easy

computation to get the MDE all. For example, we can use a non—linear least

. f
squares procedure to find 0n. If h = (Flew—If”, then the distance is called
0 O

the Anderson—Darling distance. In this case, we have
n
an) =i21(2i-1) (ln F0(0 Xm) + 1n (1— F0(0 X(n+l—i))) —1.
Parr and De Wet (1981) pointed out that the derivatives of Dn(0) generally
possess simple forms so that the Newton-Raphson routine can be used to compute
the estimator. Thus, the MDE are essentially no more costly to compute than the

usual M—estimators.

§ 2.2 Expansion.

The basic assumptions that will be assumed to hold without further

reference, are the following:

(I) IF0(1- F0) (It < oo.

(2) f0,

(3) an) and tf(t) are L2(dx) — functions.
(4) The weight function h in the definition of the MDE is

the density of F0, has continuous first derivative f0.

non—negative, Lebesgue integrable and absolutely
continuous. There exist numbers a and b (>a) such
that

{t: f(t)h(t) > 0} 3 [a, b].

The ac. derivative of h satisfies, for some constant M,
(2.2.1) sup Iniml < M.
tell

Remark. From the robustness point of view, weight functions that have compact
support are usually considered (See Parr and DeWett ( 1981).) In this case,
obviously, (2.2.1) is also satisfied. If the parameter space is a subinterval of R+,

then the condition "h is Lebesgue integrable" can be weakened to "h is bounded".
Assumption (1) guarantees that 111/2 (FH — F0) is in L2(dt) a.e. (P) for all n 2 1.

We introduce the following notation:

FM) = F0, he: 01/2h(a),
6 F0 _
"a: ‘57“ "a: We

and L2(dt)—norm will be abbreviated by ||-||. M and C will always denote
positive constants, c and (1() usually denote small positive constants sometimes we
write C(fl) or M(fl) to emphasize the constants only depend on [3. Throughout
this paper we may take P 0 to be the product probability induced by the sequence
{Xn, n 2 1} and E 0 to be corresponding expectation. Let E denote the
expectation under the distribution F0. Note also that assumption (1) is equivalent
to E | X1 | < 00.

The following result will allow us to restrict our attention to a neighborhood
of 00.
Pmposition 2.2.2. Let 00 be the true paramater. Then for any 5 > 0, there exists a
positive constant C(00) such that

P00“ 63,— 0I/a0 2 e] s 1’00“an - F00” 2 C(00)].
The next theorem is the main result of this chapter which gives a local almost sure
expansion of the MDE. We shall use this result later in Chapter 3.

Theorem 2.2.3. Let 00 be the true parameter and an be the minimum distance

estimator. Then

. n
Jfi (0n — 00) = J-g—igl 112(Xi, 00) + Rn a.e. (P 00)

where

,0 = _ -F b ‘ d ‘ 2
we 0) Hum} 000)) 00m 17900) t/ llnooll
and R11 satisfies

anl .<. Moo) «60an — F90“? + Ian - Fool'3/2) a.e. (P90)

on the set [I in -- 00l/0035] for some ((00) > 0.

For the purpose of showing asymptotic normality, it suffices to have
Rn: op(1). Many researchers have established such results. For instance, see
Pollard (1980). The result shown here emphasizes the convergence rate of the
remainder which is crucial for obtaining the optimal sequential procedure later. The
next result gives a convergence rate of lan—FOII, the L2—distance between F11 and

F

0.
Pmposition 2.2.4. Assume EIXIr < on for some integer r 2 1. Let {nd, d > 0} be
a sequence of positive integers such that nd = O(d)-2. Let t d = d25 with

A ~2A
0 S 6 < min { %, 37—1} where A1 20, A2 2 1 satisfy 2A1 < A2. Then there

exists (10 > 0 such that, for d < dO’

2rA 4rA
-2r —3r 1 —4r 1
A1 A2 12— 7 + 1;— ); ‘2”12—

where M is a constant depending only on 0 and r.
As a direct consequence of Theorem 2.2.3 and Proposition 2.2.4 we obtain
the following corollary.
Corollary 2.2.5. Under assumptions (1) — (4),
«a (in — 00) 11 Na. «72(00»
where 02(00) = (bdo)2 and

2 2
b2 = (“Po (3 A t) — F0(s) F0(t)) sh2(s)f0(s)th2(t)f0(t)dsdt)/ (f(th(t)f(t))dt) .

10

§ 2.3 Lemmas and Proofs.
In the sequel, we shall normalize 0 and in as follows:
Fawn/00
and
u=./fi( 011—00)] 00.
Accordingly without loss of generality, we may assume 00 = 1.
Proof of Proposition 2.2.2. It suffices to show that for some
6 > 0
[lulNfiz e] c [ || Fn-Fon > 61.
By the triangle inequality and the integrability of the weight function h we
have
2 ||( F1 + u/Jfi— F0)h1 + u/JH” a.e.(P00) for any n.

It then follows that

(2.3.1) ||(Fn—Fl+uNfi)hl +uNr—l||+M||Fn—F0u

0n [lul/Jfiz 6]-
For the right hand side of the above inequality we have

”(F1 + 11/7? Fo)h1 + no" = U (Pom - Fob/(1 + u/m))212(.)dt)1/2

2 (1:11:00) — Fan/(1 + u/«fi)))2h2(t)dt)1/2

2 (b' — a’) inf (F0(t) — son/(1 + u/Jfi)))2h2(t)
a’$t_<_ b'

where b’ > a’.

11

By taking infimum over [III I / 15 2 e] on both sides of above inequality we get

(232) lull/1:5 2 e "(F1 + 11/75 ' F0)h1 + UNIT"

. . 2 2
2(b'-—a') inf inf (F (t) —F (t/(l+u/JIT))) h (t).
lul/Jn' > e a'StSb' 0 0
Let [a,b] be the one in assumption (4). Choosing [a',b'] C [a, b] and combining
(2.3.1) and (2.3.2) , we get

(2.3.3) ”(Fn — F0)hl + 1Win + M lan — F0"

,_ , - - _ 2 2
ac a)|u|;31fi>ca’gtl§b' (F00) F0(t/(1+UNE)))h (t)

2(b' —a') inf h2(t)min {(F0(t) — F0(t/(1 + c)))2,
a'gtgb’

(Foo) - Fons/(1 — em?)

= C(c) > 0.

That C(c) is positive follows from the fact that
2 . 2 2

h (t) m (20(1)- Foe/(1 + e») . (F00) — F0(t/(1- 6)» }
is positive and continuous on [a’, b'].

Finally, replacing u by 13, we get, by definition of {1, boundedness of h
and (2.3.3), following inequality:

2 M lan - F0)” 2 "(FD _ F0) h ll + M lan " F0”

.2 ”(Fn - F1+{,/yfi)h1+{1/yfill+ M lan ’FOIIZ 0(6)
on [|u|/Jfi> c].

12

Therefore the Pr0positon follows.
Lemma 2.3.4. There is a c > 0 and M > 0 such that

lfiI/Jfi s M "F, - F0" a.e. (P00)

on set [IuI/v‘fis 6].
Proof: Applying the mean value theorem to F 0 with respect to 0 we get

(Foal +11/mt1-FF0(1))2111 1 {Mm

2 2 22
=t f 0(t(1 + 7(u)))(u/Jfi) h t) a.e. (P )
1 + u/vfi( 00
for anyuwhere 7(a) is between u/Jfi and 0.
Let [a, b] be the interval in assumption (4). It is easy to see that there

exists 6 > 0 and a subinterval [a’, b’] of [a, b] such that

22 .
inf tf tl+u 1nf t C>0.
a, <1< b,{ 0(( )1} .1515 b, Hum )1
'“"‘ IuI/JESc

Thus, for every t,

(F1 + II/JI-l (0— F02(t‘)) h] + Il/JII- (t)
213311 (1 + 201)» hi 1 11/76“) (In/m2 (M (1)

20(00) (fi/m21[,.,b.](t) a-e- (P00)
on [luI/Jfis e]. Then
(2.3.5) II(F1+{,H5 -F0)h1 1W," 2 (b'—a')C(00) Ifil/Jfi
a.e. (P0) on [lul/Jfis c].
By the triangle inequality, the definition of {1 and the boundedness of the weight

function h, we also have

13
(2.3.6) ”(F1 + {1 / .5 — F0)hl +{1/,/fi"
s no, — F1 1 {W521 11,511+ "(Fn - Foul +11%“
5 ”(FD — F, 11%», 11,151 + M Ian — F0"

_<_M lan—FOII a.e. (P00) on [nil/fig 6].

Finally the result follows by combining (2.3.5) and (2.3.6).

Lemma 2.3.7. There exist 6 and M > 0 such that
.. ~ 2

on [nu/.55 e] forevery 11.

Proof: Using Taylor expansion, assumption (3) and Lemma 2.3.4, we have

"(F1 1 fi/fi‘ F0 — (filmnllzs Manna/«6| urn-F0121

0 .
I] n(12 f(st))2ds dt|
”o

A

. 0 .
s M luNfil urn-FOP II “ s‘5m2 1(1))231 dSI
00
4
g M “Fn_F0" a.e. (P 00)

on [ lu/y‘fil 5 6]. Therefore the lemma is proved.

Before stating the next lemma, we introduce following notation:

Dn<u1= II (F. - F1 + 3.51111 + W. "2-

1311(3)..- II (F, — F0 411/7622"?
5F 0
Here 1} = W— .

l4

Lemma2.3.8. There exist 5 and M(e) such that
' 4 3
811 ID(11)-D(11)|5M(6)(||F -F H +||F -F II)
11115an IIF,—F,II “ n n 0 n 0

Proof: By the triangle inequality we get
I Dn(u) - 13n(u) I s I + 11 + 111
Where
I = n I 1(F, — F, — Month? 1 ,,1,— h2)dtl

II = nII(F1 1 ,,,,—F,—(u/Jfi)n)2 hf 1 ,Nfidtl
111 = 2|](Fn ---F—(u/,/fi)n)(sl + u/Jfi—FO—(u/Jmn) hf + 11/1/11 dt.

By assumption (4), we have

“111 ,N,(t)-h(t)| s 1(1 + u/ml/2-1||h((l+u/«/fi)t)l

+ |h((1+ wot) -h(t)|s IuNfi IM +11]+ ““5 M/s dsl

s we IM + |(1—(1/(1+U/~/fi)2l|s luNfilM+ luNfilM(c)-
Using the above inequality and noting |u |_<_ MJfi ||Fn— F0“ we get

1 = n |I(F, - F, -(UNIT)17)2(h1 1 ,N, + ma, 1 ,Nfi-hwtl

s n M IU/Jfil 1((F, — F,)2 + (Mocha

s n M Iu/(fil (n F, — F,"2 + 111/512 122cm

:11 M II F,-F||3-

15

For term 11, using differentiability of F we get
4

IISnM || Fn—Foll .
For term III, applying Cauchy—Schwarz inequality and using the same argument as
that in I and II, we get

2
III 5 nM {](Fn—F0—2(u/;52)n) dt [(Fl + “gm—F—
(u/(nn) at} / snM uF,-Fu .

The Lemma then follows by combining the bounds of I, II and III.
Now we are in the position to prove Theorem 2.2.3.
Proof of Theorem 2.2.3: Let 11 = (Jain—00V 00) and {i be the minimizers of D11

and D 11 respectively. Without loss of generality we assume 00 = 1. It is easy to

seethat
~ 1121: ¢(x0)
u=— ., .
451:1 ‘

By simple algebra and the definition of {1 we get
" " “ " ‘2 ”2 2 2
ID,(U) -D,(u)|=|l{-2(u-U)n(F,-F0) + (u —u )2 }h at

= I11 — {1 | | ](—2(Fn - F0)nh2dt + ((11 + u)n2h2dt]
=|{1 — {1| (—2{1 ](nh)2dt + ({1 + {1) ](nh)2dt|
=11} — {1121(nh12dt.

Thus

111,12 = 1 11—1112 = (I(nh)2dt)'lll3,({1) - 13,601

5 113,01) — 13,1501 + ID,({1) - 13,601 = A + B.

16

Choosing e such that Lemma 2.3.4 holds and applying Lemma 2.3.8 we then have

' 3 4
A s sup 10,011 — 13,0111: n M (11F,—F,11 + 11F,—F,11 1.
l11 l SMt/filan_F0ll

It remains to handle term B. By Lemma 2.3.7 we have, for some M,

1111 s M a 11F,—F,11. a.e.P. on [lullzltfi 5 e1.

By definition of u we also have

1111 #321; ax,,111= fil(F,-F,)h20dt/Ilhn||2

S M Jfilan—Foll a.e.P.
Thus it follows

B = | inf Dn(u) -— inf Dn(u) |
InlsMJfillF,-F,|| IUISMJfillF,-F,II

S|u|<M./1Ll]]Fn —F0| D,(u)--D “(UN

The last inequality follows by the fact |inf f—inf g] Ssup]f—g|. An application of
Lemma 2.3.8 then gives the desired result.
Following notations will be used in the next lemma and in the proof of Proposition
2.2.4.

K,(x) = 1(11x s 11- F,(t))2dt,

K,(x,, x2) = MIX, 5 t1 — F,(t)) (11x, 3 1]— F,(t))dt,

and

V: 2_ 2K2(Xi,Xj.)

17

It is easy to see that U11 and VII are U —statistics and Vn is stationary of order
one.
Lemma 2.3.9. IfEIX]r < on for some integer r 2 1, then

(1) E|K1(X)|r < 00

(ii) E 1K,(x,, x,112r<oo
where X1, X2 are independent and identically distributed.
Proof: For part (1), we first bound E |1<1(X)|r asthe following:

F |K1(X)|’ = 1 11(11x 511 — F,(t))2dtl" dF,(x)

.<.! I! I 11x 5 t] - F,(t)ldtlr dF,(x) = A + B
where A = I (] |1[x g t] — F0(t)|dt)r 1[x 2 0]dF0(x)

B = 1 (1 111x 5 t] - F,(t)ldt)1r 1[x < 01dF,(x)
For term A, we have

A = 13(1:(1—r0(1))dt + jfmro(t)dt)’or0(x)

= [‘31 13(1—r0(1))dt + 13F,(t)dt+ 19m F,(t)dt )rdF0(x)
Since E|X| < co, j3(1—F0(t))dt and ffmF0(t)dt are finite. Thus, there are
constants M1 and M2 such that

A g 13 (M1 + |x| + M2)rdF0(x) s 2’(Ml + M2)‘ + 2rE|X|r < .0.
One can similarly argue B is finite. Therefore part (i) is proved.

For part (ii). By the Cauchy—Schwarz inequality we have
I lllxl S t'l _ F0(t)l lllxg Z tl - F0(t)ldt

s (1 (11x, s t] — F,(t)12dt)1/2 (11111123 t] _ 20(1)]? (ml/2

5 (1 111x, 5 1]— F,(t)1dt)‘/2(I 111x, 5 t1 - F,(t)1dt11/2.

18

Thus

13111201,, x1112r = 11 111111111 11— F,(1)1(11x, s 11 — F,(1) «1112‘

E {(11 11x, .<. 11 — F,(1)1<11)’u 111112 5 11 — F,(1)1<111‘1

= (Fumx, s 11 - F,(1)Id1)’12.

The last equality holds since X1 and X2 are independent and identically

distributed. Using result of part (i) we get E|K2(X1, X2)|2r < 00.

Proof of Proposition 2.2.4: In the sequel we shall simply denote nd by n. We

2

write ”Fn—FOH asfollows.

11F,—F,112 = 1 11-15011, 1 s1 - F,0(s))211

=il§(§l [(1[xi <s]——F00 (3))2 ds

D

+ 2 2 [(1[xi < s]- F00 (s)) (1[xj < s ]- F00 (s))ds
i<j

_1 n—l
_HUn+—n_vn

raj
=_—fll(Un—Eun)ALEl—llvn + n"

Note that IE Unl is bounded we then have for small (1,

 

*1 "2 2 td
(2.3.10) 11,0111 IIF, — F00” > 1d] = PIIIF, — F 0011 > (T)
n

2
7S1

19

2 1,
sF,01,1,1U,—FU,1>11—T1r1+F11v,1>,1—§—1 1.

Applying Markov inequality to first two terms of the right hand side of
(2.3.10), noting that Emma 2.3.9 holds under given conditions and applying
Lemma 5.2.A and Lemma 5.2.13 (Serfling (1980)) to E W, — EUnlr
E |Vn|r respectively, we get following inequalities:

0 2

F,01%1U, —EU ,1>,11— dT1 21

n1

2
To. 1
<1- 1 £171 1 F1U,—FU,1

n
2

td—X2-rr/2
116113—3111

11
and

2
1 td 12
Pdollvn—Evnl>2(_XI) )l
n
2

2
d—X2'2r E21, 22r+1

1111:1111?) 1 F1v,- Ev,12‘<M121—X—1 1 n
n
where [x] denotes integer part of x.

Therefore

r t
I17 +( d —X—l') 2 n—2I').

.1...
tow-1

"1 A2 td
12,0111 11F, - F,011 > 1,1 s Maj—1

11 Il

20

Proof of Corollary 2.2.5:
By Pr0position 2.2.2 and Theorem 2.2.3, for any 6 > 0, there exist ‘1 and 152 such

that
PllR,l > 61 s F11R,1> 6, 111/151: 111+ Pllfi/Jfil > 1,1

3 PM 111111F, - F,112 + 11F, — F,113/21 > 11 + F111F, — F,11 2 1,1
2 3/2
5 PIMJfi 11F,—F,11 > 1221+ F1M¢611F,-F,11 > 6/21

'l' Pllan - Fol]? 6gl-
Using Proposition 2.2.4 with t 11 being constant, one can easily check each term on
right hand side of above inequality tends to zero as n -+ 00. Thus 11 has same

asymptotic distribution as $2 w(Xi,l). The latter has limiting distribution

N(0, b2) by Central Limit Theorem.

CHAPTER 3: SEQUENTIAL MINIMUM DISTANCE ESTIMATION
PROCEDURE

§ 3.1 Preliminaries

We list below, for the sake of completion, some notations and facts that will
be used in the sequel. Proofs will not be given since they can be found in the
indicated references.

Let (Q, .9; P) be a probability space and {53; n 2 1} be an increasing
sequence of sub—o—algebras of .9.’
Definition 3.1.1. A random variable N is said to be a proper stopping time with
respect to {55; n 2 1} if and only if N is positive integer valued and
{Nzn} 6 5:1 for n2 1.

The following two well known Lemmas can be found in Landers and Rogge
(1976).
Lemma3.1.2. Let {an n 2 1} and {Ynz n 2 1} be two sequences of random

variables. Assume that
SUP IPIX St]-¢(t)| =0(a)
telR ‘1 n
where (b is standard normal distribution, and
P[|Yn| > an] = 0(an).
Then
2161p |P[Xn + Yn 5 t] —¢(t)| = 0 (an).
Lemma 3.1.3. Let {an n 2 1} and {Ynz n 2 1} be two sequences of random

variables. Assume that
sup IPX _<_t — (t =O(a)
tell [ , 1 <1 )I ,
and

P[|Yn—1|> an] = can).

21

22

Then

:23 1F1x,s 111,1 —111111 = 0 11,1.
The next theorem given by Landers and Rogge (1976) will be the major tool for
establishing the convergence rate of the coverage probability result in Section 3.3.
Theorem 3.1.4. Let {Xn: n 2 1} be a sequence of independent and identically
distributed random variables with mean u, variance 02 > 0 and finite third

moment. Let {5!}: n 2 1} be a sequence of positive integer—valued random variables

and assume that

5n
Plln—T-1l> in] = 0 (JED)
for some constant 1' > 0 and sequence {cnz n 2 1} satisfying :15 ‘11 -1 0 as n -+ 00.

Then

E
111 supIPl 2’1 1x1-21/avn—rs11-111111=o1¢z,1,
tER v=1

1
1111 MM 2n1x1—21/on s11—111111 =o1r1,1.

tell v=1
§ 3.2 Sequential fixed—width interval estimation procedure.

Let in be the minimum distance estimator (MDE) based on the sample
Xl"”’xn as we have defined in Section 2.1. From Corollary 2.2.5, we know, when
the true parameter is 0, J11 (in—0) converges to N(0, 0%) weakly as 11 tends to
infinity. Then it follows that

 

2 o
P0[|0n—0|_<_ 0Z0} e1—2aasn-1m
Jfi

where 20 is the (1—a)th quantile of the standard normal distribution.

It is easy to see that to obtain an interval estimate with width 2d and

o
approximate coverage probability 1—2a, a sample size nd = [ (Jag—)2] should be

23

taken. Here [x] denotes the integer part of x. Since 0 is unknown, so is 11d.
Hence one cannot specify a fixed—width interval estimate by using this fixed sample
size procedure. To overcome this difficulty, one is naturally led to explore a
sequential scheme to achieve the objectives. We define our sequential procedure via

the following stOpping rule. Let

. za 2 ‘ 2 —1
(3.2.1) Nd = nun {n 2 m0, n2 (T) ((bdn) + n )}.
Our confidence interval for 0 is then
I = [b —d, b + d].
Nd Nd Nd

In (3.2.1), m0 is an initial sample size, the term 11—1 is used to prevent early

A

stOpping, 011, of course, is the minimum distance estimator defined in section 2.1

and b2 is a constant given by

 

( ) b2 [(F0(s A t) — F0(s)F(t))sh2(s) f0(s)th2(t)f0(t)dsdt
3.2.2 = .
2 2
1111111111,1111 11)
§ 3.3 Optimality of the Sequential Procedure.
The following are the main results for the sequential procedure defined in the
last section. Theorem 3.3.1 and Theorem 3.3.4 give pr0perties of the stopping time

N d' Theorem 3.3.2 provides a convergence rate of P 0 [bNd —0 5 dzzlx] to its limit

in the sup—norm sense as d approaches zero. In the sequel, 111 denotes the
standard normal distribution function, 11d as we mentioned in Section 3.2, is the
Optimal sample size for the fixed—sample procedure and b2 is the same as defined
in (3.2.2). In the following results all limits are taken as d —-+ 0. E denotes the

expectation with respect to distribution F0.

24

Theorem 3.3.1. Assume E|X| finite. Then, for each 0,
(i) for fixed (1 > 0, Nd isfinite a.e. (P0).
(ii) as (1 decreases, Nd isnon—decreasing a.e. (P0).
(iii) Nd-oo a.e. (P0).
Nd
(iv) n—d-il a.e. (P0).
N
d
(v) E —-11.
0nd
Theorem3.3.2. IfEIXIr<ao forsome r23, then
sup 126,121N d—dgdz;IX]—¢(x)| = om“),

xeR
where 0 < 6 < min {481.7316’4'}

Corollary 3.3.3. If E|X|r < on for some r 2 3, then the sequential procedure is

asymptotically consistent. In fact,

PondN d—0|<d]=1—2a+0(d6)
where 0< 6<min {gig—,1 4}'

Theorem 3.3.4. If EIXIr < 00 for some r 2 3 then
N —n
d d £2. N(0, 4112).

in11
Remark 3.3.5. The above results are actually proved under the assumption that h

 

is bounded. When h is integrable Theorem 3.3.2 and Corollary 3.3.3, under the
weaker condition E | X] < 00, can be strengthened in the sense that 6 can vary from
0 to 1/4.

Csenki (1980) considered fixed—width interval estimation for the population
mean in the case where the p0pulation variance is a nuisance parameter. He gave a
convergence rate for the coverage probability of the sequential interval estimate
which is based on sample mean and sample variance. Ghosh and Dasgupta (1980)

extended Csenki's result to U—statistics. Geertsema (1985) obtained a convergence

25

rate of the coverage probability for the problem of a estimating the center of
symmetric distribution. His sequential procedure is based on the sign test. The
convergence rate of the coverage probability shown in Corollary 3.3.3 is different
from that reported in Csenki (1980), Ghosh and Dasgupta (1980) and Geertsema
(1985).

§ 3.4 Lemmas and Proofs

In the sequel we shall denote (b70102 by 0112 . We first establish the following
Lemma.
Lemma 3.4.1. Suppose E|X|r< no for some r 2 1. Let t d=d
r1d =[(00Za/d)2]' Then there exists (10 such that, ford < d0,

251111111 0 g 651/4 and

n—3/2r -2r 1.

‘ 2 2 -2r

Pdland—oal >td]SM(0,r)[tdrn +tdr nd
Proof. By simple algebra

. . t . t

2 2 2 d d
Palland - 00] > td] 5 P0[|0nd—0| > 75] + Pddldnd—fl > 75].
The result then follows by application of Proposition 2.2.2 and Proposition 2.2.4.
Proof of Theorem 3.3.1. All limits are taken as d -1 0 or n -1 00.

(i): By Lemma 3.4.1 and an application of the Borel—Cantelli Lemma, we have
{In -1 1'70 a.e. (P 0). It then follows directly from the definition of N d that N d is

finite a.e. for each d > 0.

(ii) and (iii): For (11 < (12 and alln21,
111N111,+n‘11>1%—,2—2112, +111...1ae1F,1
Thus, by definition of Nd’ N d 2 Nd a.e.. (iii) follows easily from the fact that
l 2

A -1A
an 0036

26

(iv) Again from the definition of the stopping time Nd’ it follows that, for small (I,
—l 2 2
(Nd—1) + ”Nd—1 l> (Nd —l) (—)2 2 (Nd —1)Nd l1‘7(oNd+ Nd 1).

 

 

 

 

Thus
2 (2,2 —1
(Nd—1)l+ ”N -1 N N —1 + Nd
d > d— N ( d )2> d . N11
2 N; d z or - N 2
00 a 0 d 00

Since N d -1 no and 613d -1 00’ it then follows from the above inequality that

lglimNd/(T)<l1mNd/(z—aa——)<1ae(P0).

Nd
Hence n—d—1 1 a.e. (P0).

(v) Let c> 0. Define “Id: [nd(1- 6)] and n2d= [nd(l+c)].
Nd
Express n_ —l as follows:
11

Nd _Nd Nd

N
d

where [A] is the indicator function of event A. By taking expectations on both
sides of (3.4.2), we get

Nd Nd
(3.4.3) |E0(——1)| <E0—d[Nd (n1d] + Edl— d—ll [n1d< NdSn2d]
ddN
+ Baird [Nd > n2; + P91Nd 5n1d1+ P,[N, > 112,]-

It suffices to show that each term on the right side of above inequality tends
N
to zero as d -1 0. For the first term, since -n—d -1 1 a.e. P 0, we have:
nd
ENd
Ea—[Ndnild] <hd_d PolNd(“1dl <(1—c) P0[—<1—e]-+0

27

For the second term on the right hand side of (3.4.3), we get
N
d
Baird—1' ["111 < Nd 5 “211]5 ‘ Polnld < Nd 5 “211]S ‘5

uniformly in d.

N
By definition of n and n and the fact that -—q -1 1 a.e., it follows
1d 2d 11 d

easily that the fourth and fifth terms on the right hand side of (3.4.3) vanish as 11
tends to zero. It remains to deal with the third term of the right hand side of

(3.4.3). This we express as follows:
N d n2d+1 1
d d d n>n
2d+1
n2d+1
It is easy to see that

 

dP0 [Nd > n.2d] -+ 0. We then only need to show

)3 PolNd > n] <00.
n>n2d+1
Let n 2 n2d. By definitions of N d and nd, it follows that, for sufficiently small

(1,

2
dn
—1 22 n2(1-1 2
P0[Nd>n]S P0[n +0n>£zl—2_]-<- P6102 —oa> —z-2——n2d— 00]
Zn
2

22 2 “’0
SPoflon—00I>T].
An application of Lemma 3.4.1 with t (1 replaced by constant gives

22 2 “’3 —s
Pdldn— 00] > T15 C(e,0)n
where fl > 1 and C(e,0) > 0. Thus 2 P 9 [N d > n] converges.
n>n
2d+1
Before proceeding to the proof of Theorem 3.3.2, we first establish following

Lemma concerning the convergence rate of the stopping time N d‘

Lemma3.4.4. Assume Eler finite for some r 21, then, for

28

o s 6< min {$152, ,1},
2

Pled/nd—ll > d ’1: 0(115).

Proof. Define §d=[nd(l + (125)]. We have the following inequality:
Nd 211) _ 2
(3.4.5) P0(|n—d—l| >11 —P0[Nd>§d]+P0[Nd<nd(l—d )1.
We shall show that the two terms on the right hand side of (3.4.5) have order
0(d5). By definition of N d’
-l 2 22 -1
Pg [Nd > 6d] S PolCd < (Zad ) (06d ‘1' {d )l

g P 015d — :3; < (2 011492;]

 

2 26 d 2 d 2 22
SP0[00(1+d Hag-2911011) < 05d]

”3‘12 6

22 2
_<_ P0[|o€d—ool > —2—.
An application of Lemma 3.4.1 gives then

2 26
a - —
Poll 52d - 17%| > '4‘] S M”, r) {d—4r66d3/2r + d—8r66d2r}.

Thus, for d small, we get
(3.4.6) P 0 [N d > 5o] 5 11(0, r) {cl-4’5 “3’ + d‘8‘5+4r}.
We now deal with the second term of the right hand side of (3.4.5)

Let a - [Z] — [n (1426)] and a - 33d where [x] denotes the
111-313311— 11 211—2“

integer part of x. It is easy to see Nd 2 a1d a.e. for any 11. Since

a1d < a2d < 33d eventually, we can bound, for sufficiently small 1], the second

term as follows:

29

(3.4.7) P0 [Nd < nd (1425)] = P0 [21d 5 N(1 < nd (1425)]

a a
3d 3d 2
_<_ 2: P0[Nd=k] < 2 124153—1131 > o3—953]
k=a 1d k=a111 Zn

a 02 a
2d 3d 2
< 2 (1H,”? —o3|>—g]+ 2: P0[|o 3— o—31>o3dfi.
k=a1k=
a2d
Applying Lemma 3.4.1 to the first term in the right hand side of (3.4.7), it follows

that
3’2d 02 3'2d

(3.4.3) 11:2 P0[|&k —o3| > ]<” 2 M(0, 1)(k‘3/2r +112“)
31d k=ald

50

5 M09, r)d .
where 0 < 30 < 3:- 1.
Applying Lemma 3.4.1 to the second term of the right hand side of (3.4.7 ). We get

a
3d
(3.4.9) 2 P6,“?!k - 0%] > ogdzfi
d

a
3d

k=“‘2d+1

—46r+2,61 —86r+2fl2
_<_ M(0, r) {d + d }.
where 0331 <gr-l and 0$fl3<2r—1.

We may now conclude from (3.4.6) — (3.4.9) that, for 11 small,

N
d 2
3.4.10 P ——1 >d

30
—46r+3r —86r+4r 60 —46r+261 —861+2fl2

5M(0, 6){d +d d +d +d }.

Finally we need to find the range of 6 such that the right hand side of
(3.4.10) is 0(d6). Since 6 satisfies the inequalities:

—46r+3r 2 6, -86r+4r 2 6, -46r-1-2fll 2 6, —86r+2fl2__ > 6
with fll< —l and [32 < 2r — 1, some simple algebra leads to 0 < 6 < 8—1"
Therefore the lemma 1s proved.
Proof of Theorem 3.3.2:

We examine the convergence as d -1 0 through any sequence {dv} such

that dv-10 as v-ioo. Let xeR. FromTheorem2..,23
R

 

<x].

- -1x
P 0 —0Sdz =P TQ—N—i N2

We then only need to show:

 

 

 

 

(3.4.11) sup [P a( 2 + ) g x] —¢(x)| -_- 0(115).
4%TT1010 0 0 d
To prove (3.4.11), by Emma 3.1.2, it suffices to show the following:
Nd
0' 111(X 2 00)
(3.4.12) sup |P0[-—d—— 02“ i231 i < x] —¢(x)| = ()d( d6)
and
zaRN 6
(3.4.13) P0 [1 1 > o5] = 0(d ),

d
where 0 < 6 < $373. For (3.4.12), by Lemma 3.1.3, it suffices to show the

following:

N—
(3.4.14) P0[| iii-1| > d6] = 0(d6)
and

N
d “x. ,0)
1
(3.4.15) sup IP0[— 2 00

 

s 21—1112111 = 01151.

31

Nd— Nd
Since [ I j — II > d6] C 3[|—&— II > dfi, (3.4.14) follows immediately from

Lemma 3. 4. 4. Since E|X|3 < m and since nd2< <d25 for 6 <1 and small d,
the conditions of Theorem 3.1.4 are satisfied. Then (3.4.15) follows directly from
Theorem 3.1.4. Thus (3.4.12) is proved.

We now prove (3.4.13). It follows from Lemma 3.4.4 that

zaRN

 

aNdR Nd
J:(_i|>d5]<P0[|———za d|>d5, |—-1|<d26]
ma

Pall

N zaRN
+P0[|-fi—1| ”2515p,“

 

(1

1d 5 Nd 5 a2d]

+ 0(d5).

2 26
Where 31d = [nd(l_d 6)1, 32d = [nd(l+d )]
z R
It remains to show that P[|a(-’

 

Nd 5
|>d , aldSnga2d] hasorder 0(d5). It
1'(1

z
is easy to see Egg-S M(0) for n 2 a1d and small (1. Here M(0) is a constant.

We then have
R

 

Nd |>d5 a <N < ]<
JN; "' , 1d“ d“a2d "

aldSnsa2d|R I‘m;

Z
Fellag

P0[ >d5}]

     

S E
a‘1d5”‘5"‘2d

            

32

Applying Theorem 2.2.3 and Proposition 2.2.2, we get, for
6 .
313 s n s 323, PglM(0)anl > d”) s PglM(0)anl > d . Ion —
0| 5 «011+ Pollon— 01> cm]

s PAM(0)(JE(IIFn-F0II2 + 11F,—F0113”) > 36] + P.9 [11F,—Fan > 0(0)]

g 131 + 132 + B3,
2 d5 3/2 d6

where 131 = P 0[M(0),fii "FD-F0” 33—], 32 = P 0[M(0)fi1' "FD—F0" > 3—1

and B3 = P 0[M( 0) ”Fn_F0” 2 0(0)]. Applying Proposition 2.2.4 to B1, B2 and

B3, it follows that

131 + 132 + 133 5 M(0, r) (c1““5’/311‘5’/6 + (1‘85” 311—2” 3)
for small (1. Here M(0,r) is a constant which only depends on 0 and r, the order
of existing moment.

Using the above bounds we get

IR

 

z |
a n
a <§< Pol 411 > dfi
ld- -a2d

ald5n5a2d

231— 43/3 232 — sat/4

S MW, lf)(d + d ),

where 0 < 61 < i; - 1 and 0 < 62 < g5 — 1. Collecting together the inequalities

on ,61, ,62, r and 6, we get 0 < 6 5 $613-— with r 2 3. This completes the proof

of (3.4.11).

33

Proof of Corollary 3.3.3.

Since za satisfies ¢(za) - ¢(—z a)=1_2 a, it follows that

IPAIZ’Nd-éll gal—(Han

=|P0[6Nd—0$ d] -—P0[3Nd—0< —d]-¢(za) +¢(—za)|

s IPgIPNd—osdl—Mzan + IPflNd—k—dl—M—zan
523:1) (deNd— 0$dz;1x]-¢(x)|.

The corollary then follows directly from Theorem 3.3.2.
Proof of Theorem 3.3.4.

From Theorem 3.3.2, we have
Jadde - 0) la N(0, 020).

(b6n )2 it then follows that

Note that 00: (b0)22 dand 02
(3.4.16) fidw Nd —a%)n LN (,0 4b2a4 0).

Similarly we have

- 2 2
(3.4.17) ,lfidwal — 00) 2.. N(0, 490%).

By the definition of the stOpping time Nd’

(3.4.18) zid—ZUINd— 00)n1/2<N ”2

2
(N d— zad 200)nd
2 -2 . 2 2 —l 2 —1 —1 2 —1
Szad (”Nd—l—Uflmd/ +nd/2 +nd /N (N d—l) a.e. P0
Applying Slutsky's Theorem and (3.4.16) to the left hand side of (3.4.18), we get
2 —2 21/2 L 2
zad (0N N-d 1-00)nd N(0, 4b).
Likewise for the first term of right hand of (3.4.18), we have
Z2 —2 2 2 -l 2 L 2
zad (30—111,, — 00)nd / —-. N(0, 4b ).
The second and third terms of the right hand side of (3.4.13) obviously tend to zero

34

almost surely. Thus, from (3.4.13), it follows
—1 2 2 2 —2 L 2
Finally, by observing
—1 2 2 2 2 —1 2 —
and Ind — z: 020 d—ZI S l, we conclude that

1131/ 2(N d—n (1) £4. N(0, 4112).

1/2 2 2 (1’2)

(“d " za ”0

CHAPTER 4: BEHAVIOR OF THE SEQUENTIAL PROCEDURE IN THE
PRESENCE OF CONTAMINATION

§ 4.1 Introduction

We now consider a more general situation which differs from that of the
previous chapters. We shall study the behavior of our sequential procedure
when the observations X1,...,Xn... do not come pure from the distribution
F0, as we had before, but are contaminated by another distribution G in

the sense that Xl,...,X are now independent and identically distributed

n'"
with mixed distributionzz

Fifi (1‘7)Fg+ 7G,
where 0 < 7 < 1 and G is an unknown distribution. We shall refer to
G as the contamination distribution. This kind of contamination is usually
referred to as the gross error model. As is customary in a robustness study,
we shall explore our asymptotic robustness pr0perty for our sequential

1/2

procedure on local neighborhoods of F 0 that shrink at rate n— to zero.

In our situation, we have Nd 3: nd = 0(d_2) and 7 = 0(IIF: — F0").

1/2

Thus, to have the shrinking rate n— , 7 must depend on d and

7(d) = 0(d). In the sequel, we shall assume 7(d) satisfies

lim d =a>0.

d-10
For sequential confidence interval procedures, one is mainly interested in

the asymptotic coverage probability and the asymptotic mean of the stOpping
time N d' The subject of this chapter is to see how these two quantities are
affected in the presence of contamination. We shall also compare the
minimum distance sequential procedure with the Chow—Robbins procedure in

a Special case.

35

36

§ 4.2 Main results

Throughout this chapter we may take P d to be the product probability
induced by the sequence {Xi’ i 2 1} of independent and identically
distributed variables with distribution F; = (1-7(d)) F0 + 7(d)G. The
symbols P 0, Nd’ nd, 0% and INd have the same meaning as used before,

E d denotes the expectations under probability measure P d' All limits are

takenas d-oo.

Theorem 4.2.1. Suppose for some r 2 3, HxlrdF0 and I|x|rdG are
finite. Then

 

N P
(i) n: 32 1
N
(ii) E d 11% -1 1
(iii) Pd [0 e INd] .. ¢(z a(1-aL(G))) - ¢(—za(1+aL(G)))

d

where a = lim and

d-iO
MG) = “G - F0) he fig (it/".59”

Nd’nd

 

(iv) A; 1:.» N(2abzaL(G), 4b2).

From Theorem 4.2.1 we may draw the following conclusions.
(1) If the contamination is negligible in the sense that

lim 1551)- = a = 0, then the asymptotic coverage probability of the procedure

37

and the asymptotic distribution of N (1 remain the same as in the case of
without contamination.

(2) If a > 0, then the asymptotic coverage probability is biased.
From (iii) of Theorem 4.2.1 and Taylor's expansion,

¢(za(l-aL(G))) — ¢(-z,(1+aL(G)))

=1-2a+ 2317 (2041140))?(Wig-W,»

 

where Ea is between za and z a(1—aL(G)); 2a is between za and
-za(1+aL(G)). Since

|L(G)l s IIG-Fglhg ii) cit/115,,“2 s M.
we get that the asymptotic coverage probability is approximately
l—2a - aZC for any contamination distribution G. Here C is a positive
constant.

The variance of the asymptotic distribution of stopping time N (1
remains the same. Its asymptotic mean is affected. But the bias of the
mean is bounded in the sense that, as viewed as a functional of the
contamination distribution G, L(G) is bounded in G.

(3) If the contamination distribution G is a degenerate distribution,
then L(G) becomes the influence curve of the minimum distance estimator

A

0n. From Theorem 4.2.1 one can see that the boundedness in G of the
influence curve of the minimum distance estimator implies the boundedness of
the asymptotic mean of the stopping time and the stability of the asymptotic
coverage probability.

In order to compare our procedure with that of Chow—Robbins' in a
particular case, we need to explore the behavior of Chow-Robbins' procedure
under contamination. Jureckova and Visek (1984) studied this problem for

the case where the underlying distribution is symmetric and the

38

contamination is degenerate. Geertsema (1987) established the asymptotic
behavior of Chow-Robbins' procedure for single outlier as well as symmetric
contamination when the underlying distribution is a normal distribution.
Along the lines of their proof, we can extent their results to the following
general theorem.

Theorem 4.2.2. Assume I |x|4dF0 and I |x|4dG are finite. Denote the
mean and variance of F and G by p, o, ”G and 0G. Then, for the

Chow-Robbins procedure, under the gross error model , we have

 

N P
(i) n: J41
N
(ii) E d ‘3:— __t 1
(iii) Pdlu 6 INd] —-+ ¢(za(l-a(nG-It))) - ¢(-za(l+a(uG-u)))
where a = lim 3551-).
Ndmd 22

 

—-» N(az: (cg-02+(p—pG)2)/ 0, 2o za
”5

where nd = [(oza/d)2].

Example 4.2.2. Let the underlying distribution F 0 be exponetial
F0(t) = 1 — e—t/a and the contamination be single outlier at x0. We
then have a = 0, o = 02, ”G = x0 and ”G = 0. By Theorem 4.2.2, we
have, for Chow-Robbins' procedure,

(111:5: PdioeIN d1 = wan—4w») — ¢(—za(1+a(x0-0)))

and

39

Nd'nd

 

M ——+ N(az:((—02+(0—x0)2/02), 2022,21).

It is easy to see that (111m Pd[0 E INd] vanishes as x0 -1 co and
-+0

asymptotic mean of the stOpping time tends to infinity as x0 -» 00. However
for the minimum distance estimation procedure, by Theorem 4.2.1, we obtain
that the asymptotic coverage probability is bounded away from zero and the

asymptotic mean of stopping time is bounded in x0.

§ 4.3 Lemmas and Proofs

We first give some preliminary results that will be used later in the
proof of main Theorem 4.2.1. We begin with a remark.
Remark 4.3.1. Under the gross error model, Proposition 2.2.2 and Theorem
2.2.3 still hold. The proofs are similar to those in Chapter 2.

Lemma 4.3.2. Assume I |x|rdF0 and I |x|rdG finite for some integer

r 2 1. Let {kd’ d>0} be a sequence of positive integer such that

kd = 0(d‘2). Let A1 and 22 satisfy 0 g 2A1 < 12. Then, for 6 > 0,
there exists (10 > 0 and M > 0 such that, for d < do,

A A —3r/2 + 2rA /A —2r + 4rA /A
l 2 1 2 1 2

Proof. In the sequel k (I will be abbreviated as k. By the Cr—inequality,

it follows that

A A A A
"Pk-F011 2 s 2 2(1 new)” 2 + n F,(d)—F)u 2).

40

Since A2 > 2A1, k = O(d—2) and 7(d) = O(d), we get

k 11F,”)— F," = k (7(d)) IIG - F," . o, as d .. 0.

Thus, for any given 6 > 0, there exist (1 6 such that, for d < d 6’

A A A A A
I 2 2 I * 2
Pdlk “Fk‘ F0” > b] S Pdl2 1‘ "Fk‘ F7(d)“ > 5/21-

As in the proof of Proposition 2.2.4, we can write ||Fk— F ’K d)ll2 as a

linear combination of two U—statistics, Uk and Vk where

k 2 k

Uk $11,121 K1(Xi,7(d)) and vk = EFT) 13) K2(Xi,Xj,7(d))
where

K1(x,7(d)) = I(IIX<t] — Ff,(d)(t))2 at
and

*
K2(X,y,7(d)) = Illxst] - Ffid)(t))(llystl - F7(d)(t))dt-
Proceeding as in the proof of Proposition 2.2.4, we get

A A 1 -2 1
Pd[k 1”Pk—F0" 2> 3] g (3/k)(6/k 1) / 2)‘ EdlUklr

1 —2/1
+(3-(6/k1) 2Pius,Ivk—Edvkl‘”).

Lemma 5.2.2.A and Lemma 5.2.2.B of Serfiing (1980) can be used to bound
E(1|Uk--EdUk|r and Edlvk—Ede|2r. But the constant factor in the

41

bound may depend on (1. More explicitly, we have the following:

— 2
EdIUk—EdUklrS CEd|K1(X,7(d)) " EdK1(X,7(d))|rn r/
and

2 —2
EdIVk—Edvkl r g CEd|K2(X1,X2,7(d)) - EdK2(X1,X2,7(d))|rn ‘.

In order to bound P d[kAllle—F 0|A2 > 6] uniformly in a neighborhood of
d = 0, we need to bound the right hand sides of the above two inequalities
uniformly in d. It suffices to show that Ed|K1(X,7(d))|r and
Ed|K2(X1,X2,7(d))|r are uniformly bounded in d. It is easy to see that
lllx s t] — Fth)(t)| Sg(x,t) uniformly in (1. Here
g(x,t) = ((l—F0(t)) + (1 - G(t)))[x$t] + (F0(t) + G(t))[x>t]. By the
moment condition of F0 and G one can easily show

I Ig(x,t)dt dF7(d) _<_ IIg(x,t)dt dFo + I Ig(x,t)dth < 00.
Hence Ed|Kl(X,7(d))|r is bounded uniformly in d. Similarly one can
verify that Ed|K2(X1,X2,7(d))|r is bounded uniformly in d.

The proof of the Lemma is then completed along the lines of the proof
of Pr0position 2.2.4.
Lemma 4.3.3. Assume I|x|rdF0 and I lxlrdG are finite for some r 2 1.
Then, for 6 > 0 and some ,6 > 1

pduIrfi -o%|>6] 5 Mn_fl.
Proof. Apply Proposition 2.2.2 and Lemma 4.3.2 we get

‘2 2 2 “ "
Pd[|an - ”0' > 6] g Pd[b |0n-0l >6/2]+Pd[20|0n-0|>6/2]

s 2Pdllan-F0II > C(20)] s M(n’3r/2)-

42

Thus the result follows.

The following theorem is an extension of Anscombe’s Theorem.
Lemma 4.3.4. Let {Pd,d e [0,d0]} be a family of probability measures on
(0,3) and {€n(d), d€[0, do]} be a family of firnctions on [0, do]. Let
Y , n 2 1, be a sequence of random variables on ((2,3) and {Wn(d)}°1° is

n
a sequence of positive fimctions in [0, d0]'

Assume:
. 1
(1) For xcR ,
P d[(Yn—§n(d))/Wn(d) S x] —-1 G(x) uniformly in (1.
(ii) For any given 17 > 0, there exists C0 > 0 such that,
for C < C0
P sup |Y.—Y /W(d)>nsrl
d[n':|n'—n|< C n 11 Ill 11 ]
uniformly in d in a neighborhood of d = 0.
(iii) Pd[|rd/nd—l| > 6]»0 as d-iO
where, for each d > 0, rd is an integer-valued
random variable and nd, an integer valved constant such
that

11d I 00 as d I 0.
Then, for xEIR,
Pd[(Y1,d — §n(d))/Wrd(d) S x] -1 G(x) as d -i 0.
The proof of the Lemma can be found in Jureckova and Visek (1984).

We now start to prove the main result Theorem 4.2.1.

Proof of Theorem 4.2.1.
(i) Let n1d = [Zn/dl’ n2d = [nd(1—6)] and n3d = [nd(1+6)] with

42

6 an arbitrary positive constant (< 1).By definition of N d and 11d and

43

Lemma 4.3.3, it follows, for small (I,

‘2 —1 2 ‘2 2
s Pd [anad > (za d) (n3d — 1/n3d)] + Pd[on s d n/2za, nldSnSnZdl

n
. 2d .
2 2 2 2 2 2
3d n—n
1d
n2
5 C(0,6)n—fl + 2 C(0,6)n—fl
n=n
1
where 3 > 1. Therefore (i) follows.
(ii) The proof is similar to that of Part (v) of the Theorem
3.3.1.
(iii) By Remark 4.3.1, we have the expansion:

"oNd-H/uiouz I(FN&(t)-F0(t))h9(t)fig(t)dt+N51/2RNd

a.e.(Pd) on [|0Nd - 0] < c].

We first show that
(4.3.6) RN ———+ 0 as d -i 0.

It follows from Proposition 2.2.2. and Theorem 2.2.3 that, for a given

6 > 0, there exists a r; > 0 such that:

44

(4.3.7) PdHRNd' > 6]

IA

Plefi/ZlurNd—F)ll2 + IIFNd-Fgll3/2) > 41+ PdlllFNd - Fan > 0(0)]

IA

l>d[N1/2 |er dZ-Foll > 6/3] + Pd[N1/2|| FNd—Foll3/2 > 15/31

+ Pd[||FNd-F0|| > 0(0)] = Tl + T2 + T3.

We must show that Ti —-+ 0 for 1: 1,2 ,3 For T3, we have

Pdllan F9“ > 0(0)]

3 ”11221ng

Applying Lemma 4.3.3 to P d[lan—F0I| > C(0)], it then follows that:

T3 5 2 M(n’3/2’+n'2’) —-» o as d a o.
n2[Za/d]

To show T2 —-1 0, we define n1d = [nd(l—c)] and n2d = [11d (1+c)]
with e is in (O, l) but is otherwise arbitrary. We then have following

inequalities:

1 2 3 2
(4.3.8) T2 5 Pd[Nd / IIFNd-Foll / > 6, n1d 5 Nd 5 n2d]

2
+ PdHNd/“d‘ll > 77] S Pdlnéé: 1:23“;an - F0H3/ ]
l - _

45

+ Pd[| Nd/nd - II > 1)].

Using Lemma 4.3.3, we bound the first term on the right hand side of (4.3.5)

as following:

2 2
(4.3.9) may, max urn—120W > o]
n1d5n$l12d

5 id Pd[(1+n/l-v)1/2n1/2||Fn-F9ll3/2 > 41

n=n1d

S néd M(n—3/2r + 3r/3 + n—2r + 4r/3).

Since —3r/2 + 2r/3 < —1 and —2r + 4r/3 < -1 for r 2 2, we obtain
that the right hand side of (4.3.9) tends to zero and hence

P [n;/2 max ||F —F0||3/2] .... o.
d d n <n<n 11
1d- - 2d

That the second term on the right hand side of (4.3.8) has zero limit is the
result of Part (i). Thus we showed T2 goes to zero. T1 can be handled in
the same way. Therefore (4.3.6) is proved.

We now show:

(4.3.10) T4: = 1/u 370))? ,IN; I(FNd — when, dt

Ld - — 2
__. N («02, alG-Fol hm dis/Ila)". 4,).

T4 can be written as

46

(4.3.11) Til/"50”? 4N5 I(FNd — F 7((1)) has, dt

+ JN; I(F7(d) - F0)h050dt)= T4,1 + T42.

By Part (i), we have

P

T4,2 _Sl ooza aI(G—F0)h0n0dt.
Now (4.3.10) will follow from Slutsky's Theorem if we show T41 converges
to N(0,o20) weakly. To show this and to apply Theorem 4.5.3, we need
to verify the asymptotic normality of VE- I(Fn - F 7((1))h 050 dt and that
(E I(Fn - F7(d))h0-’Id (It is U.C.I.P.. Let
vim) = I(1(xi 5 t) — Ffld)(t))h0(t)fio(t) dt. Since I|x|3dF0 and I|x|3dG
are finite, Ed|1/)i(d)|3 is finite for every (1. It then follows from the
Berry—Esseen inequality that
n
d 1/2
(4.3.11) suplPdll/rrq ,2, (¢i(d)/(Vard¢,(d)) ) s x] — ¢(x)|
x
1 2

sC/n/ Edlvl(d)l3/(Vardvl(d)) ,
where C is a constant.
Proceeding as in the proof of Lemma 4.3.2, we can show

Edlvl(d)I’ -» E|¢1(d)|’-
Therefore the asymptotic normality of vii— I(Fn - F ’K d))h 050 dt follows
from (4.3.11) and Slutsky's Theorem. The fact that
45 I (Fn — F 7((1))h0'fi0dt is U.C.I.P. can be proved by standard arguments.

An application of Theorem 4.3.5 now gives

Ld — 4
T5,] _" N“), Var ¢1(0)/“770" )
Finally, noting dv/Na —-i 0020 in Pd and using (4.3.10) and Slutsky

47

Theorem, we get
Paws) = Pd w; is, - ill/oi s «N; w
— Pdch—l (iNd - one, < — 4N; d/eg

——» ¢(za(1-a “Gr-WW, dt/llfiolm
— cream l(G-F0)hyfig dt/lliiglli

Thus (iii) of Theorem 4.2.1 is proved.

(iv) Since
. L
Nfl/szd—r) J!» N(oaza aL(G), 0%),

we get using arguments, similar to the proof of Theorem 3.3.4,

. L
(4.3.12) N31” (afid— (:20) —d—+ N(2bza aL(G), 4b2)
and
. L
1 2 2 d 2
(4.3.13) Nd/ (aNd—l— 00) _e N(2bzaaL(G), 4b ).
From definition of Nd’
1/2 1/2 .
(4.3.14) Nd (dsz/za - 0%) 2 N(1 (efid + 1/Nd - 02)
and
1/2 1/2 -1
(4.3.15) Nd ( d2Nd/zz —o%) 5 N(1 ((Nd-l) + o§d_l- 020)
1/2
2 2
+ d Nd /za.

1/2
Since d2Nd n: -» 0, it follows from (4.3.12),(4.3.13),(4.3.14) and (4.3.15)

48

that

1/2
4
Nd (d2Nd/z: - 0%) —» N(2bazaL(G), 4boo).

The result then follows from part(i) and Slutsky's Theorem.

§ 4.4 Simulation Study.

In order to study the finite sample behavior of the procedures pr0posed
in the preceeding sections, a series of Monte—Carlo studies were done on a
IBM 3090 computer. The purpose of the study is to compare the minimum
distance estimation procedure and Chow—Robbins procedure in the case of
contamination and to see how the ratio of the percentage of contamination to
the precision of the confidence interval, a = 7/d, influences the procedures.
The comparision of the MDE procedure and C—R procedure was made with
respect to coverage probability and expected stOpping time.

Data are generated under the following gross-error—model:

F’f,(t) = (1-7)F(t) + 7G(t)
where F(t) = 1 - e—t, the unit mean exponential distribution, and
G(t) = 1[x0 5 t] the distribution of single outlier x0. The percentage of
contamination was chosen to be 5 or 10. The precision of the confidence
interval was chosen so that a = 7/d = 0.1, 0.2, 0.3,...,0.6. All simulation
results are based on a five hundred Monte Carlo run. Both procedures are
applied to the same sequence of data. The initial sample was chosen to be
7, the confidence coefficient was taken to be 0.9.

The simulation results are given in Table I (7 = 0.05) and
Table II(7 = 0.1). For each fixed x0 and a, we denote the estimated

coverage probability by GP and the estimated expected stOpping time by

49

ASN (average sampling number). Entries inside the paratheses are the

standard deviations of the estimators.

50

 

 

 

TABLE I
r = 0.05
MDE C—R

x0 a CP (S.D.) ASN (S.D.) CP (S.D.) ASN (S.D.)
.1 .89 (0.014) 17.52 (0.36) .85 (0.016) 14.47 (0.36)
0.2 0.82 (0.017) 68.44 (0.91) 0.61 (0.022) 63.21 (1.17)

5 0.3 0.73 (0.020) 154.34 (1.4) 0.41 (0.022) 158.94(l.42)
0.4 0.60 (0.022) 275.39 (1.83) 0.19 (0.018) 287.40(1.88)
0.5 0.52 (0.022) 427.60 (2.20) 0.08 (0.012) 453.14(2.30)
0.1 0.90 (0.013) 17.45 (0.36) 0.83 (0.017) 22.30 (0.69)
0.2 0.78 (0.019) 67.49 (0.79) 0.37 (0.022) 100.86(2.27)

7.5 0.3 0.72 (0.020) l54.77(l.39) 0.11 (0.014) 274.03(2.64)
0.4 0.65 (0.021) 270.70(1.87) 0.02 (0.006) 500.77(3.12)
0.5 0.52 (0.022) 426.47(2.26) 0.004(0.003) 785.74(4.00)
0.1 0.90 (0.013) 17.77 (0.37) 0.74 (0.020) 27.14 (1.13)
0.2 0.80 (0.018) 67.54 (0.97) 0.23 (0.019) l62.81(3.78)

10 0.3 0.72 (0.020) l51.56(1.43) 0.03 (0.008) 437.53(5.06)
0.4 0.67 (0.021) 269.70(1.82) 0.002(0.002) 814.15(4.57)
0.5 0.58 (0.022) 420.71(2.39) 0.00 (0.000) 1277.2(5.65)

 

51

 

 

 

r=0.10 TABLE 2
MDE C—R

x0 a CP(S.D.) ASN (S.D.) CP (S.D.) ASN (S.D.)
0.2 0.81 (0.018) 20.67 (0.48) 0.71 (0.020) 19.93 (0.47)
0.3 0.67 (0.21) 45.03 (0.84) 0.44 (0.022) 46.27 (0.94)

5 0.4 0.59 (0.022) 81.42 (1.13) 0.22 (0.019) 88.16 (1.36)
0.5 0.47 (0.22) 127.86 (1.43) 0.12 (0.015) 148.06 (1.41)
0.6 0.32 (0.021) 187.99 (1.62) 0.03 (0.008) 219.28 (1.37)
0.2 0.82 (0.017) 19.73 (0.47) 0.53 (0.022) 33.44 (1.04)
0.3 0.70 (0.020) 45.05 (0.81) 0.20 (0.018) 90.10 (1.97)

7.5 0.4 0.56 (0.022) 79.74 (1.19) 0.09 (0.013) 179.33 (2.65)
0.5 0.48 (0.022) 128.33 (1.42) 0.01 (0.004) 304.90 (2.26)
0.6 0.37 (0.022) 185.35 (1.69) 0.00 (0.000) 447.01 (2.34)
0.2 0.82 (0.017) 20.22 (0.46) 0.40 (0.022) 56.78 (1.84)
0.3 0.68 (0.021) 43.75 (0.86) 0.16 (0.016) 149.32 (3.66)

10 0.4 0.53 (0.022) 82.17 (1.17) 0.03 (0.008) 323.11 (4.13)
0.5 0.50 (0.022) 126.87 (1.33) 0.002 (0.002) 536.00 (3.55)
0.6 0.35 (0.021) 187.69 (1.62) 0.00 (0.00) 783.45 (2.95)

52

n... u . .... ... a
mo: 0.16.19 mu II... 333..
ox

I I v v

v r v v

v v I v v v v l
‘- V)
O Q

I
II)
o

I V V V

 

I v v

0111111111111111111111-1111r..-31111111111111-3111? m . o

no

53

7° 0 o ..o u .
we: 9-9-6 mu ...Itl... 333..

ox

ox m> zw< .N mmDGE

ow
on
on
as
co
co
oo—
c:
of
on—
c:

on—
zm<

54

m u 3
mg: 9.916 3 41.1.... 3.3“:

on... aria on... mic cw... mné an... n«.o o~.o

EELE ,

 

e 9 do .m #59

T
v
v
v
V
_l
v
v
v
v
.I
V
v
v
v
.I
v
v
v
v
.I
v
v
v
v
I.
v
v
v
v
I
v
v
v
V
_l
W
I

 

1

o .

FIGURE 4. ASN vs a

55

 

'V'VV'VV'V"V"I'V'V'U'V'IVVVVUVVVV'V'VVUVVVVUVVVVW

2° 0 O O O O O O O O O 0
0'5") '- a: h m n v- a. h n n .—

0.25 0.30 0.35 0.40 0.45 0.50 0.55 0;60

0.20

0-9-9 ll 0 E

sis-H08

LEGEND

X0 8 5

56

DISCUSSION:

From the simulation results, we may draw the following conclusions:

(i) For the MDE procedure the coverage probability remains the same
as x0 increases. For the Chow—Robbins procedure there is a clear decrease
in coverage probability as x0 for the MDE procedure. But for the
Chow—Robbins procedure the estimated sample size has a definite increasing
trend as x() increases (see Figure 2).

(ii) The estimated expected sample size is little affected by changes in
x0 for the MDE procedure. But for Chow—Robbins' procedure the estimated
expected sample size has a definite increasing trend as x0 increase (see
Figure 2).

(iii) As increases the coverage probability decreases and the estimated
expected sample size increases for both the Chow-Robbins' procedure and the
MDE procedure. But the MDE procedure has higher coverage probability
and lower estimated expected sample size than that of Chow—Robbins'

procedure (see Figure 3).

10.

11.

12.

13.

14.

15.

57

References

Anderson, T.W. and Darling, DA. (1952), " Asymptotic theo of certain
oodness of fit' criteria based on stochastic processes", nn. Math.
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Blackman,J.(1955), "On the approximation of a distribution function by an
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Boos, D. (1981). "Minimum distance estimator for location and goodness of
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Ghosh, M. and Dasgupta, R. (1980), "Berry-Esseen theorem for U—statistics
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Koul, H. L. and DeWet T. (1983:. "Minimum distance estimation in a linear
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16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

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29.

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31.

58

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