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970

 

IHESI.

LIBRAR Y

- MichiganState

‘ University

 

This is to certify that the

thesis entitled

NEIGHTED EMPIRICAL-TYPE ESTIMATION
OF THE REGRESSION PARAMETER

presented by

Mark Allen Williamson

has been accepted towards fulfillment
of the requirements for

Ph.D. degreein Statistics and
Probability

M

Major professor

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HEIGHTED EMPIRICAL-T¥PE ESTIMATION
OF THE REGRESSION PARAMETER

By
Mark Allen Williamson

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

1979

ABSTRACT-

NEIGHTED EMPIRICAL-TYPE ESTIMATION
OF THE REGRESSION PARAMETER

By

Mark Allen Williamson

We consider three estimators for the slope parameter 3
in the simple linear regression model, each of which is based on
the minimization of a statistic for testing H0: 8 = 0 versus
the alternatives H]: s f 0. The statistics include a Cramér-
von Mises statistic and its rank analogue, and the Kolmogorov-
Smirnov statistic.

Invariance and symmetry properties of the estimators are
studied for finite samples, and their asymptotic distributions are
derived. The Cramér-von Mises-type estimators are shown to be
asymptotically normal, while the asymptotic distribution of the
Kolmogorov-Smirnov-type estimator is expressed in terms of func-
tionals of a Brownian bridge. _

The Cramér-von Mises-type estimators are compared with some
common estimators of B by an examination of asymptotic variances
at various underlying distributions. Comparisons for the Kolmogorov-
Smirnov-type estimator are made via a Monte Carlo study and by
comparing asymptotic upper bounds for the lengths of associated con-

fidence intervals.

ACKNOWLEDGEMENTS

I wish to thank Professor Hira Koul for his guidance in the
preparation of this dissertation. The advice and encouragement he
gave are greatly appreciated.

I would also like to thank Professors Dennis Gilliland, Roy
Erickson, and Joel Shapiro for their review of my work. The
excellent typing of the manuscript was done by Mrs. Noralee
Burkhardt.

Finally,_I would like to thank my parents, Mr. and Mrs.

A. Rex Williamson, for their constant encouragement during my years

as a graduate student.

TABLE OF CONTENTS

Page
INTRODUCTION AND SUMMARY ................. 1
Chapter
1 CRAMER-VON MISES TYPE ESTIMATION OF B;
THE RANK ANALOGUE ............. . 6
1. Notation and Preliminaries ....... 6
2. Finite Sample Properties ........ l0
3. Asymptotic Behavior of M (A) ...... l3
4. Asymptotic Distribution of 0 C81 . . . . l6
5. Asymptotic Efficiency of Bl ...... 20
2 CRAMER-VON MISES TYPE ESTIMATION or e . . . 23
1. Notation and Preliminaries ....... .. 23
2. Finite Sample Properties ........ 28
3. Asymptotic Behavior of M2(A) ...... 29
4. Asymptotic Distribution of OcBZ . . . . 42
3 KOLMOGOROV-SNIRNOV TYPE ESTIMATION OF B. . . 48
l. Notation and Preliminaries ....... 48
2. Finite Sample Properties ........ 56
3. Asymptotic Distribution of o €33 . . . . 58
4. Interval Estimation of B ........ 6l
5. Asymptotic Efficiency of the IC a . . . 66
6. Monte Carlo Study . . . . . . . c: . . . . 69
BIBLIOGRAPHY ....................... 7l

INTRODUCTION AND SUMMARY

1. The Model. Consider the simple linear regression model

Xm. = 80 + 8cm. 4- Eni’ l 5 i in, where cn1 gcnz 5..._<_ cnn are
known constants, not all equal, 80 and B are unknown parameters,
and the eni are iid F for F an absolutely continuous distribu-
tion. We regard 80 as a nuisance parameter and consider three
methods for estimating 8.

Throughout this paper we will, for the sake of convenience,
suppress the dependence of 'Ixni}. {eni}’ and '{cnii on! n. The
vectors (c1,c2,...,cn)' and (X1,X2,...,Xn)' will be denoted by
g and 5, respectively, d1, 1 5_i 5_n, will denote the centered
ci's, and we will take CE = fg=1 dg. Furthermore, many of the
statements made are true w.p. l, even though it may not be stated

explicitly.

2. Cramér-von Mises Type Estimation of B; the Rank Analogue,
As given by Hajek and Sidak (l967, p. 103), the rank analogue of
the Cramér-von Mises test for H0: 8 = A, where A is a given

constant, is based on the statistic

_ T " 2
MM) - f0 [1;] diI(RniA int)] dt

where Rni is the rank of X1 - Ac.

1 among {Xj - ch, l §_j §_n}.

A
Since H0 is rejected only for large values of M](A), it seems

I

I

reasonable to attempt to define an estimator B] for 8 based on
the minimization of M1 in A.

In section l.l we propose a unique definition for such an
estimator and give a numerical example to illustrate its computation.
When so defined 61 is translation invariant and has a distribution
which is symmetric about the true parameter provided the underlying
distribution is symetric or the centered regression constants are
skew-symmetric (section 1.2). Section 1.3 contains intermediate
results which are used to derive the asymptotic distribution of the
normalized estimator in section 1.4. Finally, in section l.5, we
consider the asymptotic efficiency of 3]. With asymptotic variance
as the basis for comparison, 6] performs remarkably well against
some common estimators for B, particularly when the underlying dis-
tribution has heavy tails. Comparisons are made with the Wilcoxon,
median, normal scores, and least squares estimates at the normal,
double exponential, and logistic distributions. At the double
exponential, B1 out-performs all of the above but the optimal
median-type estimator. Similarly, at the logistic,.only the optimal
Wilcoxon-type estimator is more efficient. At the normal, B1 beats
only the median-type estimator, but shows only a slight loss of

efficiency against the other estimators.

3. Cramér-von Mises Type Estimation of B. We base our second
estimator on a statistic which is similar to M1 of the previous
section, but which uses the observations themselves rather than

their ranks. Here we consider the process

3

n
M2(A) = 1me 2 diI(X1. - Ac. __<x)32dx, A e R,
‘ i=l ‘

and seek to define an estimator §2 for 8 based on the minimiza-
tion of M2 in A.

In section 2.l we propose a unique definition for such an
estimator and give a numerical example to illustrate its computation.
50 defined, 32 possesses invariance and symmetry properties
analogous to those of B] (section 2.2).

When specialized to the two sample location problem (i.e.,

=c =l;l§m<nL

c1= c2 =...= c"I = O;cm+1= cw2 =... n

n
CIc2 f-wETTI—m .2 I(Xi—ix + A) '15

T=m+l i

I(Xi _<__x)]2 dx.

"Ma

M (A)
2 1

Thus oEM2(A) represents the squared LZ-distance between the
empirical distribution of one sample and a shifted empirical distribu-
tion for the other sample. Fine (l966) showed that in this situa-
tion the Wilcoxon-type estimator éw for 8 satisfies M2(§w) =
-w:2:w M2(A). This raises the question as to whether §2 = fiw in
general. In section 2.1 we give an example which shows that this
is not the case. It is true, however, that the two estimators are
asymptotically equivalent, as shown in section 2.4.

It should also be noted that 32 is related to the weighted
median estimators for 8 considered by Scholz (1978). In section
2.1 we show that

M(A)=- dd.X.-X.-A(d.-d.).
2 l<iz<i<ni3|3 ‘ J ‘I

The (a.e.) derivative of M2 is

T] + 1-23- didjfl - 21(Xj - X1. - A(dj - d1.)].

where n is a constant which is independent of A. This statistic
(is analogous to that upon which the Scholz estimates are based. We
remark that the weights 'didj(dj - d1), 1 5_i < j g_n, satisfy the
Scholz optimality condition and that oc(B2 - B) achieves the
minimal asymptotic variance for his class of estimators. Our re-
sults do not follow from Scholz's work, however, since the weights

here need not be nonnegative.

4. Kolmogorov-Smirnov Type Estimation of B. Since the Kolmogorov-
Smirnov test for H0: 8 = A, A fixed, is based on the statistic

n‘ ’
D (A) = 'sup | Z dlI(X. §_x + Ac.)|,
c MN” i=1 1 1 l

with small values of Dc(A) favoring H0, we seek to define an
estimator B3 for B which satisfies
o(§)= inf D(A).
c 3 qw<A<m c
We note that when specialized to the two sample location

problem

n m
min-m)ln1m 1 Z I(X,i §_x + A) -.%. Z I(Xi 5_x)|,

DC(A) = sup
=m+l i=1

.m<x (m

which is a constant multiple of the sup-norm distance between the
empirical distribution of one sample and a shifted empirical dis-

tribution for the other sample.

In section 3.l we propose a unique definition for B3 and
illustrate its computation with a numerical example. So defined,
BB agrees, in the case of the two sample location problem, with the
estimator of location proposed by Rao et al. (l975).

Section 3.2 establishes that the invariance and symmetry
properties enjoyed by B] and B2 are valid for B3 as well.

The asymptotic properties of B3 are discussed in section
3.3 where it is shown that the asymptotic distribution of the
normalized estimator can be expressed in terms of functionals of a
Brownian bridge.

In section 3.4 we consider l00(l-o)% confidence sets for

B of the form
{A; DC(A) :Yc,a}

where Yc a is the critical value for which one rejects H0: B = A

whenever DC(A) > y We derive asymptotic upper bounds for the

c,o'
lengths of such intervals, both in probability and w.p. 1.

Finally, in sections 3.5 and 3.6, we consider the efficiency
of B3 and the associated confidence intervals. Comparisons are
made for B3 versus B] and BW via a Monte Carlo study, while
the confidence intervals are compared to normal scores and Wilcoxon-

type intervals using the upper bounds of section 3.4.

CHAPTER 1

CRAMER-YON MISES TYPE ESTIMATION
OF B; THE RANK ANALOGUE

1. Notation and Preliminaries. To the assumptions of the model

introduced in section l of the introduction we add the following:
(l.l) F has a continuous bounded density f satisfying
f(x) > 0 a.e. on '{x; O < F(x) < l}.

(1.2) lim 0" max |d.| = 0.

hr» lffign 1

In what follows let the vectors 5 and x be given ahd
define the quantities B = If“ f2(x)dx and K = fjm f3(x)dx. For

each real A and for each t 6 [0,1] define
n
S(t,A) := X diI(R . 5_nt)
i=l
where

n
RniA := .2

I(X. - Ac. 5 X. - Ac.) .
J l 3

We also define, for each real A,
W1(A) := I; S(t,A)dt

and note that

7

-1 n
”1(0) “-n iEI diRnio 0

Next consider the process
{MI(A)’ '°° < A < m}

where
,_ l 2
M](A) o- [0 S (t,A)dt o

If we let (0 ) denote the vector of anti-

nlA’DnZA""’DnnA
ranks for (x - Ag)’, it is interesting tO note that;

n-
-l -2
I

o 2
c .

‘i
[ d J
li=l DniO

-2 _
0c ”1(0) - n

This is the Cramér-von Mises statistic (Hajek-Sidak, l967) for test-
ing the regression slope parameter B 8 0 against the alternatives
8 f O.

For a fixed sample, M1(A) is a step function (in A) whose

points of discontinuity are contained in the set
I] = {(Xj - X1.)/(cj - c1); i < j and c1 < cj}.

Set A0 = miniA; A 6 P1} and A1 = maX{A; A 6 r1}. Then for

c1 < cj, A < A0 implies A < (Xj - Xi)/(cj - Ci) and hence

R Thus the residuals '{Xi - Aci, l 5_i‘5 n} are

niA < anA'
naturally ordered and therefore (w.p.l)

 

( ' . .
g d. for t e [13 1:1), l < j < n-l
S(t,A) =<
o for t e [0,‘10 u {l}.
g n
Hence,
_ n-l j
M1(A) = I; 52(t.A)dt = n ' Z T 2 di]2
i=1 i=l
and thus
M ( ') '1 nil i d 2
A =n [ .]
1 ° i=1 i=l ‘

Similarly, for A > A], the residuals are in a reversed natural
. n _ . J 2 _
ordering. U51ng' Zi-l di - O one obtains M](A)=n'12j;][fi=1 d J
+
As A crosses A0 only one pair of adjacent residuals cross.
Let ck < ck+l denote their respective regression constants. Then
n-l '
-l g 2
n X E d 3
i=1 ‘

i=1

"1(A5) ' Ml(AO)

1% k“] 2

n' {3&1 [ H: d 32 + Idk+l + jz d1] }

jfk
k-l

n-l 2 2

n{[1: d. J -Tdk+I + ig} di] }.

Now c1 §_c2 5,..§_c and ck < ck+1 imply

k-l
+ Z d. 5.0 .

k .
Z d. < d
= “i=1

1 l k+l

' . . . ~l- ..
Thus M1(Ao) > M](A3). Similarly it follows that M1(A]) > M](A]).

As a result, the following quantities are finite:

*

B1

min{s e r]; M1(S+) = inf M](A)}

C
AEI‘.l

** -
B] = max{s e I]; M1(S ) = inf M1(A)} .

c
AEI‘1

We now define our estimator B] for B by
A - L * **
B] ‘ 2(81 + B] ) °

Numerical example

By the preceding remarks we may determine the value of B1
by identifying the set of slopes P1 and computing M](A') for
each ‘A 6 F1. Computation of M1(A) is facilitated by using the

formula

-1
(1.2) M](A) n 1<i§j<n didleniA ' anAl 0

We consider here a two-sample problem; that is, we take

c1 = c2 = O and c3 = c4 = c5 = c6 = l for the data

160, -26, T7, -l50, -30, 12.

TABLE I
Values of M](A-) for A 6 r1
A e r -310 -190 5l48 -143 -124 -4 3B 43
M1(A-) .6296 .3519 ‘ .1852 .l296 .1852 .1296 .l852 .3519

 

10
From the above table it is clear that
* _ *1: _
B] - -148, B] - -4
so that B] = -76.

2. Finite Sample Properties
(a) Invariance
A useful property of the estimator B] is its translation

invariance; that is, for all real 7,
(2.1) 81(5 + is) = §1(A) + Y-
TO verify (2.l) we note that

mu-Yxp

1 n n . 2
IO{1§1 diltj§11(xj ‘ (A " “Q; 5. X1- - (A - Y)°i) 5 mm dt

- 1 n 'n 2

M] (A) ()5, + YE)

Thus for all real 7

§Q+Y9=mmB€P+Y3M¥H£+m)= MfCMMH1+mH
A£(P+Y)

mmeer+y;me-yfnp= in CmA-nmn
A€(I‘+Y)

min{s + y; s e r, M(s*)(x) = infc M(A)(x)1
AEI'

*
B (25,) +Y-

ll

**

Similarly, B**(1 + yg) = B (5) + y for all real 7, and (2.l)
follows.

From (2.l) we conclude that
(an Pgé-egn=Pyégn.

for all real 2, where PB and PO indicate that the true parameter
is assumed to be B and 0, respectively. Because of (2.2) we may
assume throughout the rest of the paper that B = 0 without loss

of generality.

(b) Symmetry

Theorem 2.l. B is symmetric about B if one Of the following con-

 

ditions hold:
(i) F is symmetric

(ii) d. = -d

l n-i+l’ ‘ 5-1 5-"°

Proof. Assume, without loss of generality, that B = 0.
Proof of (i). Since for any real number a, M](A)(1) = M](A)(g + al),
we may assume that F is symmetric about 0 without loss of generality.

Now A ~ -5 so that B](§) ~ B1(-§) and hence it suffices to Show

that B1(-g) ~ -B1(g). Let 8 = {A; x.

, some
J )

' Xi = A(Cj - Ci

1 5_i < j 5_n}. Then for A e @C.

12

I
I'M:

jl

u
er1:
—l

J I(Xj + ch g_Xi + Aci)

if

j l {1 - I(Xj + Ac. §_Xi + Aci)}

J

+
j

"M:

. + . = . + .
1 I(XJ AcJ X1 ch)

= n + I ‘ Rni('A)(z) .

Using (1.2), (2.3) and the fact that B = I] w.p.l, we have (w.p.l)

M,<-A)(A) ”“13, d,d,IR,,(_A,(A) - an(_A)(z)|

-1
-n is d,d,IR,,,,<-A) - anA(-A)|

M](A)('l)

for A 5 Pg. But this implies that B1(-£) = -§](£). Completing the
proof Of (i). D

*
Proof of (ii). Let 5 denote (xn,xn_,,...,x1)'. Since B = O
we have A ~ 1* and hence 61(3) ~ §1(l*)- We show that
A * A
81(l ) ~ ~81(x). Now

* n * *
RniA(A ) = j§1 I(Xj - ch _<__X.i - Aci)
n
= 321 I“(n-3+1 + Acn-j+1 f-xn-i+1 * Acn-i+1)
=R

n,n-i+1,-A(A)°

13
Thus,

s<t.A><A*)

II
II M:

diI(Rn,n-i+1,-A(A) 5-"t)

i l

"M:

dn-i+TI(Rn,n-i+1,-A(A) 5- m)

i l

-S(t.-A)(A).

It is now clear that M](A)(x*) = M](-A)(g) so that §1(l*) = -§](x).
Since 5* ~ 5 we have §](x*) ~ -§](x) and the proof of (ii) is

completed. B

3. Asymptotic Behavior of M](A). Throughout this section we re-
tain the notation of section 1 and assume that (1.1) holds. The

following theorem is proved as in Koul (1977).

Theorem 3.1. Let 0 < a < w. Then

 

p
(3.1) sup o"|S(t,Ao") - S(t,O) - Ao f(F"(t))l +0 o .
O<t<l C C C

IKIEa
A consequence of the above theorem is

Lemma 3.1. Let D < a < w and T](A) = fg[S(t,O) + Aocf(F'1(t))]2dt.
Then

p
-2 -1 0
(3.2) IzTBa oc |M1(Aoc ) - T1(A)| +~ 0 ,
. P
(3.3) sup o'2|w$(Ao") - (f[S(t,O) + Ao f(P"(t))Jdt)2| +0 o .

14

Proof. To establish (3.2) we note that

(3.4) sup o;1|S(t,0) + Aocf(P"(t))| 30;]

IAISP
Ogtg]

sup |S(t,0)| + allfll=0 .
Ogtgj

-1
But oc

sup |S(t,0)| is the Kolmogorov-Smirnov statistic which
< <1 v
has a limififig distribution (Hajek and Sidak, 1967). Thus the LHS

of (3.4) is bounded in probability so that in view of (3.1)

p
sup o'2|52(t.Ao;‘) — [S(t,O) + Aocf(F-1(t))]2|‘+0 O
IAISA c
Ogtg]
and (3.2) follows.
To establish (3.3) we note that
-1 -1 ' 1 -1 P0
(3.5) sup oc [W1(Aoc ) - fo[S(t,O) + Aocf(F (t))]dt| +1 O

IAISP

follows from (3.1). We also have (recall B = f:;f2(x)dx)

(3.6) sup o"|;‘[5(t,0) + Ao f(P“(t))3dt| §_o-]|W (O)| + a8.
|A|<a C 0 C C I

But -o;1W](O) is the Wilcoxon statistic and hence has a limiting
distribution (Hajek and Sidak, 1967). Thus the LHS Of (3.6) is
bounded in probability so that (3.3) follows from (3.5). D

For fixed 0 < a < m and O < d < m, define the event

-2 2
N1

En1(a,d) = {oc (0) < d, inf 022W§(Ao;1) 3_d} .

|A|=a
Lemma 3.2. For every e,> 0 there exist positive real numbers

N,a and d such that whenever n 3_N,

15

P0(En1(a,d)) 3,1 - e.

1

Proof. Since a; W1(D) has a limiting distribution, there exists

a positive real number b such that
”DIG HIW (0)I_ < b] > 1 - e V n .

2

If we also take d > 2b and choose a so that

a > b + (so/2)15 8" we have

2N1(O) §_b2 < d/2

and

inf o 22(f0[5(t, O) + Ao f(P"(t))1dt)2
IAI =5

nin{(oj W1(O) - aB)2, (o"w1(0) + a3)2 1

IV

(aB - og‘lw1(0)|)2 3_(a8 - b)2 3_3d/2

on {o;|W1(O)I_< b}. Choosing N according to (3.3) completes
the proof. U

Lemma 3.3. For every 2 > O and d' > 0 there exist positive real
numbers a and N such that n 3.N implies

-2 2 '1 g
IATEa oC W1(Aoc ) 3_d ) 3_1 - e .

Po(
Proof. In the proof of lemma 3.2, take d > max{d',2b2}. The proof
is completed by using the fact that W1 is nonincreasing in A

(Hajek, 1969, p. 35). D

16

4. Asymptotic Distribution of OCB1. Throughout this section we
retain the notation of sections 1-3 and assume that (1.1) and (1.2)
hold. In addition we assume, without loss of generality, that

B = 0.

Lemma 4.1. For 0 5_t §_l and n 3_l define

an(i,t) O i 5_tn

i - tn tn §_i 5_tn + 1

= 1 tn + 1 5.i

Then the process

{Zn(t) := 0;] 1:] dian(R1,t), 0 §_t 5.1}
converges in distribution in (B, C[0,l]) to the Brownian Bridge
{3(13). 0 _<_ t :1}.
Erggf. Héjek and Sidak (1967), Theorem V.3.5. D
Remark, {Zn(t), O §_t §_1} is a process with continuous sample
paths which is related to {S(t,0), O 5.t 5_l} in the following

manner:

sup |2n(t) + oE‘S(t,O)| 5_o-] max Id.

|.
city ° 1_<_i_<_n ‘ .
For y a bounded integrable function on C[0,1J define

-1

W) = K I}, y(t>f(r“<t))dt

where K = ffgf3(x)dx.
Note that since h is a continuous functional on C[0,1]

we have

17

Lemma 4.2. L0(h(Zn)) =>N(O,o§) where
o? = K'thg 62(t)dt - (I; G(t)dt)2]
and
_ t -1
G(t) - f0 f(F (s))ds, 0 5_t 5_1 .

We now define

2 ._ -1 .
ocB1 .- —h(oc S( ,0)) .

For 0 < b < a and p > 0 define

6 (a,b) := {lo 3 1 §_b, inf _ M (A) > inf _ M (A)}
n1 C 1 IAIZaoc] I |A|<aoc1 1
Al?

-2 2
H (a,p) := { sup 0 M (A) - T (A0 ) > Kp /2}
n1 IAls-aog" C A I 1 C I —

.= _ 3 . c = -
An1(a) . supiacls1 81I . 81 e r1. M1(B1) (ATan" M1(A)} .
Aer1' C

Lemma 4.3. Let s > O and 0 < b < a. Then
P0({An1(a) > e} n Gn1(a,b)) + O.

2599:. Suppose there exist y and p positive such that
P0({An1(a) > 2p} 0 Gn1(a,b)) > y for infinitely many n. By (3.2)

there exists an N > 0 such that n 3.N implies
P0(Hn1(asp)) < Y/2

and hence

18

P0[{An1(a) > 2o} n Gn1(a,b) n H:1(a,p)] > 1/2 .

Since the above event is contained in {An1(a) > 2p} n Gn1(a,b),
we can find for any given 5 e {An1(a) > 2p} n Gn1(a,b) n H:1(a,p),

1

* c - -1 . . B *
a A 6 P1 n (-aoc , aoc ) satisfying 1B1 - A1| > p and

*
(4.1) M1(A1) = inf _1 M1(A) .
|A|<aoc

Because we also have x 6 H:1(a.p) it follows that
-2 * * 2
(4.2) 0c |M1(A ) - T1(ocA )1 < Kp /2 .

Noting that T1(ocA) is quadratic in A with leading coefficient

2

Koc and minimum occurring at A = B1, we conclude that

.. * 2
(4.3) oczri1(ocA ) - T1(oce1)1 > K02

1

A * . C ' ’1
from 13] - A | > p. Since for any AIE F1 n (-30C 3 3°C )2

-2 2 -2
-2 A 2 A
+ oc |T1(Aoc) - T1(B1oc)l < 15K6 + IT1(Aoc) - T1(B1oc)|.

**
by the continuity Of T1 in A there is a A 6 P1

1, aoE‘) for which

n (-aoc
-2 ** 2 2
(4.4) cc |M1(A ) - T1(B1oc)| < Kp l2 .
But combining (4.2) - (4.4) we see that M1(A*) > T1(ocA*) -
2 2 2 ** . .
KO /2 > T1(OCB1) + KO /2 > M1(A ), contradicting (4.1). D

2
Lemma 4.4. L0(ocB1) =»N(O,o1).

19
Proof.

loc§1 - h(Zn)l = |h(o;‘S(-.O)) + h(zn)l

_<_ K-nmugioysnm + Zn(t)ldt s (16,) Till, max Id I + o

°°1<i<n
* 2
Thus L0(ocB1) =>N(O,o1). 0

Lemma 4.5. Given a > 0 there exist positive real numbers a,b

and N with a > b such that n 3.N implies
POIGn1(a,b)J 3_l - e .

Prpgf. Since OCB1 and o c2M1(O) have limiting distributions there

exists a positive real number b such that
2 -2
POEOCIB1I §_b, oc M1(O) 5-b1 3_1 - 6/2

for all n. Taking d > b and noting that M1(A) 3_W§(A) by the
Cauchy-Schwarz inequality, it follows from lemma 3.3 that there
exist a > b and O < N < m such that

inf _1 o “M1(A) > d] > 1 - 5/2

P I
O |A|>ao;

for n 3_N. But then

022M1(0) _>_ inf _1 022M1 (A)
|A|<aoc
A£F1

implies

20

a -2
POEGn1(a.b)J 3_P0[oc|81| 5_b, °c M1(0) §_b,

lin _1 o;2M1(A) 3_d] 3_1 - e. D
A>aU
- C

Theorem 4.1. L0(OCB1) +-N(O,o§).

 

x P
Proof. We prove that oclB1 - 811 +0 O. The theorem them follows

from lemma 4.4. Let c > O and 6 > D be given. By lemma 4.5
there exist positive real numbers a,b and N1 with a > b such

that

POIGn1(a,b)] 3_1 - 6/2 V n.3 N1

Now use lemma 4.3 to choose N > N1 such that

P0({An1(a) > e} n Gn1(a,b)) < 6/2 V n 3_N .

Then for n 3_N we have

s 3_sup{o 1B - g | : M (B ) = inf _ M (A)}
c l 1 1 1 |A|<ao 1 l
A¢r1°

~

~ 2 * 2
= sup{ocIB1- e11 :M1(B1)=22;M1(A)}Z*20c[131‘ B1|
** 2 A
+181 “811]:Oc181'811

on the event {An1(a) §_e} n Gn1(a,b). Since P0(£An1éa) §_e}
n Gn1(a,b)) 3.1 - 6, we have established oc|B1 - B1| +0 O . D

5. Asymptotic Efficiency of B1. We define the asymptotic

efficiency of B1 relative to any other estimator B Of B as

A

the ratio 01/02 of the asymptotic variances of ocB and OCB.

21

The following tables indicate that 81 does quite well relative to
some common estimators for B when the underlying distribution has

heavy tails.

TABLE II
Asymptotic Variances of Various Estimators for B

2 2 2 2 2
1

 

 

OW ° °M 01-1 °Ls
O. Exp. 1 333 1.2 1 n/2 = 1.5707 2
Logistic 3 3.0357 4 n = 3.1416 Ila/3 = 3.2899
Normal H/3 = 1.0472 1.0946 n12 = 1.6707 1 1
TABLE III
61/02 for the Variances hiTable II
2 2 2 2
“w “M “1 -1 °LS
0. Exp. .90 1.20 .7639 .6
Logistic 1.0119 .7689 .966 .9227
Normal 1.0453 .6969 1.0946 1.0946

Computation of oi, 01 can be computed from either of the formulas

 

2

(5.1) 01

= K‘chg Gz(t)dt - (I; G(t)dt)2]

where G(t) = 73 f(F'1(s))ds, o §_t §_l or

= K‘277[P(x) x F(y) — F(x)F(Y)Jf2(x)f2(y)dxdy .

2
(5.2) 01

22

For E double exponential or logistic, (5.1) was used
since f(F'1(s)) is easily Obtained.

For F = 9, (5.2) implies that
a? = (4r)'lK'2ETo(x x Y) - o(X)o(Y)i
For X and Y independent N(O,.5). One then uses the fact that
Eo(x x Y) = P(Z1 5_0, 22 5.0)

where (21,22) has a bivariate normal distribution with
E(z1) = E(Zz) = 0, 02(21) = 02(Z2) = 3, and o(Z1,Z2) = 2.

CHAPTER 2-
- CRAMER-VON MISES TYPE ESTIMATION OF B

1. Notation and Preliminaries. To the assumptions of the model
introduced in section 1 of the introduction we add the following:

(1.1) F has a finite mean and a continuous density f satisfying

f f2(x)dx < w.

(1.2) H(w) :

f F(w + x)F(dx) has a positive derivative H'(0)
0.

m

d

t
N

(1.3) lim 0'] max |d1| = O.

0*” ljjgn
-4 2 _ + m
(1.4) cc 1;. |d1dj|(dj - o1) - 0(1) as n .
.1
-5 11 T1 2

We retain the notational conventions of preceding sections

and define the additional quantities

dij = -didj(dj - d1), 1 :1, j < n
d+ = {o d } l < i ' < n
ij max g .ij 9 _ 9 J _
dij = maX{09 -d1j}, 1 :19 J i n

23

24

+ -4 +
K -0 Z d 0(d0 " d.)
n c i<j i3 J l

.) .

- -4 -
K - o 2 dij(dj - d1

n c i<j
Remark. Note that (1.4) implies that K; + K; remains bounded.
Furthermore, both (1.4) and (1.5) follow from the more common
assumption
(1.6) 25'021 max 1d11 = 0(1) as n +-m

lfjjm

since

o'4 I |d1dj|(dj - d1)2 _<__4o;4

c max d? 2 Ididjl
i<j

lgjgp 1 i<j

 

2 max d? I d2

5- 4"; i d?
lgjgn i<j i<j

-2 2=
_<__4noc max d1 0(1)

lgjgn
and
no;6 E E dgdg(d. - d1)2 §_2no;6 max d? if ldid.|(d. - d1.)2
i=1 i=1 3 J lgign i<j J J

§_8[no;2 max d?)2 = 0(1) -

In what follows let the vectors g and x be given. For

each real A and each real x define

n
U(x,A) = Z

1 1 d1I(X1 5_x + AC1)

and consider the process

{M2(A)s “0° < A < 0°}

25
where
M (A) - I” uz(
2 - _m x,A)dx .

Recall that in the case of the two sample location problem,
M2(A) is a constant multiple of the squared Lz-distance between two
empirical distribution functions. Thus we are led to estimate 8
in the general regression problem by attempting to minimize the
quantity M2(A) in A. TO this end we investigate the properties
of M2 as a function of A.

Taking X1(A) = X1 - Ac1 and using the identity

2 max(a,b) = a + b + Ia - bl we see that

(A) m 2
(1.7) M2(A) #:1231111 [1.21 d1I(X1(A) _<_x)] dx
X (A)
_ n
- 121 j211dd1j fmax(X1(A),Xj(A)) 1 dX

n n
-121 121 d1 d Jmax(X1(A). X3 (A1)

- a.e. x - x. - A(d. - 11.).
1§i<j§n ”'3 ‘ J ‘I

It is immediate from (1.7) that, for fixed 5 and g, M2(A) is
piecewise linear in A and that any changes in slope occur at

points contained in the set

={(X11- - X1)/(c11- - c1);1gi<j_<_n and c1< C11}.

Now consider the set

26
A = {A3 M2(A) = igf M2(s)}.
If we establish that
(1.8) lim M2(A) = +on
IA|+°°
it will follow from the piecewise linear nature of M2 that A is
a nonempty subset of r2. It can be seen from (1.7) that the slope
of M2(A) is X didj(dj - di) for A < min P2 and

l§j<j§n
- Z didj(dj - di) for A > max P2. Computations similar to
l5j<j§n

those of (l.7) yield
w' 2
- d d.(d. - d.) = f [2 d.I(x > c )3 dx .

Since c1 5_c2 5,..5 c and the ci's are not all equal, d1 f 0

n
and there is a K.S." such that dK # d1. Let K* denote the

first such K. For x between c.l and c * we have
K

n: diI(x _>_ c1)]2 = (K* - l)2d$ > 0.

Hence

- d d (d. - d.) > 0
lgigjin 1‘1 J 1

This establishes (1.8).

We now define

82 = ave(A).

Remark. In the two sample location problem

27

82=mEd{xj ' xi311isjins C1- =0, Cj = 1}.

A regression example
That 82 need not agree with the Wilcoxon estimate 3w in
the general regression problem is illustrated by the following

example. Here we consider the sample
-l, -2, l0, -3, 15, -28

with the weights Ci = i, l §_i §_6. As‘was indicated earlier in
this section, we can determine §2 once we have computed n'1M2(A)
for each A 6 r2. To compute 3”, it suffices to calculate nw1(A')
for each A 6 P2 (Adichie, l967).

 

TABLE I
Values of n'1M2(A) and nw1(A') for A’E r2
A 5 r2 n'1M2(A) nw1(A‘)
-43 454.3333 17.5
-13 93.0833 16.5
-12.6667 89.0972 15.5
-12.5 87.3125 12.5
- 6.5 18.0625 10.5
- 5.4 10.8667 6.5
- 1.0 27.9167 1.5
- .6667 28.7917 .5
- .5 29.4375 -2.5
2.5 42.5625 -4.5
4.0 49.8750 -6.5
5.5 64.6875 -10.5
5.6667 66.1944 -12.5
12 137.7083 -15.5
18 203.9583 -16.5

28

Here we see that

82 = -5.4; SW = -.6667 .

Although §2 and éw may differ for any given finite sample, as

the above example illustrates, we prove in section 2.4 that

A A

oclBZ - Bw' converges in probability to 0.

2. Finite Sample Properties

 

(a) Invariance. A useful property of the estimator §2 is its

translation invariance; that is,
(2.1) §2(£ + Y9) = §2(g) + v for all real 7.

To verify (2.l) we note that

M2(A - Y)(X) — Z didjlxj - X, - (A - v)(dJ - d1)!

i<j

-i§j djdjl(xj + ch) ' (xi + Ycl) ' A(dj ' di)l

M2(A)(£ + Y9)-

Y; a e A(£)} from which (2.l) follows. As

4.

Thus A(5 + 79) = {a

a result of (2.l) we have
(2.2) PB(§2 - B §_z) = Po(§2 §_z) for all real 2.

(b) Symmetry. We assume, without loss of generality, that B = 0.

Theorem 2.l. 32 is symmetric about 8 if one of the following
conditions hold:

29

(i) F is symmetric

(ii) di = -d l §_i 5_n.

n-i+l’
Proof of (i). 82(5) ~ §2(-x) follows from a proof similar to that
of theorem l.2.l. Using (1.?) one obtains M2(-A)(g) = M2(A)(-£)
and hence A(-L) = -A(g). Thus §2(5) ~ §2(-l) = -§2(£), completing
the proof of (i).
x x *
Proof of (ii). As in the proof of theorem l.2.l, 82(5) ~ 82(5 )
'k *
where )5, = (xn,xn_],...,x1)'. From (1.7), M2(A)(x ) = M2(-A)(x)
* A * A
and hence A(g ) = -A(g). Thus 82(5 ) = -82(5) and hence

 

82(5) ~ 32(5*) = -§2(5), completing the proof of (ii).

3. Asymptotic Behavior of M2(A). Throughout this section we re-
tain the notation of section l and assume that (l.l) through (1.4)
hold. Assume, without loss of generality, that B = 0. For con-
venience we introduce the following additional notation; for

-co < A < co define

1*2'(A) = Z d:j1(xj- x1. 5%.)

i<J

-)

T§(A) = Z d'..I(Xj - Xi E-Aia

i<j 13

12(A) = 1;(A) - 15(A)

V+(A) = z (didj)'I(Xj - Xi §_A1j)(Xj - x1)

i<j
- +
V(A) = V+(A) - v'(A)

30

where

Remark 3.l. In terms of the notation introduced above, we show

that for A 6 Pg
M2(A) = M2(0) - 2[V(A) - V(0)] + 2A[T2(A) - 12(0)]
+ 2AET2(0) - EOET2(0)]] .

Proof.

- 2 d1 d. ”Ix - x1

M (a)
2 i<j

Aijl

-i§j d. 1d1[1 - 21(x. -x1 5_A11)1(x1 - x1 - A11)

-i§j1 d. d. (x. - x1) + 21§j1d1d 11(x1 - x1 5_A11)(x1 - x1)

- 2 f d.d A .I(x. - x. 5_A..) + 2 d1 (111 .A
i<j 1 j 13 J 1 13 i<j ij

- 2 d1 d. [1 - 21(x. - x1< 0)](x. x1)
l<j

+ 2 Z d1dj I(O < X
1<j

- X1< A1H)(X X)

J

+ 2A 2 d1jI(0 < Xj -X. 5_A..)
i<j 1 13

+ 2A 2 d1H[I(X - X1< 0) -
l< <3

= M2(0) - 2[V(A) - V(0)] + 2[T2(A) - 12(0)]

+ 2A[T2(0) - E(T2(0))J- D

31

In lemmas 3.l-3.3 we investigate the asymptotic behavior of

T and V.

2
Lemma 3.l. Let 0 < b < w. Then under the assumptions of section 1,

-3 Po
Ispr lac [T2(A/oc) - T2(0)J - AH'(0)| + 0 -
A <

+

Proof. The lemma is proved by combining analogous results for T2

and T5. We consider only TE; the proof for T5 is similar.
Fix A e R. Using (1.4) one proves as in Scholz (l978)

that

(3.1) E0{o;3[T;(A/oc) - 1;(0)1 - AK;H'(0)} + o .

We next show that
-3 + +
(3.2) Var0{oc [T2(A/oc) - T2(0)]} + 0 .
Set
Z11 = I(O < X1 - X1 5_A11/oc) , l 5_i, j 51n,
5+ = 0'3 Z d+

. z.. ,
n c 1<1 i3 13

n
.+ = -3 +
Sn oc k2] Eo(Sn|Xk) .

One shows as in Scholz (1978) that
+ a+
(3.3) Var0(Sn) - Varo(Sn)

-6 + 2
§_oc igj (d11) [EoVaro(Z11|X1) + EOVar0(Z11|X1)].

32
Thus

LHS (3.3) §_%-o;6 Z ((111)2

i<j
-6 2 2
< 0 max d d.d. (d. - d.)
" ° l§k§p k 1§j I ‘ 3| 3 1
-2 2 + -
= 0 max d [K + K J
c 1SKSP k n n

o(l) as n +00.

For each 1 §_k §_n we have

+ _ -3 +
E0(Snlxk) ‘ Uc 1E1 dij E0(Zijlxk)

-3 + +
o E Z d. E (2 IX ) + Z d . E (2 |x )1
c 1<k 1k 0 ik -k . k<1 k1 o ki k

-3 +

-3 +
+ 0c kgi dk1[F(Xk + A1k/oc) - F(Xk)].

Using f2” f2(x)dx < w and the Cauchy-Schwarz inequality it follows

that F is uniformly continuous and hence that

+ + 2
VarOEE0(Sn|Xk)J1§_E0[Eo(snlxk)l

IA

-6" 2
0 E 2 Id l sup |F(x + A. lo ) - F(X)IJ
C i=l 1“ i,j,x ‘3 C

026 sup |F(x + Aij/Oc) - F(x)|2n E dgk
i,j,x i=1

IA

- -6 n 2 +00
- noc 1;] d1k o(l) as n .

33

Applying (1.5) we obtain

A1 -6 n n 2 -
VarO(Sn) §_[noc kg] 121 d1kJo(l) = 0(1) as n +~w.

Combining this result with (3.3) yields (3.2). From (3.1) and (3.2)

we conclude

P
(3.4) lo;3[T+(A/oc) - T+(0)] - AK; H'(0)| +0 o .

We next verify that

P
-3 + + + 0
(3.5) sup 0 [T (A/o ) - T (0)] - A H'(0) + 0 .
IAISP I C C Kn I

Let e,6 > 0 be given and let
‘b = A1 < A2 <...< Ak(€) = b

be a partition of [-b,b] such that

+ -l
max (A. - A.) < E{sup H'(0)] .
l§j5k(e)-l 1+] 1 2 n K"
By the above we have

P
-3 + + + 1 +0
Ioc [T (Aj/oc) - T (0)] - KnAjH (0)] 0

for each l 5.j 5_k(e). Thus we may choose 0 < N < a» such that

-3 + + +
1-6 < PE max 0 [T (A./o ) - T (0)] - A-H'(0) < e]
19:k(e)l C J C K" J I

whenever n 3_N.

Now suppose that A0 6 (A1, A ) for some 1 5_j 5_k(e) - l.

j+l

34

Then since o;3[T+(A/oc) - T+(0)] and K: AH'(0) are nondecreasing

in A we have
-3 + + + 1
Cc [T (AD/ac) - T (0)] - Kn AOH (0)
-3 + + + 1
-<-°c [T (Aj+l/°c) - T (0)] - K11 AjH (0)
+

-3 + + .
10c [T (AjH/Oc) - T (0)] - KnA1+1H (0)

+ I

and

;3[T+(Ao/oc) - T+(0)] - K;A0H'(0)

0

-3+ + ‘+, +,
3_oc [T (A1/oc) - T (0)] - KnA1H (0) + KnH (0)[A1 - A111]
3_-e
Hence
-3+ + +1
|oc [1 (AD/0c) - 1 (0)] - KnAoH (0)] < e
and it follows that

POE sup Io‘3[1+(A/oc) - T+(O)] - K; AH'(0)| < e] 3_1 - a,
IAlsb °

completing the proof of (3.5).

In a similar fashion one can show that
-3 - - - P0
(3.6) sup lo [T (A/o ) - T (0)] - Kn AH'(0)| + O .
lAisb c c

Combining (3.5) and (3.6) completes the proof of the lemma. U

35
Lenma 3.2. Suppose X ~ H. Then

lim t'zEH{X[I(X §_t) - I(X §_0)]} = H'(0)/2.
t+0

[£599f. We consider only the right limit; the proof for the left
limit is similar. Let s > 0 be given. Since H'(0) exists there
is a to > 0 such that [H(y) - H(O) - yH'(0)| < Zye whenever

0 §_y §_t0. Thus 0 §_t §_t0 implies that

It“2 131H(y) - H(O) - yH'(O)deI

5_t'2 [3 2y 2 dy §_e
and hence

lim t'2 It [H(y) - H(O) - yH'(0)]dy = o .
t+o 0

Defining X = XI(0 < X §_t) we see that

t

lim t'zEH{X[I(X §_t) - I(X §_0)]}

t+0
. -2 t
= 11m t f P (X > y)dy
t+o 0 H t
-2 t
= lim t f0[H(t) - H(y)]dy
t+0
-2 t t u
= lim t {f H(t)dy - f [H(O) + yH (0)]dy
t+o ° 0

- f3[H(y) - H(O) - yH'(0)de}

lim {t“[H(t) - H(O)] - H'(0)/2}
t+0

H'(0)/2 . D

36

Lemma 3.3. Let 0 < b < m. Then under the assumptions of section 1

p
sup lo;2[V(A/oc) - V(0)] - %-A2H'(0)| +0 o .

Algb

Proof. Fix 0 < A 5_b. The proof for -b 5_A < 0 is similar. Then

E0{o;2[V+(A/oc) - V+(0)]}

E0{o;2 Z (d1d1)'1(o < x1 - x1 5.A11/oc)(x1 -x1)}

i<j
Azo'4 Z d (d - d.)E {(A ./o )‘21(o < x - x < A /o )(x. - x )
c 1<1 ij j 1 0 i3 c j i - ij c J i
d11>o
- H'(0)/2}
ll 2 -4 g
+ 2 A oc 121 d11(d11 d1)H (0) .
d11>0
Now
1 2 -4
E-A 0c 1;. d11(d1- d1 )H (0): fix H (0)
J .

and by lemma 3.2 and (1.3),

IA A2 0C4 igj dij(d j “d- H)Eo{(A1j/C )- 21(0 < X1 -X1< Aij/OC)(xj- xi)
d11>0
- H'(0)/2}I
2+
< A Kn max E { A / ) 21 0 < x. -x1< A / (x. x.) - H'(0)/2}
i<jl ° ( 15 CC ( 1i ° c) 1 l

< AZK+ o(l) = 0(1) as n + m.

Thus

37
(3.7) Eo{o;2[v+(A/oc) - v+(0)3 - %-K;A2H'(O)} + o .

To establish the lemma for V+ we show that

(3.8) Varo(o;2[v+(A/oc) - v+(0)3) + o .
Set

Zij = 1(0 < Xi ' xi S-AiJ/°c)(xj ' Xi)

+=-2 -

sn oc igj (d1d1) 211
and

§+ = ‘2 E E (s+|x )

n 0c 0 n k '

It follows as in theorem l of Scholz (l978) that
+ +
Varo(Sn) - Varo(§n)

-4 - 2
§_oc 1E: [(d1d1) J [EOVaro(z11|x1) + EOVar0(z11|x1)J .

Hence, by (1.2) and (1.5),

(3.9) Varo(5:) - Var0(§;)

- 2 2
Z [(d1dj) J 50211

i<j

-4
§_Zoc

-4 - 2 2
5'20C 1§j [(didj) J (Aij/OC) [H(Aij/OC) ' H(O)]

2 -6 + 2
2A 0c igj (dij) [H(Aij/UC) ' H(O)]

2 -6 + 2
2A a Z (d. ) max [H(A lo ) - H(O)].

IA

IA

For each 1 §_k §_n we have

38

O’cz .-
Ho(5 lxk) .5 (didk) E0(Ziklxk)
1<k
+ °;2 2 (didk)-E0(Zkilxk)
k<i
where

E0(Ziklxk) = IMO E xk ’ .y f. Alk/OCka ’ .Y)F(d}')

x
fx

" (x

- )F(d )
k‘Ai k/Oc y y

IA

(Aik/OC)[F(XK) ' F(xk ' Aik/OC)J’ i < ks
and

E0(zkilxk) _<__(A1k/oc)L'F(Xk + A1k/oc) - F(Xk)], i > k.
Therefore

a+ n +
Varo(Sn) =k21VaroEo(Sn|Xk)

< o ;4[ sup |F(x + A1k/oc ) - F(xm2 {If Id1dk|(A1k/oc )32

x,i<k

(#2206121. d1ko(l)= 0(1) as 11+».

Combining this result with (3.9) yields (3.8). Using (3.7), (3.8)
l
1 2
nondecreasing (nonincreasing) on 0 §_A §_b (-b 5_A §_0) it follows

and the fact that a;2[v*(A/oc) - v*(0)1 and KTAZH'(0) are both
that
P

(3.10) sup lo‘2[v+(A/o ) - v*(0)1 - %-KTA2H'(0)| +0 o .
IAl1b ° °

In a similar fashion one shows that

39
-2 - - 1 - Po
(3.11) sup Io (v (A/o ) - v (0)] - E-K A H'(0)| + o .
(Alsb c c

Combining (3.l0) and (3.ll) completes the proof of the lemma. U

Theorem 3.l. Let 0 < b < m. Then

PO

sup [02 2[M2 (A/oc ) - M2(0)] - A 2H (0) + 2A0 312(0)| +-° o

|A|<b c

where 12(0) = -(12(0) - E0(T2(0))J-

3399:, The theorem is a consequence of remark (3.1) and lemmas
(3.1) and (3.3). D

We conclude the section by proving three lemmas which will
be useful in showing that the sequence {ocfiz} is bounded in
probability.
Lemma 3.4. The sequence {022M2(0)} is bounded in probability.

-2 -2
2522:, Eotoc M2(o)] EOE-ac 1) d1djlxj - x1|1

i<j
-o CZEOIX] - le1 {j d1dj

= 1

Since 022M2(0) is nonnegative for each n, application of the
Markov inequality completes the proof. 0
Lemma 3.5. Let

H 2(A) =1) d1 fx:(g§A ) VfTYT dx. ~w < A < we

where X1(A) = X1 - Ad1. Then

40

(i) 1112 is a nondecreasing function of A.

P
O. -2 0
(11) [0c [N2(A/oc) - w2(0)1- A E0 Jfli’lll -> o v A e R.

(1'1‘1') N§(A):M2(A) v aeR.

Proof of (i). Let A < A' and set A = max(xzn)(A), xzn)(A')).
Then

. " A A
new ) - 112(1)) 1%] diUx;(A') mx dx - In“) #136 dx]

n + X%(A) n X'(A' )

3_0 .

Proof of (ii). We consider the case when A 3_0; the proof for

A < 0 is similar.

o;1[wz(A/oc) - "2(0)]
n X' (A/o )

xi 1 c
oc [12 d: Ix. .(A/Oc ) Jflx 5 dx + Z d; Ix X1 Jflx 5 dx]

'1 E d+ (Ad+/ )"ix1 mu - M‘X'TJAdV
o i=l i[ 1 CC X§(A/Oc) x x i i 0c

C

n

+ 0'1 Z d ”E(Ad Ioc ) 1
° i=l

2 " 2
2 ”1' “mi”
i=l

fii(A/O'c) _
Jflx) dx — JTIXiJJAdiloc

Therefore, by assumption (l.l) and the above,

4l
Io-][w (A/o ) - w (0)] - Ao'2 E d2 Jflx 5|
c 2 c 2 c i=1 i i

X.
§,A max I(Adi/oc) 1 IX}(A/o ) JTIx) dx - JTIXiJ| = 0(1) as n + an
1 c

ljjfm
We complete the proof of (ii) by showing that _
" 27‘7 mpo

leEz
l 1

But this follows from the NLLN since
VarOUfTX-T] ) g ; f2(x)dx < co.
Proof of (iii). By the Cauchy-Schwarz inequality

11
142(1)) = [{f>o}{ci§1 dim. - Adi g x)32/f(x)}f(x)dx

Iv

a n
(Id, 12 diI(Xi - Ad]. 5 x)/"(‘Tf x dx)2
=1

X' (A) n 2
(IXE:;(A) fig] d11(xi - Adi §_x)/f(x5 dx)

I
A
M

O.
_l.
H
x
A:
Dv
V
v
'0.
X
D.
X
v
N

Before stating our final lemma we define, for each
0 < a < m; 0 < d < m and n 3_l, the event
En2(a,d) = {0;2M2(0) < a, 11?; ogzwgwoc) 3 d}.
Lemma 3.6. For every 6 > 0 there exist positive real numbers

N,a and d such that n 3.N implies

42

P0(En2(a,d)) 3_l - 8.

Proof. Since 0 §_W2(0) §_M2 (0) for all n, the sequence
{a 21H2(0)} is bounded in probability by lemma 3. 4. Hence for fixed

M! < “’9

(3.12) (c‘zw 2

NZC(A/o)- [02 112(0) +AE “MYTH IP +0 o
by lemma 3.5. Now let b be such that
9010;2M2(0) §_bJ 3_1 - e v n 3_1 .
If we take d > 2b and choose a so that
a >125+ ammo #1171)"
we have
022w§(0) §_o;2M2(0) §_b < d/2

and hence

liT:a to; w2(0) + A 50 Jflx‘712> [a £0 J?TX‘7 og‘lwzwm2 3_3d/2
A

on {022M2(0) §_b}. The proof is completed by applying (3.l2). D

4. Asymptotic Distribution of océz. Throughout this section we
retain the notation of sections l-3 and assume that (l.l) - (l.4)
hold. In addition we assume, without loss of generality, that

8 = 0.

Lemma 4.1. {ocgz} is bounded in probability.

43

Proof. Let a > 0 be given. By lemma 3.6 there exist positive

real numbers a,d and N such that
POEEn2(a,d)] 3_l - e V n > N.
By parts (i) and (iii) of lemma 3.5

-2 -2 2
inf o M A o > inf w A > d
1A1.>.a ° 2(/‘)‘IA1:a°‘ 2W9“

and
-2 2 -2
oc ”2(0) f-Oc M2(0) < d
on En2(a,b) whenever n 3_N. Thus

,. -2
{locezl §_a} : {oc M2(0) < d, I2T:a M2(A/oc) 3_d} 2 Enz(a,d) V n 3_N

implies

Ptlocézl 1a]: PEEn2(a.d)1:1 - e v n :N . o

The result of theorem 3.] suggests that an approximating
statistic for chz is 023T;(0)/H'(0). The next lemma gives the
asymptotic distribution of that statistic.

Lemma 4.2. Under assumptions (l.l) and (l.3) L0(JT2'o;3T;(O) =>N(0,l).

Proof. Since

*

*
the projection of T2(0) into the family of linear rank statistics

is

44

*,- -12 "
wz .- n 0c .2 diRniO
1-l
(Hajek and Sidak (1967), p. 61). Since
P

-3 * 'k 0

(Sievers, l976), the proof is completed by noting that
L JT? ’3 w* N o 1
0( 0c 2) => ( . )

under assumptions (l.l) and (l.3) (Hajek and Sidak (l967), p. 163). D

For 0 < a < b < w define

-3 * . .
G a,b = { T 0 /H' 0 < b, 1nf M A/ > 1nf M A/ }
"21 > lac 21) ()|__ |A|>a 21 Cc) Mia 21 0c)

and

An2(a) = supue“ - of T;(O)/H'(0)l; W1 5 a. M2(A*/oc)

= inf M (A/o )}.
lAls.a 2 °

Lemma 4.3. Let c > 0 and 0 < b < a be given. Then
»P0({An2(a) > e} n Gn2(a,b)) + 0

as "+00.

Proof. Suppose there exist 81 and 6 positive such that
Po({An2(a) > e} n Gn2(a,b)) 3_6

for infinitely many n. For each such n there exists a

lel g_a such that

45

(4.1) 1A0 - og3 T;(0)/H'(O)I z.e(

and

(4.2) M (A /o ) = inf M (A/o )
2 0 c IALSP .2 c

on Gn2(a,b). Since
._ '2 '3 * 2 l
Q(A) .- oc M2(0) - 2Aoc 12(0) + A H (0)

is quadratic in A and achieves its minimum at A = 023T;(0)/H'(0),

(4.l) implies that

(MAO) - (no? 12(01/11'1011 _>_ Hume? .

By (4.2) we also have

M2(o;4 12(01/11'1011 _>_ "2<Ao/°c> .
But

-2
su o M (A/ ) - 0(4)
IAIE§.| c 2 0c I

_>_ maxnogznzmoxoc) - omen. (a;2M2(o;41;(o)/H-(o))
- 0(og3 T;(0)/H'(0))I}

3_H'(0)c§/2

on ‘Gn2(a,b) since
(022M2(A0/oc) - Q(A0)| < H'(0)e§/2

implies

46
-2 -4 * . -3 * .
0c M2(°c 12(0)/H (0)) - 010C 12(01/H (0))
3,022 M21A0/oc) - 0(o;3 1:101/H'10))
> 0(A0) - H'(0)e§/2 - 0(o;3 T;(O)/H'(0))
.3 H'(0)e§/2 .
Thus

lim sup sup [022M2(A/oc) - Q(A)| 3_H'(0)e¥/2 ,
n+on IALga

contradicting theorem 3.1. D .

We are now ready to give the asymptotic distribution of
o 82. '
Theorem 4.1. L0(oc§2) +'N(0, (12H'(0)2)").

p .
Proof. We prove that IOCBZ - 023 T;(0)/H'(0)l +0 0. The theorem

then follows from lemma 4.2. Let c > 0 and 6 > 0 be given. By
lemma 4.2 and the proof of lemma 4.l we may choose 0 < b < a < w

and N1 > 0 such that
P0[6n2(a,b)3 3_l - 86 for n 3_N1 .
Now use lemma 4.3 to choose N > NI such that
Po({An2(a) > e} n Gn2(a,b)) < 56

whenever n 3_N. Then for n 3_N we have

47

sup{|A - o C3T2(0)/H (0)): M2(A*/oc) = inf M2(A/oc )}

c §.A (a)
n2 [Alia

sup{|A* - 023T;(0)/H'(0)| : M2(A*/oc) = igf M2(A/oc)}

|v

Iocéz- og312(o)/H (O)!

on {An2(a) 5-5} n Gn2(a,b). Since
P({An2(a) 5_e} n Gn2(a.b)) 3_l - a

for n large,

P
Iocé2 — 0312101/H (0)I +0 o o

The asymptotic relationship between éz and the Wilcoxon-
type estimator fiw is established in the following

p
Corollary 4.1. Under assumptions (l.l) - (l.5) ocléz "gwl +0 0

 

Proof, An immediate consequence of the asymptotic uniform linearity

(in A) of w* is

. P
(212'623 w; - o e |.+°

CW 0'

Since lemma 4.2 and theorem 4.l yield

p

-3 'k v: 0

and
A ‘3 * 1 P0
lacs2 - 0c 12(01/H (0)) +- 0.
we have
A A P0
0c'82 ' BwI + 0 U

CHAPTER 3
KOLMOGOROV-SMIRNOV TYPE ESTIMATION OF B

l. Notation and Preliminaries. In chapter 3 we retain the nota-
tion of previous sections. To the assumptions of the model intro-

duced in section 1 of the introduction we add the following:

(l.l) F has a continuous bounded density f satisfying

f(x) > 0 a.e. on {x: 0 < F(x) < l}.

(1.2) lim 0'1 max Idil = 0 .
new c lgjgn '

In what follows let the vectors 3 and x be given. For

A real define

(l.3) DC(A) = sup |U(x,A)|.

-a<x<m

Remark. We now verify that

(1.4) DC(A) = sup |S(t,A)| V A.€ R,
0<t<l '

a result which will be useful in establishing the asymptotic distribu-

tion of our estimator.

and
4B -

49

SM) = i an . j/n:t< (1+11/n.
i=l ni
Since we also have

U(x,A) = o , x 1! [Xm(A), X(n)(A))
and

S(t,A) = 0 9 X6 (09 '3'") a

(l.3) is established.
To aid in investigating the properties of Dc as a function
of A we define, for A e R,

D:(A) = sup [U(x,A)]

and

D;(A) = - inf [U(x,A)].

Lemma l.l. 0: (DE) is a left-continuous non-decreasing (right-
continuous non-increasing) step function in A whose points of

discontinuity are a subset of.
P3 := {(xJ. - X1)/(dj - d1); dJ- _>_o. d1. < o, 1 :1 < .1“ in}.

Proof. We consider only 0:; the proof for D; is similar. That

D: is a step function follows from the fact that D:(A) is a

function of the ranks of {Xi(A), l §_i §_n}. To establish its

non-decreasing nature we make use of (1.3). Let A1 5.A2 5,..5_Am

denote the ordered members of {(X3 - X1)/(dj - di); 1 5_i < j §_n,

di f dj} and set A0 = -w, A +w. For 1 §_j 5.m choose any

m+l=

50
A', A" such that

A.

J-]<A (Aj<A <A.

1+1 ‘

Now Aj = (Xi - Xk)/(d£ - dk) for some k < t such that dk < dt'

Hence
xk1A') < x,(A')
x£(Au) < xk(An)
X£(Aj) = Xk(Aj) .

For A real let X0(A) = (Xk(A) + X£(A))/2. Then

(1 5) D:(A') “ max{ sup U(x,A'), U(x0(A'),A'). sup U(x,A")}
. x<Xk(A'). x>X£(A")

and

(1.6) D:(A") = max{ sup U(x,A"),U(x0(A"),A"), sup U(x,A")} .
x<X£(A") , X>Xk(A")

Note that as A E (Aj_], Aj+]) crosses A., only the residuals

J
Xk(A) and X£(A) cross. The other residuals remain distinct and
in their same relative order with probability one. Hence the

following are valid:

(l.7) sup U(x,A') = sup U(x,A") = sup U(x,A.)
x<Xk(A') x<x£(A") x<x£1Aj1=xk1A31 J

(l.8) sup ' U(x,A') = sup " U(x,A") = sup U(x,Aj)
x>X£(A ) x>Xk(A ) x>Xk(Aj)=X£(Aj)

(1.9) u1x01A').A') = u1x01Aj).Aj) - a,

51
(1.10) U(x0(A"),A") = U(xo(Aj),Aj) - dk .

Ne complete the proof by considering three cases:

Case I. If dk < d£ 5_0 then by (1.9) and (l.lO)

sup U(x,A') 3_U(Xk(A')',A')
x<Xk(A') .
= U(xo(Aj),Aj) - dk - d2
3.max{U(xo(Aj),Aj) - d2 , U(x0(Aj),Aj) - dk}
= max{U(xo(A'),A'), U(xO(A"),A")}.
Thus

D:(A') = max{ sup U(x,A'), sup U(x,A')} = 02(4")
x<xk(A') x>X£(A')

follows from (1.5) - (l.8).

Case II. If O'idk < d2 then

sup U(x,A')
x>X2(A')

|V

U(x,<A')*.A') = U(x01Aj).Aj)

|v

max{U(xo(Aj),Aj) - dk’ U(x0(Aj),Aj) - dz}

max{U(x0(A'),A'), U(xo(A"),A")}
by (l.9) and (l.lO). Thus

+ 1 = I 1 3+ 11
DC(A ) max{x<§:pA')U(x.A ). x>§:?A') U(x.A )} DC(A )

follows from (1.5) - (l.8).

Case III. If d §_0 §_d£, d < d£, then by (1.9)

k k

52

sup U(x,A') _>_ U(X£(A')+,A') = U(xo(Aj),AJ.)

x>X£(A')
.3 U(x0(Aj),Aj) - d2
= U(x0(A'),A').
Thus
D:(A') = max{ sup U(x,A'), sup U(x,A')
x<Xk(A') x>X£(A')
§_max{ sup U(x,A'), sup U(x,A'), U(x0(A"),A")}
x<xk(A') X>X£(A')
= D:(A")

follows from (l.5) - (1.8).
To establish left continuity, first note that

+ - ,
Dc(Aj) - max{ sup U(x,Aj), U(x0(Aj),Aj), sup U(x,Aj)}.

x<X2(Aj) x>Xk(Aj)

Applying (l.7), (l.8) and

x>§:?Aj)u1x.Aj)_>_ U(kajifiAj) = U(xoujmj).

we have

+ _ , , = +
Dc(Aj) - max{x<§:?A')U(x,A ). x>§:?A')U(x.A )} Dc(A)'

Finally, since D:(A') = D:(A") in cases I and 11, the
points of discontinuity of D: are seen to be a subset of r3. D
Before defining 33 we need one additional

+ - - + - - n +
Lemma 1.2. DC(A1) = 0 = Dc(Am) and Dc(A]) = Z d =

+ +
i=l i Dc(Am)°

53

Proof. Note that for A < A]. l §_i < j §_n and di < dj imply
Xi(A) < Xj(A), so that the {Xk(A), l §.k §_n} are naturally ordered
with respect to d1 §_d2 53"5-dn' Using the monotonicity of the

.d

1's and {'13:} d1. = 0 we have, for 1: k gn-l,
k
U(x,A) 8 2.] di 3 09 X G [xk(A)9 Xk+1(A))°
1:
Since
U(x,A) '-' 0 . X e H(O)“), x(n)(A))
we have
sup [U(x,A)] = 0
-ao<x<oo
and
n _ n +
-inf [U(x,A)] = Z di = X di'
-co<x<oo i=l i=l

Because 0: and D; are constant for A < A1 it follows that

+ — = + =
DC(A1) DC(A) 0
and

.. .. - n +
Dc(A]) = DC(A) = .2] d, .

'l

n
The proof that D:(A;) = 1;] d: and D;(A;) = 0 is similar and uses
the fact that the residuals {Xk(A), l §_k §_n} are in a reversed
natural ordering with respect to the di's when A > Am. B
Lemmas l.l and 1.2 guarantee that the following exist and

are finite:

54

. ,+ -
1nf{A E R, Dc(A) Z-Dc(A)}’

m
(.0
II

sup{A e R; D;(A) 3_D:(A)}.

ID
(A)
ll

Note that by the monotonic nature of D:(A) and D;(A), B;* 3_B;
w.p. l.

We are now ready to define the estimator
A - 'I * **
B3 ’ 2(83 + B3 ) °

Lemma 1.3. Dc is nondecreasing for A 3_§3 and nonincreasing for
A:%.

Proof. Note that
(1.11) DC(A) = max(D:(A), o;(A)) v A e R .

By the definition of 5;, A > 8; implies D:(A) 3_D;(A) and hence
+ ** **
DC(A) = Dc(A). By the definition of B3 , A < 33 implies
D;(A) 3_D:(A) and hence DC(A) = D;(A). Thus, since
* A ** + A ..
33 5_B3 5.83 , DC(A) = DC(A) for A > 83 and Dc(A) = DC(A) for

A < B3 and it remains to show that

A . A- A+
(1.12) DC(B3) 5_m1n(Dc(83). DC(B3)) .
But by lemma l.l,

+ A + A- + e+
DC(B3) - Dc(e3) §_DC(B3)

and
D’(“ ) - D'(“*) < D'(“')
c 83 - c 83 - c 83 '

Therefore,

55

(1.13) DC(§3) = max<n§1é3). o;(§3))
:,maX(D:(§3). D;(§g))
= oc(§;)

and

(1.14) ”C(33) = max(o:(§3). 02(33))

_ max(o:(§§). n;(§;))

A

e+
Dc(e3) .

Combining (l.l3) and (l.l4) establishes (l.l2), completing the
proof of the lemma. D
Remark. Lemma l.3 shows that

nc(§3)= inf o (A) .

-°°<A<oo C

Numerical Example

According to its definition, 33 can be determined by
identifying the set P3 and computing D:(A') and D;(A-) for
each A 6 P3. As an illustration we take Ci = i, l 5_i §_4 and
consider the sample

1, 5, 3, 6 .

Simple computations yield:

56

 

TABLE I
Values of DC(A') and o:(A') for A 6 r3
A 6 r3 D;(A') D:(A')
-2 2 o
.5 1.5 o
1 1.5 .5
5/3 1 .5

* ** A
Here B = 5/3 = B so that 83 = 5/3.

2. Finite Sample Properties
(a) Invariance. A useful property of B3 is its translation in-

variance; that is,
(2.1) 63(5 + Y5) = Y + §3(5) V v e R .

To verify (2.1) we note that

D:(A)()$ + Yg) sup [25(t.A)(2£ + “YEN

0<t<l‘

sup [15(taA-Y)(£)]
0<t<l

Dim-1112.1) .

Thus,

836 + is) 1nf{A: DEM-111.1) : Ugo-11(1)}
= Y + 1nf{A; D:(A)(5) 3_D;(A)(£)}

*
= Y + 83(5) V 1 E R.

57
Similarly,
s?q+yg=y+s?q1 v 16R
so that
§3(1+Yg)=1+§3(z) YveR.

(b) Symmetry. We assume, without loss of generality, that B = 0.
Theorem 2.l. B3 is symmetric about 6 if one of the following

 

conditions holds:
(i) F is symmetric.
(ii) d1 = -d

n4”,lgign.

 

Proof of (i). 63(5) ~ §3(-x) follows from a proof similar to that

of theorem l.2.l. Using the definitions of D: and 0; one obtains

D:(-A)(A) = o;(a>(-A)
and

- +
Dc(-A)(A) - DC(A)(-A) .
1k *1:
Thus the definitions of B3 and 83 yield
* *
331-5) = -83(A)
and
** **
B3 ('5) = ‘33 (1,) -
Therefore
§3(l) ~ §3(',X,) = '§3(£)s

completing the proof of (i).

58

Proof of (ii). As in the proof of theorem l.2.l, 62(5) ~ 62(§*)

 

where 5* = (Xn,Xn_1,...,X1). Using the definitions of D: and

D; and the proof of (i) one obtains

D:(A)(A*) = o;<-A)<A) = D:(A)(-A)

and

o;(a><A*) = o:(-A)(A) = o;<a>(-A) .
Thus

63(5) ~ 6311*) = 63(-A) ~ -§3(A)
as in the proof of (i). ' D

3. Asymptotic Distribution of UCB3. In this section we assume,
without loss of generality, that B = 0.

To aid in the proof of theorem 3.l we first define a class
of functionals {T2, z 6 R} on the set V of bounded functions on

'[0,l] by

12(h) = sup {1h(t) + zf(F'1(t))]1/0}
0<t<l

+ inf {[h(t) + zf(F"(t))J n 0} v h e 11. z_€ R .
0<t<l

For h,g E W and z E R we have
lwn>+nu4un1vo-mu)+fiu4un1vm

5. Wt) - oh)! 5 llh - gum v t e [0.13

and

59
l£h(t) + zf(F'](t))] A o - [9(1) + zf(r“(t))1 A 0|
:.|h(t) - g(t)| 5.uh - gum v t e [0,l].

Thus

112111) - 1.1911 5.21111 - 911,.

establishing that T2 is a continuous functional on C[0,lJ.
Remark. Let {B(t), 0 5_t 5.1} denote a Brownian bridge. Then
since 3(o+) = S(0+,0) = f(F“(o+)) = o w.p. 1, we have (w p.1)

12(3) = sup {B(t) + zf(F'1(t))}

0<t<l

+ inf {B(t) + zf(F'](t))}
0<t<l
and

1 (0"S(-.o)) = sup {0"S(t.01 + zf(F"(t))1
2 C 0<t<1 C

+ inf {0215(t90) + zf(F'1(t))} °
0<t<l

We are now ready to state

Theorem 3.l. Under conditions (l.l) and (1.2),
Pofoc83 5.21 + POETZ(B) 3_0] v z e R .

Proof. We assume that z > 0; the proof for z 5_0 is similar.

* **
From the definitions of B3 and B3 9

ink + ..
83 < z/oc c>Dc(Z/cc) > DC(Z/oc)

and

60
+ _ *
Dc(z/oc) _>_ Dc(z/°c) $83 _<_ z/oc .
* A ** . .
Thus 83 §_B3 §_B3 implies that
+ - A + -

POEDc(z/oc) > Dc(z/oc)] §_P0[B3 §_z/oc] §_P0[Dc(z/oc) 3_DC(z/oc)].
Applying theorem l.3.l and using the inequalities

-1 +

0c Dc(z/oc) 3_0,

sup {0215(t,0) + zf(F'](t))} 3_0,
0<t<l
we obtain
log‘o:(z/oc) - sup {0215(t,0) + zf(F"(t))}|

0<t<l

. P
_<_ sup Io;1[S(t,z/oc) - S(t,0)] - zf(F'](t))| +0 o .
0<t<l

Similarly,
-1 - . -l -1 P0
|oc Dc(z/oc) + 1nf {oc S(t,0) + zf(F (t))}l + 0 .
0<t<l

Hence

p
(3.1) ID:(z/oc) - D;(z/oc) - Tz(o;]S(t,O))| +0 o .

He now show that
(3.2) 10(12(a;‘s(.,o))) e L0<TZ(B)) .

Recall from lemma 1.4.] the process {Zn(t), O 5_t 5_l} with con-
tinuous sample paths which converges in distribution in (B, C[0,l])

to {B(t), 0 §_t 5_l}. Since

61

-1 -1
“2 (~) + o S(-.0)Ha,5,o max Id I.
n C C 131:" 1

the continuity of T2 yields

-1 P0
ITZ(<1c $0.01) - Tz('Zn(°))| + 0.

Since
L0(T(‘Zn(°))) 9L0(T("’B)) ~ L0(T(B)),
(3.2) is proved. Combining (3.1) and (3.2) gives

4» -
PotDc(z/oc) 2,Dc(z/oc)1 + POET2(B) z_01
and

POED:(z/oc) > D;(z/oc)1 + P0[12(B) > 0].

But Po TZ(B) = 0 = 0 by lemma l of Rao et al. (1975) and the

proof is completed. B

4. Interval Estimation of 3., Throughout this section we assume,
without loss of generality, that B = 0 and that (l.l) and (l.2)
hold.

Let Yc,a denote the critical value for which one rejects
’ Ho: 8 = 0 at level 6 whenever Dc(0) > YC,a° Then a l00(l-a)%

confidence set for B is given by

Ic,a := {A; Dc(A) s-Yc,a} .

Lemma l.l and (l.l) imply that IC 6’ when nonempty, must be

an interval. Note that

62

Ic,01 = 4’ $DC(D) > Yc,a;

hence empty intervals are obtained only on an event where one rejects

the true null hypothesis.
It has been shown (Hajek and sidak, l967, p. 189) that

10(og‘oc1011 = 10103351 181111).

Defining K to be the l-a percentile of L ( sup |B(t)|) we
a 0
Dgtfj

then have

1. -l -

1m oC y - K

"#0

Lemma 4.1. Given 0 < B < m» and e > 0 there exist b and N

positive such that n 3_N implies

. -T
POE 1nf o D (z/oc) > B] > 1 - e.

|2lzb c c

Proof. Using lemma 1.3, the proof is similar to that of lemma
l.3.3. D
Define K; = sup 7

n>l c,a
on R. We are now ready to prove

and let u denote Lebesgue measure

Theorem 4.1. Suppose that (l.l) and (1.2) hold. Then for each

n > 0, lim sup oc u(Ic a) is bounded in probability by
n 9

-l
ZKaufum + n. ‘
£3221, Let €»> 0. Hme > 6 > 0. By lemma 4.1 there exist b and
N positive such that n 3_N implies

*
P [ inf D(z/o ) > K J > T - 8/2 .

63
Choose t0 6 (0,l) such that
1114110)) 1 11111.. - a
and N1 3_N such that

-l -l
POE sup oc [S(t0,A/oc) - S(t0,0) - Aocf(F (to))| _<_ 6] >1 - 8/2 V n 1 N

IAlzb 1'

Then n 1N1 implies
OcIc,a : (A; |S(t0,A/oc)| Z-Yc,a} n {A; [AI 3_b}
-l
c {A; |S(t0,0) - Aocf(F (to))l f_Yc,a + oc6} n {A; |A| §_b}

; {m (5(t0,o) - Aocf(F"(to))l 31cm + oc5}

+ 0C6 + s(tO.O))/ocf(F"(to)) < A

= {A; '(Yc,a

:_(y + océ - S(t0,0))/0cf(F-](to))}

c,a

with probability greater than l - 5. Hence

“1 + 6)/(Hfflm - a): > 1 - e.

POElim sup Ocu(1c,a) §_2(oc c,a

11

Since 6 and s were arbitrary and lim 0'1

n+m c = KG, the proof

c,a
is completed. . D

It is possible, under more restrictive conditions, to show
that the bounds of theorem 4.1 hold w.p.l. Such a result is given
in the following
Theorem 4.2. Assume, in addition to (l.l), that f'(x) exists and

is bounded for a.a. x. Regarding g, assume that

 

64

(4-1) nlfio;1 max |d1.| =0(l) as n+m,
lgjfm
(4.2) lim inf n"o2 > o .
n c
Then

. -1
l1mnsup “c““c.o) _<_ 2KallflL° w.p. 1.

Proof. Let c > D and D < b < m. Define, for t e (0.1) and

x,A 6 R,

u*(t.A) = f] dimx.) : 111411) + Adm.
1:

n
U'(t.A) = Z diI(xi 5_x + Adi).
. i=1 .

From theorem 3.l of Ghosh and Sen (l972),

sup |U*(t,A/oc) - U*(t,0) - Aocf(F'1(t))l + 0 w.p. 1 (P0) .
0<t<l
IAl_<.b

The theorem is proved by modifying the proofs of lemma 4.l and
theorem 4.l once we show that for |A| 5_b and n sufficiently
large,

(4.3) SUP lu*(t.A/oc)l = SUP IU'(x,A/oc)l = D(A/OC) -
0<t<l xER

Let S(F) denote the support of F. By (l.l) S(F) is a real

1

interval. In case S(F) = R, F' is a bijection and hence

65

'1' -
sup |u (t,A/oc)| sup [U'(F 1(t).A/oc)(
0<t<l 0<t<l

sup lU'(x,A/oc)|
MXER

D(A/oc). IAI 5,b.

establishing (4.3). In case S(F) f R,

*
sup lU (t.A/oc)| = sup IU'(X,A/Uc)l
0<t<l 0<F(x)<l ‘

and we must prove that

(4.4) ~sup lU'(x,A/oc)| = sup |U'(x,A/oc)|
0<F(x)<l ‘ x€R

for n sufficiently large and |A| 5_b. To this end choose 0 < N1 < m

such that

-l

boc max Idil < u(S(F)) V n _>__N1 .
1

<i<n

Then for n 3_N1 and x g-xo = inf S(F),

0 < U'(x A/o ) < E d+ = U'(x+ A/o ) A > 0
_. ’ c _. «i 09 c 9 9

1=l
U'(x,A/oc) = o = U'(x;,A/oc) , A = 0.
and
" - + .
[U'(x,A/oc)| §_1£] di = |UI(xO,A/oc)| , A < 0
imply
(4.5) [U'(x,A/oc)| _<_ |U'(x3.A/oc)l.

Similarly, for n 3_N1 and x 3_x1 = sup S(F).

66
(4.6) |U'(x,A/oc)| 5_|u'(x;.A/oc)| .

Combining (4.5) and (4.6) yields (4.4) and hence (4.3). D

5. Asymptotic Efficiency of the Ic,a' Using the bounds of section
4 one can compare the Ic,a to other common confidence intervals for
which asymptotic lower bounds can be computed (Rao et al., 1975).
Koul (1971) has computed the asymptotic lengths of the normalized
confidence intervals based on a wide class of linear rank statistics.
Although his bounds were in probability bounds, they can be
strengthened to w.p.l bounds by applying the results of Ghosh and
Sen (1972). As an example of the type of results which can be
obtained and to demonstrate the efficiency of the Kolmogorov-
Smirnov type intervals, we compute bounds for the asymptotic
efficiency of the Ic,a with respect to confidence intervals based
on the Normal scores and Nilcoxon type rank statistics.

In what follows let 6 denote the standard normal c.d.f.,

let 20‘ be defined by ¢(za) = l - a and define
o(t.f) = f'(r"(t))/f(F"(t)). o < t < 1.

Assume, without loss of generality, that B = D.

(a) Comparison with Hilcoxon-type intervals. Let

-1 n
a=={A; In E

K
‘ i=l

diRniAl 5-6c,a}

where 6c is such that one rejects H0: 8 = 0 at level 6

whenever In"1 X?=] diRniAl > 6c,a and accepts HO otherwise.

Under the assumptions of theorem 4.2

67

. 2
11m 0 u(K ) = 2 //S f f (x)dx w.p.l.
C Caa a/Z

Thus,

. “(1C a) _ J17 2
(5.1) 112+:UP 3112‘37'5-*6(F) .- KG f f (x)dx/za/zllfuco w.p.l.

To obtain an upper-bound on

lim sup Wa(F)
o+0

for fixed F, we investigate the behaVior of

112+;up KZa/Za'

From Hajek and sidak (1967, p. 182) we obtain

1 - o = P[ sup (3(1)) 5_KaJ 3_1 - 2 exp(-2K§)

0<t<l

and hence
2
-ln(o/2) Z-Ka .
Thus
lim sup K2 /2. 5_lim sup -[2¢-](o)]-21n(l - a)
n+0 a a n+0
= lim sup -2x'zln(l - o(x)).

x-bco

Using

(x'1 - x'3)¢(x) 5_l - ¢(x) 5_x-]w(x),

where o denotes the standard normal density, one obtains

68

lim sup -2x‘21n(1 - ¢(x)) = .25 .

x-KD

Therefore,

(5.2) lim sup 9 (F) 5,73'7 f2(x)dx/2flfflm .
mo 0.

The following table gives the upper bound wa(F) for
various choices of F and a. Here 90(F) := RHS (5.2).

TABLE I

Values of wa(F) for Comparison to
Hilcoxon Intervals

a\F Std. Normal Logistic Dbl. Exp. Cauchy

.5 3.005 2.834 2.125 2.125
.1 1.823 1.718 1.289 1.289
.05 1.697 I 1.600 1.200 1.200
.025 1.618 1.525 1.144 1.144
.01 1.548 1.459 1.094 1.094
.005 1.510 1.424 1.068 1 068
0 .612 .577 .433 .433

(b) Comparison with Normal Scores-type Confidence Intervals. Let

n
- Q -1
Jc o - {A’ '12] di¢ (Rn'lA/MI S-6C.a}

where 6
c o

9

is such that one rejects Ho: 8 = D at level
whenever If;1 d19'1(Rn1A/n)| > 6c a and accepts H0 otherwise.
Under the assumptions of theorem 4.2,

. _ l -1
A12 ocu(Jc,a) - Zza/z/fo ¢ (u)o(u,f)du w.p. l .

69

Thus,

11(1 )

. c a = 1 -]
limup mgwafi) K01 f0 9 (")‘PW’IMU/Za/znfuaa w.p.l.

The following table gives the upper bound WG(F) for various choices
1

of F and a. For a = 0, ?a(F)-:= f0 o'](u)w(u,f)du/4flfum.
TABLE II

Values of wa(F) for Comparison to
Normal Scores Intervals

a\F Std. Normal Logistic Dbl. Exp. Cauchy

.5 3.076 2.769 1.958 1.8
.10 1.865 1.679 1.187 1.1

.05 1.737 1.564 1.106 1.0

.025 1.655 1.490 1.054 .96
.01 1.584 1.426 1.008 .92
.005 1.546 1.392 .984 .90
0 .627 .564 .399 .36

 

6. Monte Carlo Study. In order to compare 63 with other point
estimators of B, 5000 samples of size 40 were generated from

each of the standard normal, double exponential, logistic, and
Cauchy (median 0) distributions. Taking c1 = i, l §_i g_40, we
then computed 6]. 63, and the Nilcoxon estimate, 6", for each
sample. The following table gives 52(ch.) for each set of 5000

samples.

70

TABLE III
Values of 52(oc6 )

2 F

 

5 Std. Normal Logistic Dbl. Exp. Cauchy
s2(oc§w) 1.0668 2.9883 1.4541 .3153
52(oc61) 1.1755 3.1985 1.4532 .3666
s2(oc§3) 1.1181 3.1138 1.5497 .3386

Each set of observations was based on a corresponding sample
of uniform (0,l) variates generated by the Fortran subroutine RANF
on the Michigan State University CDC 6500. The logistic and double
exponential variates were generated by computing F'](U) for each
uniform variate U, the Cauchy variates were generated by computing
tan[(U - .5)/n], and each normal variate was generated by computing
(-2 ln U1)15 cos(2n U2) for independent uniform (0,l) variates

U1 and U2.

BIBLIOGRAPHY

BIBLIOGRAPHY

Adichie, J. (1967). Estimates of regression parameters based on
rank tests. Ann. Math. Statist. 38 894-904.

Fine, T. (1966). On the Hodges and Lehmann shift estimator in the
two sample problem. Ann. Math. Statist. 37 l814-lBlB.

Ghosh, M. and Sen, P. (l972). 0n bounded length confidence interval
for the regression coefficient based on a class of rank
statistics. Sankhya Series A. 34 33-52.

Hajek, J. (l969). Nonparametric Statistics. Holden-Day, San
Francisco, California.

Hajek, J. and sidak, z. (1967). Theory of Rank Tests. Academic
Press, New York.

Koul, H. (1971).. Asymptotic behavior of a class of confidence
regions based on ranks in regression. Ann. Math. Statist.
42 466-476.

Koul, H. (1977). Behavior of robust estimators in the regression
model with dependent errors. Ann. Statist. 5 681-699.

Rao, P., Schuster, E. and Littell, R. (1975). Estimation of shift
and center of symmetry based on Kolmogorov-Smirnov statistics.
Ann. Statist. 3 862-873.

Scholz, F. (1978). Weighted median regression estimates. Ann.
Statist. 6 603-609.

Sievers, G. (l976). Weighted rank statistics for simple linear

regression. Mathematics report #44, Western Michigan
University.

71