ASYEWYG'E’IC YHEORY OF $9535
NQRPAIQAMETRSC TESTS

'Fhmisfio‘rfiha MmoéPH.D.

MEWGAN S'FATE UNIVERSEW

Shashkkaia 5. SukheémQ
me

This is to certify that the

thesis entitled

ASYMPTOTIC THEORY OF SOME HONPARAMETRIC TESTS

presented by

SHASHI KALA B . SUKHATME

has been accepted towards fulfillment
of the requirements for

PH. D. degree,“ STATISTICS

c1 14/; A/ AA ¢//'

Major professor

Date 621’» f any 3 ”7,40
/ ,

 

0-169

LIBRARY

Michigan State
University

THE

IRE

 

ASYH’TOT! C THEN" 0F SOHE
WARNER": TESTS

, BY
Shashikaia a. Suid'iatne

A THESIS

Suhaitted to the School for Advanced Graduate
Studies of Michigan State University in
partial fulfillment of the requirelaents

for the degree of

DOCTOR OF PHILOSOPHY

Department of Statistics

l960

; Please Note: .

r Not original copy. Indistinct type

I throughout. Filmed as received. f
University, Microfilmfl Inc. .

----- 'u.—_-_— 7 , -~—"—— ’.'

 

THE

4!...

ACKNOHLEDGHENTS

it is with pleasure that l express my sincere gratitude
to Professor Copinath Kallianpur for suggesting the problem
treated in Part II, for his stimulating advice and guidance
throughout the entire work. day sincere thanks are also due
to Professor Ingram Olkin for suggesting the problem treated
in Part I and for his keen interest and encouragement during
the progress of the work. Thanks are also due to Hrs. Barbara
A. Johnson for typing the manuscript.

The author appreciates the financial support of the

Office of Ordnance Research.

 

THE

ASYHPTOTIC THEORY OF SOME

HOHPARAHETRIC TESTS

By

Shashikala B. Sukhetme

AN ABSTRACT

Submitted to the School for Advanced Graduate
Studies of Michigan State University in
partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Statistics

Year l960

APPROVED gag/SA N /d’ 07///,

 

we:

 

SHASHIKALA B. SUKHATHE ABSTRACT

This thesis consists of two parts in which two different problems
are treated. Part l deals with some nonparametric tests for location
and scale parameters in a mixed model of discrete and continuous varia-
bles. in Part ll we consider asymptotic theory of modified Cramir-
Smirnov test statistics in parametric case.

The following problem is studied in Sections i - 6 which constitute

Part i. Let Z' , ..., 2 with ZI - (X', Y.) be independent observa-

H
tions from a bivarlable population. Assume that the random variable

x takes only two values 1 and O with probabilities p and i - p
respectively. Let NY 5 y|x - j) - r] (y) , j - o, l . lie consider
the problem of testing the hypothesis H: F' . F0 against the alterna-
' and F0 are assumed to have the same

tive A: F, i F0 where F
functional form except that they differ either in the location or the
scale parameter. Two sample median test and Hilcomon test have been
considered for testing the differences in location while two sample

rank test and run test are studied for the differences in scale. The
problem has also been generalized to the case when the random variable

X has a multinomlal distribution. In the case when p is unknown, the
test statistics are modified by replacing p by its usual estimator

and we investigated whether the tests based on the modified statistics
are asymptotically distribution-free.

In Part ll consisting of Sections 7 - l0 , we consider the follow-

ing problem. Let x‘, ..., Xn be n independent observations‘with a

SHASHIKALA B. SUKHATHE ABSTRACT

continuous distribution function 6(x) . For testing the hypothesis

H: 6(x) - F(x, 0) where the functional form of F is known, but the
value of 9 £1 , an open interval in ii' is unknown, Darling modified
Cram‘r-Smirnov to: test by replacing O by its estimate 3n obtained
from the sample. He obtained the asymptotic distribution of the modi-
fied test statistic under the hypothesis and studied its properties.

in this part we extend Darling's results to the case when 9 - (0', 92)
is a point belonging to an open interval in R2 . we obtain the asymp-
totic distribution of the modified Craer-Smirnov test statistic under
the hypothesis. The limiting distribution is found to depend on the
properties of the estimators of (9', 92) . Two different cases are
considered according as the estimators are superefficient or regular,
jointly efficient in the sense defined by Cramér. As the characteristic
function of the limiting distribution is the Fredholm determinant of a
symmetric, bounded, positive definite kernel of a particular form,
methods of finding the Fredholm detenminants of such kernels are given.
Lastly we study the k-sampie CramEr-Smirnov test in parametric case

for testing the hypothesis of goodness of fit and investigate its

asymptotic distribution under the hypothesis.

PART I.

PART II.

TABLE OF CONTENTS

uouranaustnic 15515 FOR LocAiiou nun SCALE
rmnns iii A MIXED noon. or oiscners
AND continuous VARIABLES

l. lNTRODUCTlON

2. m smelt ieomc TEST

3. m mu vitcoxou 1:51

a. c-SAMPLE rnosten

5. RANK TEST roe DISPERSION

6. THO SAMPLE RUN TEST

ASYMPTOTIC THEORY OF MODIFIED CRAMER-SMIRNOV
TEST STATISTICS

7. INTRODUCTION

8. THE cumin-salami 1551' ll THE M-
PARAMETER case

9. LIMITING DISTRIBUTION OF C: - CASE
OF EFFICIENT ESTIMATORS

to. k - smut CRAMER-SMIRHOV TEST in
THE rmmc case

PAGE

lb

38
#5

#7
SI

60

 

T”? c‘i

 

Part I

NONPARAMETRIC TESTS FOR LOCATION AND
SCALE PARAMETERS IN A MIXED MODEL OF DISCRETE AND CONTINUOUS VARIABLES

I. introduction

Let 2', ..., Zn idiere Z‘ - (X', Y') be ii independent observations
from a bivariate distribution. He assume that the random variable X
takes only two values, I and O with P(X . I) . p and P(X - O) I-
l - p - q . Let the conditional distribution of it given it .1 0 - o, i),
be PIT 5 y | X n j) «- i-‘J(y) . The problem considered is that of testing
the hypothesis H: F' - F0 against the alternative A: F. f F0 .

He divide the observations 2', ..., ZN into two groups according
as the observed value of X is l or O . Let U', 02, ..., U" , (n >0)

and VI, ..., V denote those values of Y for which the corresponding

N-n
X is observed tobe l and O , respectively. Since 0', ..., ”n and
V', V2, ..., V...“ are independent, the problem of testing the hypothesis
ii is equivalent to the problem of testing the hypothesis'that the two
independent samples come from the same population. However, the problem
differs from the usual two sample problems in that, the number of observa-
tions in each of the two samples is a random variable.

In that follows, we assune that F' and F are absolutely con-

0
tinuous, having density functions F; and F6 , respectively. lie
further assure that F' and F0 have the same functional form except
that they differ either in the location or in the scale parameter.
Several two sample nonparametric tests have been proposed for test-

ing the differences in location, especially those by Hilcoxon [l],

Mood [2], Held and Holfowitz [3], and Lehmann [ii]. More recently some
nonparametric tests have been proposed for testing differences In dis-
persion by Mood [S], Sukhatme [6, 7], and Xamet [8].

In Section 2 we consider the median test and in Section 3, the
two saqiie Hilcoxon test, with reference to the'probiem considered here.
In Section ii we generalize the median test to a c-sample problem. In
Section 5 we consider Mood's rank test for testing differences in dis-
persion, and Section 6 is devoted to the run test. For convenience of
exposition, the cases where p is known or unknown are treated sepa-
rately. In the former, both the exact and asymptotic properties are
investigated. In the latter case, the various test statistics are mod-
Ified by replacing p bylts usual estimator 3 , and we investigate
whether the test based on the modified statistics is esynptoticeiiy

distribution-free.

2. Two Sale Median Test
Hithout any loss of generality we assume that the sample size M

is odd, say M - 2k + i. Let G’ denote the median of the combined
sample of 03's and Vj's , and let m be the number of U.'s which
are less than H . The hypothesis II: F a F

. i 0
either too large or too small. First consider the case when the distri-

is rejected, If m is

bution of X is known, i.e. when p is known. In Section 2.l, the
exact distribution of m is derived and in Section 2.2, its limiting dlf‘
tribution both under the hypothesis and the alternative is obtained. The

consistency of the test is proved in Section 2.3, and Its asymptotic ef-

ficiency

with respect to the corresponding parametric test based on the correla-
tion coefficient is determined in Section 2.4. The case when p is not

known is dealt with in Section 2.5, were it is shown that the test

based on the statistic (m - I6) [Jkpi’i is asymptotically distribution-

free.

2.1 Joint and Marginal Distributions of m and ii .
Menceforth f(-) denotes the probability density function of the
random variables written In the parentheses. He first prove the follow-

ing lane idiich gives joint distribution of m and 'H’ .

Lena 2.l.i. Thejoint distribution of m and AH is

(2:. + i): [p marinate)“

(2.i.i) f(m, 3’.) - I... (k-an RT

 

[Hi we - q roan" [9 ms) + q ram] .
m-O, I, 2, ..., k .

Proof. Observing that n is a binomial random variable b(II, p) , we
have

(2.I.2) f(n, m, 75) . f(m, 'w'In) f(n) ,
where
(2.1.3) fin) - (,2) p" 4“"

and from Mood [2],

(2.I.Ii) f(m, zlnln I‘m" ir-m
MIIFAUJW [VI—(”5"

 

 

WI (71 m 1)!

fl—m

.fl1[r;(q;)']w[i—.F5(-sifl 5 _(_N-n>1[rcm] LI»F(w):I

’mI (”m-7n)! <k—vnfl (N‘W'I‘HW’ ”I

 

N—n-irm

' ..,., 41v

(N—mJU-gcwil [vi—gm] few?
(krm)I(N-'fi"'h+’n*,}I

N fl- I‘HW'I

i:

h

To obtain f(m, 'w) , sue (2.l.2) over all admissible values of n , re-

calling that NOZR+I, J
_m _ r—m

'“ . "31" " (9”
Thai") 2'— N'! [kgl'fjj [TIE ”LI X

'm! (k-mvg w

 

‘Yl-Wi'i
b-Ti m A,’->)vkz+n4r

uni-I'M WI

('3’)

F162

 

 

1) i» I," WWI] firfihi’fl L
2—- (7) “m 0’ (N Yi-Irz+w)Iai i -

lizmil
N W N'h'kwfl"
~— T’m N' 7’ Rim ~\" ’-
+ > ..., Lb “CI/H [I- F( AI] I-I'gf‘ Wit/h -..-_ I:(\V)
“T" (7“ WII ’N ”n Iri‘m-UI
73177)

.. ”3%“ W2" «Um [12(ij
I’m/(If..- I‘lI kI. x

U - we - rote)" [afl'h'il + «5(a): . H
Under the hypothesis II: F' . F0 , (2.l.l) reduces to,

(24.5) fin. 33) - 8121.5}? p" q""'ir(&ii"iI-rim" r'm .

m-O, I, ..., k.

7
I

/.

To obtain the marginal distribution f(m) integrate (2.i.5) over the
duein o 5 F0?) 5 i :

(24.6) fin) - (3;) p' a“.

so that m is a binomial random variable bIk, p) . The marginal dis-
tribution f(w) is obtained by swing (2.I.5) over m :

f(i'r') . STE-$131 [won‘t i-riiin" mi.) , -m <w<m .

2.2 Mtotlc Distributions
Let g denote the median of the distribution of Y , i.e. § is

the root (assumed unique) of the equation
(2.2.!) pf,(§) + «Gigi - in.

As before let m denote the nimber of li"s that are less than II ,
the samle median of the combined sample of Il"s and V 's . The fol-

.i
lowing theorem gives the asymptotic joint distribution of m and H .

Theorem 2.2.I. Let
1) :. ”flr'NFFCi}. '7] :JFJI‘II'W
1—3531”??? J
where g satisfies (2.2.I). Assume that in some neighbourhood of
the density function f'(x) - FIIx) (i - 0, I) has a continuous de-
rlvative. Then the asymptotic joint distribution of (v, ii) is bi-

varlate normal with zero mean vector and covariance matrix 2 - (on)

given by

 

 

02., 2 _. .

or. -.-. 0‘32»): c1503) + i-E-IJI‘KHE(VIT)TQ£MTH7[I'd] .,
2 5m [ht/animal

2r ) '
' C“: '-
11 _. i _ W. ...,: _-_ - _-_....‘....\ ’2- - J
itrfit’wte’w"

t. ixz‘flfl-P- 5F“)

we; [Wei-i tit-w"

21.9.9.9 Throughout this proof for slqlicity write F'Ig) - FI and
mg) - r, . Thejoint probability density function. of m and II is
given by (2.l.l). Substitution of expressions for v and q in (2.l.l)
yields

.5 CG\— NIP ‘51“- I :. 'I \T’
(2e2e2) i (7?)? j" M‘“"’“"" “Hi-I [III (3" ng)’ [F0(§+ I)- I) I x

Expand FII‘grrfl) In Taylor's series about 2 :
a N J

fiiffil
1".“

) :2 EoiflI. --:- 0(315), 5:0,!-

.-. I.
IN )

pa 2.
I-PE(§I%)—ie(€+?fi :1 i-%(P£I+CLTO)-o(%_) e

 

Using these expansiais in (2.2.2) and noting that I - 2k 4- 'l , we get

f-(mjgv) : INYIRQLI I k! ”WI, (2pfi)w(26) Fo>k‘mf X

" LII? MICk-W‘l.
I):+'1%+o(gs)] b.3323) ”(3’) x
2— k 7
I" % (5*,Tfiio)+°(%)I fIIH‘rlfiIW‘Ii )3

am m m {a} -

Let 5-,I(v,n): e$v_<_b,c.<_ngdf share a, b, c, d, are

finite. Iiow using Stirling's formula for n! we have

Az does not depend on ‘I and due to convergence of binomial distribu-

tion to normal, uniformly In S we havé

 

A “z
‘2” W ”(WI '
Newconsider

l0. ‘3 - (”'T" M” IWI+1:.£L;_+C(129

:2 2
+ (R-pr' -v hf.) ioin-rlieearc LI)
m F5
+ klosIz—m .

54‘ [" P-0(n: /°
‘I

Using series expansion for log (l 4- x) it is seen that for all (v,n) e S

2 .1 ;‘ ‘f- , \v;
a I 7; ./)., ‘ ,I‘.‘ . “gi'TT‘T/f p
log A3 - ELL -7“ 4 rt dried . s‘ +::'/1I:2{it._-jctutor;
-' "" ._ " . .I ‘
I 0 i- '
I w)

Using continuity of f' at " it follows that for all values of m and

H for which (v, 1') e S

'7 l m- ”. r 6 "I
"J \ / i, 4-» '1) ' -. . ' v ei— I_ ‘ .. 5, I I
+6er 4/ (".1 1,,_L._" .-__"._ - .: fl é/t -;-A I If f L. ‘ J 'I.’ 11157," J Le
- '4". T' S " ' L. ‘ ' ‘
TI f‘ I_; 1‘ ' [Tb a) .,/ '\ i ’-
2. —— i r. 9 I
r ’- - ' P + d. ‘O
J' I I; - c I
a.“ 7‘5 "‘ i I" I :1- "’ i e
s. 5 r' I
e' r I Al 'I
/,
. ~
Mow after making the transformation from (m, H) --> (v, ll) we get
P.*a<v<b,c< <4:
i - - - - ‘
. 7f -
f ’1 f I. I/ I :d P. l r I
’ FT "' :’ - r I ' K "J ' ~—
~ I r) I - I c f L’ ‘ ‘T . '
(J -~ s’ / t ’I“ I . r/ T 7—1 '7‘ ‘ I!” J '1’ /
'3' / A V J ¢__ k: it "'
i i ’1 C " fl
/ 7
I x” “' I _- . r , d I 1’ '
/ , .. ..- . f I- ‘_f__§ «f 2 '4 (LIL, 1 '~ ’ 'T o i j I
-. ~ 5.. , . , ’ i
v f ‘ l V K *1 ’x A d"
A! is)"; L'

Hence 9. a_<_v5_b,cs_n$d£) -—-->'}, 1’ fiwnidvdn.

.1 “,
Ct C

where f(v, q) is the density function of the bivarlate normal distribu-

I

tion stated in the theorem. . II

.9-

Remark. it follows from the above theorem, that idien the hypothesis
I ll. F‘ "o is true, on- 0 , so that v and n are asymptotically

independent. v is asymptotically on(o, q) and ‘l is asymptotically

6mm i/M’ign -

2.3 Consistency of the Test

Consider a two sided test of the hypothesis ll: F. . F0 against

F. f F0 , for which the critical region is given by

Hie-hp) I Um)": I > tli,a . The sequence tn,“ is chosen so that

u 11; O tn,“ - ta , man to satisfies l - §(ta) act/2 , and

fit 3 is the standardized normal distribution function. Then the power

of the test is given by

+ M13591-
N51571:) F; (*3)

Ami/ow < cw »_ + kEKI—1€(“%>)

 

-Q— '4

3N<Pi§ffififéf¥p fofi-é—fiflD liflofiqfi (i;

if F'(§)fl12,thepomerapproeches l,as li—>oo, andhence

the test is consistent.

-l0-

For alternatives F. > F0 ,

too large (small), and prove in a similar manner that the test is con-

sistent if F'(§) > i]: , (Fl(§) < in) .

(f'<l-'o) wereject ii if m is

2.10 Asmtotlc EfficiencLof the Test

Definition: Given two tests of the same size of the same statis-
tical hypotheses, the relative efficiency of the first test with respect
to the second is given by the ratio nzln' , where n. is the saqle
size of the first test required to achieve the same power for a given
alternative, A , as is achieved by the second test with respect to A ,
when using a sepia of size ii2 . For a sequence of alternatives
changing with n in such a vmy that as n ->o the power of the cor-
responding sequences of the tests converge to some number less than i,
the relative asmtotic efficiency of the first test with respect to the
second is defined as the limit of the corresponding ratios nzln' .

Let F'(y) . Fo(y - O) , then ll: F. :- F0 is equivalent to
ll: 0 - o . For alternatives 0 > 0 , we evaluate the relative asymp-
totic efficiency of the median test with respect to the corresponding
parametric test when P, and F0 are normal, with means ii' and “0 ,

respectively, and a couon variance 62

. Then FICy) I Fo(y) if and‘
oniyif u' - "0 , id'iich is equivalent to f I f (X, Y) c 0 , more
/0 is the correlation'coafficlent between x and Y , Tate [ii].
Let 0 - (il‘ - isoHa , then we are testing the hypothesis 0 - 0
against. 0 > o . The test is based on the tapia correlation coeffi-

cient r , defined by

Tate [ll] proved that r is asymptotically normally distributed with

asymptotic mean and variance given by

’/2_

(2.1m) “9") a “(4573:...
i+p19

O‘QET);11L(I+I°?,9 -9 (6:03 i)
G 4NCHP19")3
The critical region for this test is given by rfi > t"‘. a , idiere

' . . - .
{t".’af is such that wig. tr,“ ta and f(ta) i a

the power of the test is given by

’i'i'“, " '9 {VW Noi %J=I—Pif_____—l‘<9<r)> %;54'NN9(»~)
6;”) J? 03”) . .
Since r is asymptotically normally distributed,
0. - "a
m N“) _ "I . tuna F u.ir)
li' ->o li —>o fir oak)

( T
liow for a sequence of alternatives 4 0"“ s idiere

 

‘—

/ o,— ...7
a". a 5' /ii' ,s'>o, lim 4/; .,.:ii' oval: .i

li'—>m *'—‘ li'

and

Tate [ll] proved that r is asymtotically normally distributed with

asymptotic mean and variance given by

’/2_

(2 ‘i I) 990') - o(_m,____
i+P19

03%); 9(Hb9’9 —-9(6I°<1~i).
G 4N(i+bie")3
The critical region for this test is given by rf— > ti." (1 , where

t. I “d! H t. I 00‘ I i - e
{ “g’af S 3 mt “I —; 0 "g,“ ta 1(t0) a

The power of the test is given by

3;.(.) . '0 {r> {’N“ %j=l‘pf_____ 73/490") {‘N’ Nd'N/wg(i")
Since r is asymptotically normally distributed,

a. _ ”a

n. 5”,“) .- 15; (”an tll ,a f— “9“)

'—>0 ll'-> J? 000’)

liow for a sequence of alternatives 50' . } idiere

 

v

f T" --j
0;. - b'fgii' ,6'>0, lim fi/j «li' 60. (r)- -i
ii'->m- --‘ u' ....

and

-12-

.0

iim fuoi (r) o0. (r) g'- b'./ pq .
li' -> a; li' ll' :
Therefore

(2.h.2) lim '.(9'.) - I (- t + b'fi; ) .
li' —> as“ l a
liow for the median test under consideration the critical region
for testing ll: 0 . o , against the alternative 0 > 0 , is given by

(Intel/W <tu’a where "ll-gut“, - -ta with f(- ta) no.

The power of the test is obtained from

 

 

 

 

 

a ._. ,c Lip J 4
f“ 7/ h "-——.._“‘ ‘-;{~ ~ "‘ ,
r ./ I; L c.
. , \
I
N " l
l .‘ I 7
:.. 0.3 f' '7” ‘ Y'LQE’. K ’1’, "ka2 " ‘\NI’9E(£)' kP/f
f, _ \ w.- ‘ _ , “
a ,r / (Ind)
, / G
idiere
/ " c - ’ .".
1 , , H f . 9 f u, -"’; fi fin: 2.1"," "S -> a ;‘
(”i j“ 1L7": ’. nP'fz ( - [ >1.
«’- fi 7‘ 5:3! Ln- (5’ "y-F’W ’t
' " '~ 9 " r o .
By Theorem 2.2.l
\\
/.0 r} ’-'--I- >" [’N I. L- [A K 1 ’ \'
l. — N”- we?! __ (.I'F‘r‘;‘.,,i__* u.‘ ,
"I fl'(9) I I { Hm .- --.c-M‘M'“ i _-_ V g .u_,._ 'h *3». / c
I ‘-9 m '\1,\I~‘},:\.l 0-;(m) /
.‘ I

( * -_.
For a sequence of alternatives {0 -- , with O" - 8/ (ll ,

a>o, lie 331—2-7— .- i . Also since is satisfies
li->o 0

ll
pFO(§ - o) + «row - m .

2‘3- .. Eo(€‘9L ,_
d9 "' pfiC‘ireHcfioM)

Hence for the semce {9.}

s 02‘
Maw/m) = N; [my gym (ta-Q + owl—7;)

so that
n. Mg£§._:§/E)n:ffim ._ ._:_ 65M )fef/Ja/
I .9 m %(m)
'N
tilidl yields

(2.1..10) lie 511“») - I(- ta+bfo(§)\/2pq ) .

ii-->oo

The two sequences is“ and {0,3,} will be the sane if "II-6'25: .

r 2.5.2- “(2.1m i i the n a e - ll ".o'.
roe( )a )tsseen til-;oo”(") ~4m3.(')

only if b'lb . H f°(‘5,) . lience the required efficiency is given
by .(n, r) - 2102“,) - ll'n' .

2.5 Case lien I is Unknown
the theory developed so far is not applicable when p in. unlmouu

The usual estleate for p is '3 . nlli , and we consider the test based

on the statistic (e-lé‘) l~(k$é)"z; lie now show that the test based

-11.-

on this statistic is asyeptotically distribution free.

Theoree 2.5.l. linder the hvathesls ii: Fl - F0 , the statistic

(nu-ti) I use)": is asmtotialiy non-ally distributed with can

zero and variance 1/2 .

Proof. Since on. 3 - o , by an application of Slutsky's theore-
[9, p. 255], pile (061m)”: . i . lience the lielting distribution of
(e - (0)1[38 is the sees as that of (n - l6)IJ kpq . lirlte

ELLE. :- 29:3,.‘1... hilt-...,?!” -_-_—_ T -T
W 9,“ v7.55: We, - ' L

The asywtotic joint distribution of (T', T2) is bivarlate noreai
"TL(0,Z) with Z Oh”) there o"-l,ou-oz'-ou-III2.
llence the required result follows. || ‘

3. Tm Sale Hilcoxon Test

As before, let 2. a (X', Y.) , i- l, 2, ..., ii , be ii independ-
ent observations froe a bivarlate population, tdtere x asst-tesonly two
values, l and 0 with probabilities p and q- l - 0 respectively.

The test statistics eay then be defined as,
N

U : --L""" :— H<2£22j)

J
N N(N")L#-J:,

where

u, if xI-Inxjoo and '3"):
”(2., 2’) - '

0, otherwise .

-15-

lf 0', liz, ..., ”n denote those Y observations for which the corres-
ponding values of x are observed to be I , and V', V2, ..., V...”
the reeeinlng observations on V , than I(ii-l) 5' is the total nueber
of pairs (0', VJ)~ such that Ii. < VJ . The'hypothesis ll: F' . F0
is rejected if fin is either too large or too seeii.

in Sections 3.i and 3.2 we obtain the eean and variance of fin ,

and the exact taming distribution of ii under the hypothesis. the

ii

asyqtotic distribution of ll" , both under the hypothesis and the al-
ternative, in the case when p is imowii, is obtained in Section 3.3.

in SectionBJi we prove consistency of the test, and in Section 3.5 find
its asmtotic efficiency. Lastly, Section 3.6 deals with the case idien
p is unimown, where it is shown that the test statistic, with p re-
placed by its estinate 3 , does not yield an asyeptotically distribution-

free test.

3.i lean and Variance of fin .

(Y

1}

.. r{x'-i,xj-o} P{Y <1! I): -i,xj-o}

- n f my) «om .

(3.i.i) 5PM") - E, "(2" 2]) - r{x‘ - i, x - o and v

J, l

To oonute the variance of l7" , write 0' as

N n 3.;
(3.i-2) U

N N(N!)JZZ¢(U {Z} )

' L=|

ii

tdiere

-16-

l , if u < v ,
’("s V) '
' O , otherwise .

Squaring (3.i.2), and taking expected values, we obtain the conditionel

eonent:
f) I“ an

(3.i.3) N (N- I) L‘p (BUN): Ep ZZWUD’)

Nn Vi N—w l“

+Ez. z 9% vmu vhf: 2: MW .13

[33,4 ' Cf’R I TL; [Li-4

N 3'

VZ; \) ((Jo V)f(Uf9vl:)

yet 69% 7m

:: 11(N'Vi) RI U5< \331’ ”(W’l)(N'n)P{U¢4\g_) (£4 \é

+M(N-n)(N-n-1)P{'UL+4\3~ 9 (15.4 vk}

+ n (n-i)(N-n)(N+n+/) Pi (254x?) Ur‘ H

:- ”°(N’W) III-45, +%(n-l)(N-h)jfiz’df;

2.

+ M (N—n) (Iv-n 4).}. [i -_- f—gf'dfi' + M (14-!) (N-l’l2CN-Yi-I)[5f;dfgj .

Since n has a binoeial distribution, b(ii, p) ,

(Ll-lo) Enifl-n) -' HOG-lira . Enin-llil-n) - ii(ii-i)(ii-2)p2q ,
Eniu-niiu-n-i) - uiu-i)(n-2)pq2 .
Enin-I) (a...) (u-n-ii - Nil-ll (vi-2) (u-zir’q’

Using (3.i.3) and (3.i.li), the unconditional moi-ant is

Kai-025,5: - liz(ii-l)zE[E'(5"|n)] - "(I-qufF‘dFo + ll(ii-l)(ii-2)pzq Isidro

‘2 2
+ li(li-l)(ii-2)pq2 f (14012“! + li(ii-l)(ii-2)(ll-3)qu2[fF'dFo

ilence

(3. i. 5) 0: iii")- 75377 [fr til-'0 + (ii-2).: fr? di-‘O + («unfit/”(i4392 or

- 20q(2li-3) Kf r. f"o>:”

in particular, under H: l-’l . F0 , (3.i. l) and (3. l. 5) reduce to

(3.1.6) s,(fi,|H) - (pailz .
(3.i.7) a:(5"|H)-ifi:1'y LIE-En - 53?”- My]a

3.2 Distribution of 6"

Define

- /

(3.2.l) TN - li(il-l) ll" ..; nuaber of pairs (2‘, 21) such that
L. .

<rv '3 .

X‘II,XJ-O and Y, J

-18-

T” takes values 0, i, ..., k , where k‘- max n(N-n) - [lzlh] , and
[x] denotes the largest integer _<_ x . Let 1' denote the value

n,li-n
of 1' when n is fixed. Clearly 1’ takes values 0, l, 2, ...,
N n,N-n

n(ii-n) , and

(322) P{T:ch—:. ZP{n affixex ’5=I quxfzs .0
and TM anti-ff

N _ ._
3 Z (3)91” n P{Tanm‘:—ij ‘

llann and Witney [i2] have shown that the probability P {In li-n- t 3‘
1

satisfies the following recurrence relation:

(3.2.3) P{ 1:1,N-ntt...‘ .3 #74711 ~_=t}+QV_-:12P{T

”IV-n: I

Substituting (3. 2 3) in (3 2. N2) we get

(3.2.1.) “7-- ij: [DZ (IV-I m'ilN-nP‘PLIM’ffj
.. 2

-t
”‘4 «’1
+12%” "n P{TNn__Jc;—= 4‘

ZFWT tjfl’é (NW/”L n N n lP£hNen~7tm

-19-

(3.2.1.) is a recurrence relation for PiTu . t} , froa mich we can
find the distribution of T” for all ll . it is} easy to prove by in-
duction from (3-2-‘0 Wt:

N

(3.2.5) '{Tu - 03%- Z 'll-rqr , for all ll .

V / n-O
The phobabiilty distribution of 1'" obtained by using (3.2.h) is given
below for ll - 2, 3, lo, 5_ .

-20-

 

 

 

 

 

r+m+q
ill-2 i no
2 o
0 r3+3q+nzsg
l qu * an:
“'3 2 'qum2
3 o
0 #.,%+,%2+"3.$
1P3q+rnq*ré
N-“ 2 ra+zruq+P9
3 33+33+r3'
u '2q2
0 is + a q + '34: + r2q3 * rd“ * as
' rhq+p3qz*r2q:;n“
2 p:q+b%2 +hzq 3+fi“
u - 5 3 *r“ q + 2r3q2 + 2p2q3 + Pd“
h . p“ q+ 2.3..2 + 2p2q3 + ,4“
S ‘ h3 + azq3
6 '42 +P23

 

-21-

3.3 Asmtotlc Distribution of I?"
Let 2', ...,z" be ll independent and identically distributed ran-

zaz, ... za‘) , s<il , be a real valued

sy-etrlc function of its arguents. iloeffdlng [l3] defines a U-statls-

dd variables; let 0 (2a ,
i
tlc as follows:

(3030') "(2" ..., z") I
3

)0}: ”(a ',z Q‘2,...,za),
C“

.u-

where C“ eeans that the smation is over all coeblnations,
(a', a2, ..., as) , of s integers chosen froe (l, 2, ..., ii) :and
proves the following theoree on the asyeptotic noreaiity of a ii-statis-

tic,

Theoree 3.}.l. (Hoeffding)
Let 2', ..., 2' be independent and identically distributedrando-

variables. Let "(2,, ... 2") be a U-statistic. if
E [“20 , za ))2 <ao , then as l —>ao , the iieiting distribution
“I

of (u -£uu)la(u”) is %(o,I .

Clearly u is a li-statlstic and hence [5" - 6'07"” I 0'07")

ii
esyqtotically in“), i) , both under the hypothesis ll as well as

under the alternative.

3.10 Consistency of the Test

Consider the two-sided test of the hypothesis ll: f, - f0 against

A: F. f F0 , with critical region, Ill)" - 5'0"] I 0'01") I > 50,0
The sequence {tine} is chosen so that lie t where to

' 9
li—>o "’0 a

I
a y l. . .
5.-...1-.. ,. V .. - : ', 7 c Njo’

_—‘. —a.
‘ . 1
. \ fl

7.: -~ .- .." ‘ .' {I '2 'I' ’0‘ . T ' I” J " ) ' 7"
’LI I ff 3’ J; ' ’ - F— 35 "" """i' ‘- i”; i'(._ N.) ”4 if "7: r.
' .____, 1 HIV
.3 V" , ‘ . ' a' .' ,f 2 i r
r; . ‘4" i J p t {f , 4“,
Proceeding as in Section 2.3, if fF'dfo f in , the power tends to
i , as ii -—> o , and hence the test is consistent. in a sieilar manner

it can be verified, that the test is consistent when P, > f0 or

F'<Fo.

3.5 Asmtotlc Efficiencx of the Test

lie now find the asyeptotlc efficiency of the test based on Ti"
with respect to the para-atric test based on the saeple correlation co-
efficient between X and Y , described in Section 2.10. He have seen

in Section 3.3 that 5 is asymptotically noruaily distributed both

I
under the hypothesis and the alternative. Proceeding as in Section 2A
it can be proved that the required relative asymptotic efficiency is

gi ven by

2.
3 ffozivldv-j , J

l - 3M “Til - 3N)

 

 

(3050') C(fi”, r) a

The asyeptotic efficiency given by (3.5.l) is a maxim, naeely 3/7‘7”
when pq - l/lI , and is a einiwula, namely 31M? when pq - 0 .

3.6 Case HIen p is Unknown

lie now estleate p by its usual estieate ’i- nlli and consider
the test based on the statistic, [0' - Cal-in] / 9‘03") , where £603..)
and 0603”) are obtained by replacing p and q by ii and a , re-
spectively, in (3.i.6) and (3.i.7). it is interesting to note that this
test is not asyeptotically distribution-free, in that it depends on the

distribution of’ X .

Theoree 3.6. . tinder the hypothesis ll: f' - F0 , the limiting distri-
bution of the statistic [0" - Edi-5'9] I 03“,") , is noreal with wean

2
zero and variance ll - E {figs-j.

Proof. because plie 3 - p , by an application of Slutsky's theorem, we

obtain pl in (palm) ‘ ,2

Hence by a theoree of Craeer [9, p. 25“]. it follows that the asyeptotic

- I , tdIich leplias um pm ”5“)? - 0'03") -

distribution of [5” - Camel/03W") is the sane as that of
[6" - eafiuua'w”) . Hrlte

 

'0 --5 J ‘ . -fi
N f.( N2 2': '94:: -:A(U :12_ EEO) N). :p‘, m
0;.(%) ' r’ U) I?" -" '°""""'

N 72(jIV)

Sim m-M-(d-rlil-Zp)43-9“.me

_, -b -. , .. _. n
u-bfv. -’/ PM “ / i)r’/- H
(3.6.i) H b‘ " __ “N "' L “ml it?“ >1 L 2 H,"
--—~ ~-- -~—~-b--___. ———-_.,___... ...--. ..... - ..-—.... .. -... -. . ...._ - .. . , . - , ......
."‘ I T- ‘. .— (7" ”T A. f r 7—}, ‘. ’, ‘ " .1},""
i “HUN 21 ’33. sz'z.~---’/._:«‘-N-'3Mr 21"
.._._..._.._.\ 5 ~.__..._ as“-..:...-... " ..
"MIN—UL g 2’ J.
, i .2
.’ r: ’c ,
i
of. ,

'd "‘ '\:- —’
P 6' " I i .9 ‘ f -' .« ”a /!/J
A5 F. f 59-" U '1” up] 9
.' Mun-u. -. -‘ ____ ...- -, I
' v. .' 2 .
" ' é "‘1 ..i

i

As [J7 (3 - ell/«Ti it bomb: in probability and plin I’p‘ - pl - o ,
the third term in (3.6.l) tends in probability to zero. By iioeffding's'
Theorem [i3, Theore- 7.2] the asymptotic joint distribution of the first
two terns in (3.6.l) is bivarlate nornei 4% (O, ‘2 ) where Z - (on)

..m. a" - l m a,,- .2, - «2,- isii -‘ api’muii - mil . this
proves the theorem. || '

he C " 2'. PM'-

Let Z - (Y, X', x2, ..., Xc) have a (c + l) variate distribution,

where XJ-O or i,

C

ZXJ-l, P{Xj-lj lip], waj-o-éf- qJ II l-pj, and

1-1 . "

‘3
f
v

up _<_y | iiJ - iJ - rjm , 1- i, 2, ..., c . The distribution
functions F], ..., F; are absolutely continuous. 0n the basis of ii
independent observations 2, . (Y', X", X”, ..., Xe.) i - l, 2, ..., ii ,

the hypothesis lie: F' - . F" is to be tested. For this purpose

 

-25-

divide the observations 2', 22, ..., 2' into c sets according as
X). . ' ’ 1", 2, eee, C e at “1', ”12, see, ujnJOIJ) 0 for
c

each 1, Z nJ-li) denotethose Y
1-!

i.’ for tdiich the corresponding
X” - l . The problem then reduces to that of testing the hypothesis
that the c independent samples of ,“'s (i - l, ..., n , j . l ,
..., c) come from the same distribution, where the sample sizes
n', ..., "c are random variables having a multinomial distribution
with parameters p', p2, ..., 'c .

He assine that the Fj's differ only in location. Let Fj(y) -
fly 4» OJ) , j. i, 2, ..., c for some arbitrary choice of real winters
9', 92, ..., 0c . Clearly a :- 0 for all 1 yields the hypothesis

J

iic . Further we denote by ii"

FJ(y) . fly 4- 011(7) , j- i, 2, ..., c , and for some pair

the hypothesis that specifies that

(i, j) 0. f O] . It is known that the median testis sensitive to
translation-type alternatives, so in this Section we generalize the two
sample median test developed in Section 2 to the c-sampie problem under

consideration.

li.l 232i. Median Test
Let [l7 denote the median of the combined saeple of' "N's

ml the nuber of "N's

A}
(i - l, 2, ... nj) that are less than H . He assume li- lit-ti .

(' - ', 2’ eee fl ’ 1", 2, eee C) .M

C

Cieariy Z .1 - it . The test statistic proposed for testing the
J-l

'hypothesls llc: Fl :- '2 - - Fc , may then be defined as

 

f ’\
.5, " v 7 it ,0 \"‘.
.' f c- ' "
(lid! e I) [/3 z; '1, .= ___.‘ __ ,;'__ . ; a
“Iv...“ P , )
L-‘f‘ r V ’K K’ /

.-m- 1.. , ‘
1 “ ' n' it '\' _ l

(1.4.2) V- : 4' {“' :.- ,

I
f
. ~-—- _
— I
’v-{ \ [bl I. ‘I
‘ ' Vi /

|
m‘ -- d

I

i}

were 31- n1] ii , when p', p2, ..., pc are unknown. The test 'con-
sists in rejecting the hypothesis "c if Ma) iseither too large or
.too smell. late that ii defined by (li.l.l) is different from the sta-
tlstic defined for usual t-tupid median test, e.g., sot Andrews [iii].
in Section li.2 we find joint distribution of m', ..., mc and ii’
and in Section ‘0.3 the limiting distribution of ii . In Section ‘6.“ the
relative asymptotic efficiency of the median test based on ll with re-
spect to the corresponding parametric test based on multiple correlation
coefficient is evaluated. Section “.5 deals with the case when I
pp '2, ..., pc are unknown end gives the asyemtotic distribution of
3 under tbt hypothesis iic , from vdiich at conclude that the tttt based

A

on it is asymptoticelly distribution-free.

(l

 

“.2 Joint Distribution of m1, m2, ..., inc and
IV

Lemee li.2.l. The joint distribution of m', m2, ..., me and U is

(ti.z.i) f(m', m2, ..., EC, ’6)

I”) .
r" .1 ‘
-’. r‘ " ‘ 5’
’1: I O. 4 l i O .. i '. ‘ .4
e _ l t. ‘ l l 7- ‘J \ I
-- .. °‘ _; - r l- c‘f- _, ° -»
—- f ( ’ . 4”“ l l
- j' i L a.“ ' .
I I 77‘: . i, i. '- ' '
..3 as- I-
:_ to r
.. l ‘v
U
C

where ml, ..., iiic is a partition of it , ij . k .
14

Proof. As in Section 2.l the conditional probability density of

ml, m2, ..., mg and H for fixed values of n‘, ..., "c is

 

ml
( e a) ('g Oee’ C, 1, 2’ ..., c)
“fl-”Ff
R r ‘. -. ' I
l” \1’ "i‘ - ) t- a. ; Tr / i ‘3‘ ’ ».n‘i' .- J
l} / a i, 'L \ g i- 'r(‘ i " I: .
« * “’ .‘ L .. . u 2 -
- i , 1' *3 ._/. , *4" ' J ’-

"l’ n2, ..., "c have the multinomiai distribution it“; p', p2, ..., pc)

given by
. .‘x. r" I. Y 3.
(4.2.3) .13. i 1.3:) 7,: ‘4 , . ,, :- “33'” -.-... L Li . .
' ‘ .{41‘1‘3'21 1,. I “
Hence using (h.2.2) and (10.2.3) we obtain
f(." .2, eee, “c a.)
I X . f(." .2’ eee, flc’ U‘ I." “2’ eee, ac) f(n', n2, eee, 0c)

“I,“2,0.0nc

 

I C l“ .v“ "I”. .0
" ' ” v’ .. _‘ .

‘ I 1 ... 1’ t . v ‘ d 1 .I . . Vt
- "v j V! r':. \ w! (’7‘ L ’7 f '- f "w \ I I
‘ " \-—— ~ -w--—" .- I " | r . f " I I ‘\l

A. , / W’- I l . ' / I /H‘

"‘ I . ’ “4 ~ ‘n——-- .._ ~ J .~ .' I l
,7? 7rd ,3. c,-.- i— (“Mn + ’rx— 7.3., : , - .
' " a t f _ A_ C \(‘I I I f ..‘I‘ ~.‘ .‘ q
‘ r d. ‘ e
' ‘ ' t l": .
' ’ .... "\
l
. W, ] /
A K} . - O .I
" ~ 2' J r a?
"TT‘ .- ) i Il ‘f
I f r / p 1’ I I i l
m!— n' ' \ ‘ .
I) A B {L . l . I ~ i .'
in” VI, a! fl'

where for each i the smation is over all partitions (n', n2, ..., nc)

of ii suchthat n'gml+l, nj>mj(jfi), idiichgivesthere-

quired distribution (h.2.l). ||
Suing (10.2.” over m', m2, ..., "c we obtain the marginal

distribution
’ i
o c "k » ~ k
; - _' P
_ , . . n ., i —
p/ w} ____ N.’ I / ' it ' ” ; .r \w r- .24 " "" . r- "
1 ./1 /. ...... _.____.v~--" > ’0 « “ \‘aii. 1 1‘ I P ’ k h l \ l 5 I
.L t .' I I 4—— l‘ u. " i L O l. / 1/ ‘ ‘ '
. K. i e-.- II - to ‘ .‘~_ I. J} ‘ __" L... e...— l l ‘
. . :vl : I : a ‘
J ‘1 .J

Under "c: F' - F2 - ... - Fc :- F , integration over the domain

0 SFJ(w) _<_l , in (h.2.l) yields the distribution of ml, m2, ...,

' l
9 \ k. “
any.) +(an- r / *3 ,. ”Wm. 11?
/ J ‘ 7m i.7w I. "Y? i -

 

idiich is the multinomial distribution 'ii’?’;,(it; 'l’ ..., 'c)

l
C

77‘

f5 "

{\
I

 

.\.

“.3 Asmtotic Distribution of ii
He first prove the following lena idiich gives the limiting joint

dlstl‘lbfltlm Of "’ .2, eee, .c M H e

Leua'i.}.l. Let
. .., i r {4 J..- .- r-x .
'9’. :‘_ Til}. ‘_HV€.:.I~ 1) , J. ', 2, eee, C; 73 :1 ‘17." (: "bl ,— {-11}
J -..:1 t ,. I.
. f1 ’4
l// {5.5 r
idiom is such that
C

1-!

Assume that in some neighbourhood of 43 the density function
Fj'(y)'- fj (y) (j . l, 2, ..., c) has a continuous derivative. Then

the asymptotic joint distribution of v', v2, ..., v and 'l is

c-l
c-variate normal distribution with zero mean vector and covariance

matrix E givenby :-'."1\.(AU) vdiere

"5%)
pcfcl’gl '

 

x“.'+ ‘-',2’eee’(C'l),

., ’ i
. {,7 ('0,
D ‘ p . \:.
I « \ l‘
> .- \\ Ft: +(' 211‘ ' . l I
m — .. -—~- ~- vi ~ / ° » 3
1’ ‘ -
I: f “4»: '
L ~' ’ i 1

 

 

 

--.» W'-
l i . p t
I [b .0. PM ' 9 ' .
A“ " l . l ‘I -.. - ‘. I, ' — i " ’ J 1’
L! - L V} L V L- —-‘ 7 I #‘j ) fl 1) v / I:
h b T: (’4. ‘i
c C ‘ 1'."
r""" / \ pav-~- ---o-v~
. z ,. . , .
\A. .— ”’4 f.“"' /1. " ’ :2" 7-' /'”’~)
LC - t. J ; -- - 7—; L _~ u , ‘ " .
I L. C - ’
Proof. Throughout this proof for convenience set Fl - F'hg) and
.6 . 4 .
f, - f‘( ...) . Using Teylor s expansion about 4 .
"II; P — T] 1 F ’k \l . i
Ft / N/ :- ' 4! ‘ r “” r0 ‘4 ;’ ’7 + D, _r I .« H m
I, A - -'- f] / J /
.. . , ,
C L , 3,
“\4 l __ / ... i I. l .
I - 5—... {-3. 'I l J — J - ‘ Y :q’.T‘.. - A ' 9
J J ft: , .— f "
and substituting these in (h.2.l) we get
.1 ‘ji '
l I ' j I! ‘ ‘ I r \
M > 1/ t’r‘ . - F ’7’ :"J/ I
m -' .--__.._.._, ’ ' w ,
(“'3'2) f(ul’ '2’ H" c’ w) ‘) k, i , n.27.. ‘ c. “‘7?“ < /
) FC, ‘1‘ I, ‘ *:T r I
‘\ ‘ ‘ / 2 1.1- m," -
, ..
x k .
. I a / .'
2 / h d ’ ‘ .5; r R . ‘1
*9 if; , e ‘__Y ‘ P ‘ ~‘ /
m) .‘- I“; I ‘ :3; w ’i f : I... '“"/"‘ z- 1': - ' I i, ‘
'. ’ — J I}. [1" '5 L. ’ I l ( (- " i/

f t I

l k . I" C . ' a

f '5‘ J . ' I .‘f “' '

1' «...-an— t ' ’ ‘y

" '3' I ' i

0
c ‘ .2
‘I

liote that v' satisfy the relation

C

(4.3.3) XVI J;;;' I 0 .

lion consider the region 5 defined by
'1’ O
S e .3 (v', ..., vc-l’ 'l)- a' 5v. 5b., a2 _<_v2 sz , ...., .c 51.5%? .
L. . _/
Using Stirling's approxieation for n!

F (J ... ‘
' War
A2 is independent of n and because of convergence of eultinouial dis-

tribution to eultivariate normal distribution, uniformly in S

g.

E C" 2‘2""
" f‘am (lficfiHIO-HF) /

2.

F H

L“!

l

i

. I 'T" \ (I ' I
3’10“ .31 "Lfij-r‘l- / ”92' ”8'55 1

1

 

 

 

 

j 3“] ‘ ft}: £2]; ¥ PC]:
Nouconslder
c
log - (v "of + Hafllos HILwE; o 27:)1
A3 32.1 i ll ii ( W7 F; + (N)

+ k logil- firél [96.8. - 4-2127

Using series expansion for log (I + x) it follows that, uniforuly in
S

logA3 --_..LZ ..-... +42 [Hey-re 20712150" /+o(1).

Using the continuity of f' we amt, uniformly in s

f(m,)m1)~-m cw) N N(Z/’F F')[kfl'(:fr) (9.}: F)I’J(1k/°Fj'/LX

(:0

exf) —J—[__‘}: (”N") +V{Z~:t F": “(éfllor )j

+: Vivi—Liéfi -— 237:2" {{oéfi )7: ::(P¢’i)/“fl

”42‘ c

Hence as in Theore- 2.2.l it follows that

P 5v Sb', azgvngz,...,ac_<_u_<_bc

5. b L
_ 2. C.
. ‘ f I ‘ f 78(7)“ 7‘1)" vo_“7)dv’clv2--~clvmd‘7 )
Q. 61 Go -

-33-

there f(v', .. , 'c-l’ q) is the probability density function of the

Iaultlvariate normal distribution described in the present theoreu. ||

Corollary. Under the hypothesis ll¢, the set (v', ..., 'c-l) and q
are asmtotiully independent.
The following leeaa gives the asymptotic joint distribution of y

and 1} under the hypothesis ll which specifies that

v" 0.0, v

. 31. .
gm r<y *fi),j n, 2, c

c-l

 

Lenaaa h.3.2. Under the hypothesis ll" the asymtotic joint distribution

of (v', ..., vb], ll) is c-variate nonaal distribution given by

 

 

w,» ,, ‘ navy) 4“” I, ., x
77 5/2. I: RAJ/J- [372-
- C ’I f. a -
_' .’ ,' “1-. .1, ”A t 7- ” 3‘1", ’19" \‘i f“; -~ ._ . , ,1: ’23.
‘x’ 7 ,‘ -+ I, I . 1' + ‘1',“- If a I "+ _:-_/I -+‘ 4-. /'L “d~'_‘ihi J .-
-* till C. Cf‘J‘ ; ’_,

Proof. It is similar to that of tens ll.3.l. ||

llow we are In a position to obtain the lieiting distribution of 14

defined by (lo.l.l) under the hypothesis ll“ .

Theoree h.}.l. Under the hypothesis ll" the asymptotic distribution of

2H ls noncentral "k2 with ’c-l; degrees of freedoe and noncentrality

paraeater

(lo-3c“) 2. - um n X ”(a -5)!

I"
C
there 0 I 2: p0 .
J J
1“
Proof. write
‘ 4' 9’ I gr" 7“ l r'
{A ’- yr -L'P’. r/f -—P."b_;rn{»,‘,-.I,} K EL’H’b I» {5/ ’ l )C
- k .. ...-. .~_,_ '- ‘ - - ~- - — .... t. 1 J
L m- ...- “...-“h“...‘fi I j
( ‘fi ',, ." If p
/ IV ‘ lb’. :' ‘

Under the hypothesis ll" using Lena 45.3.2 it follows that the asymp-

totic joint distribution of ('l’ "2’ ..., uc) is c-variate noreal with

means "i - O'F'(‘§)[—E‘ , and covariance matrix 3 - (on) of rank
(c-l) were

a I - (l - p')I2, i- l, 2, ..., c, a” - mfg?) / 2 , i 41 - l, 2, ..., c .

Hence noting that

C

X 'j FIE] - 0 it follows that the liliting distribution of
1-!

r\

2" - 2 [Pi] '4 5E0“ " 5+} ’1 ffi-‘r ‘3 lid“) . were
’l “I r‘ "v-

1' "'1

,/ u-' s n
h . ' I,” ,

A is given byo(ll.3.ls). ll

lull Asmtotic Efficim
Let Fjiy) -F(y+ej) , then He is truevd'ien 91 -o . Uenow

find the relative asymptotic efficiency of the c-samie median test with
respect to the corresponding parautric test, when F10 .- l, 2, ..., c)
is a noruel distribution with eean OJ and variance oz ; The hypothesis
ac is true if and only if fax" ..., "c’ - o , (0iitin, Tate [15])
were fvix', ..., Xe, is the nultiple correlation coefficient between
Y and X . Let it denote the sanple nuitipie correlation coefficient

betweenYandX.”

(7:. :- ( Z ”10/” .7 0;. "5-” (L: %£)/fi:)j )

 

 

L’J
then
c
, 2.
7 “(l—U
7'" E W“ J' ')
I..r ’ 1 ". 2 C; __.__ __:-~2-_“ .
‘ Z(<c":“"-’- (Ur-M.)
J‘Ju'; Jo! Jo
Also
C " __\9. 3..
f2 _ 231—(51’5/5'1/0'
7m,- xc> 4—; c. ._ 1' 7.. 2 2
2&5)» P) :3- /o"
Jrl
C
more 5 . Z p30).
1-1

Following Fisher [l6] it is seen that under the hypothesis

0 .
ii": FJ(y) - F (x :71?) , j - l, 2, ..., c, the asynptotic distribution

-35-

of (Ii - c) T2 I (c - l) is XCE'OJ) distribution where the non-
centrality parameter is given by
c
. X - z 2
it - pj(0j - O) I g . Also it is proved (Theoreie h.3.l) that
fill.

the limiting distribution of an is 'x 2

c_'().) , where noncentrality

parameter i. is given by

x . ZIF'Q )]2 E: '1“) - 5)’ . Following Andrews [iii], liaiwien [l7],
j-i
since the two test statistics are asymptotically distributed as a non-
central 7.2 variate with the same number of degrees of freedow, the
asymptotic relative efficiency is given by the ratio of the two non-
centrality parameters, i.e. the efficiency is found to be
sin, a) - 2 o’ir'wjii’ - mr

‘ioS Case Hien p', p2, ..., 'c Are "alum.

in this case we estimete '1 by $1 - njlii , j- l, 2, ..., c
and consider the test based on ii defined by (1.4.2). it is interest-
ing to note that the test of He based on ii is asymptotically distri-

bution free.

Theorem u.5.i. Under the hypothesis iic, «8 is ssynptoiiosiiy distri-
buted as a "L 2 variable with c-l degrees of freedom.

I. .
PM'e 'r‘ t. ‘; ’ll 2!],

i. L- - ’ 7 M
v. - WLE': .. .— ( CV ‘ ‘ ' / ":-" Ti: “”"’ 'ECH ( léi. til/fl). ' .‘I' “7‘40
J """ " J ,J

...— -4

\i‘ HE ‘i'r/Ff
i" i
’ ‘v

-37-

Let v- (v', v2, ..., vc) and w- (w', wz, ..., 'c) , then v-wo ,
mere 0 is a diagonal matrix with (F; /f—’§; as its diagonal
elements. Since plim $1 - p] , it follows that plim (‘5; 1/73; ) -i
and hence the matrix D converges in probability (element-wise) to
identity matrix. An application of a iemiia of Chiang [l8, Lena i]
yields that the vectors v and w have the same limiting distribution.
iioting that w) - (Wj- kpj) m1 - '4‘} "' DJ) W; s
it is seen that the asmtotlc distribution of w is c-variate normal
with zero mean vector and covariance matrix i - (on) of rank c - i
-(I-pj)flo, j-l,2' ' 'c,and oU--ffi;lu,

C

with a“

idj-l,2°°'c. liotingthat XE?) .convergesinprobe-
. J"

bility to zero as N —> m , the asymptotic distribution of

v1, v2, ..., v is given by

c-l

 

') ' ' ‘X

(mic-nu" A) (Edi/2'32

C i

- 0-!
fit: ”IL-[- 2’ V310 +kf>l £2 "’53! PzPJ'

O

 

.. i i1; (' fJ'm

-33-

 

ilence
rc-l A c-l F—T
hill .- ii 2 v: (i + 9-) + z v'vJ -—-—:"j
A M 'o isij-i 'o

 

 

has the asymptotic distribution stated in the theorem. ||

5. Rank Test for Dispersion

Let Z', 22, ..., Z", idiere Zl . “i’ Y') , be ii independent
observations from a bivarlate population. lie assi-e P{X - ii :- p ,
Hg - a} ...(i ~a); mgr I X-ii-Fjiv) . 1-0. i.
Let 0,, 02, ..., on (n > 0) be those Y observations for which the
corresponding X observations are l , and V‘, ..., vii-n be the
remaining Y observations. Let rI denote the rank of the ith
ordered ii observation in the combined sample of Li's and V's .
For testing the hypothesis ii: F, - Fo against the alternatives that
F

and F differ only ln'the scale parameter, we consider the test

l 0
based on the statistic,

which is known to be sensitive for such alternatives. ii is rejected
if ii is either too large or too small.

in Sections SJ and 5.2, the mean, and variance of H , and the
limiting distribution of Ii are obtained, when p is know, vdiiie in

Section 5.3 we deal with the case idien p in unknown.

5.l liean and Variance of H,
write ii“) -ii(li-l) ... (li-r-i-i) . Since n is a binomial random
variable b(ii, p) ,

N" .
(5 ‘ l) E[’Yl(fl(N' f]. S Nflfi‘j’N
' ' ":3: :(N-rPE-(N-n—s)

m

 

 

i
‘0'
so.

‘7
1..

First we find the mean and variance of ii under the hypothesis

ll: F, - F0 . it has been proved by iiood [S], that the conditional mo-

ments of ii for fixed n are,
E'(li|n) - n(li2-l) I i2 , o:(li|n) - n(ii-n)(ii+l)(ii2-li) I l80 .

Hence, using (5.l.l),

(5A4) spin) 4- tlgiwlnn - Mail-i) I i2 .

E[o:(H|n)] - pqii(ii2-l)(ii2-li) 1 ma .

To find 0:“) we note that o:(w) - E[o:(\i|n)] + OZIE'WInH . iience
(s.i.3) aim - muiuz-iltsuz-n / 2A0 .

Let,

”U - f [FO(Y)] i (F' (Y)]Jdr' (Y) °

40-
To obtain Epili) under the alternative note that

(5.i.ti) v - Z r?- (I'M) Zr, +4171:

and use the following results proved by Sukhatme [ 7],

\in
=0: 4

n l . n(ii-n) ii,o + [n(n+l)] I 2 ,

/

n) - 3n(li-n) Hm -Irn(ii-n)(2)ll20 + 2n“) (ii-n)”H + i- n(n+l)(‘2n+i) .

 

 

Also

Elnimlil - m + “292 + up .
and using (5sisi),

(2) p2

:[n(n+i)(2n+i)] - tizn‘” + 9n“) + 6n] - "(3),: s 9» s 6 ii p ,

(S.l.5) 5', Z til - i-: ep<iq I): - u‘z’pqn,o+-',-Im . ~52. tel .
i-l i-l
(5.l.6) ti: r,2 - £[EP (2 r? 9] - 3”,“)... 0+ "2.6)”20 .

Lit-l i-i
(2) P2

 

 

s 2p2qii(3)ii” + i- mum? + 9n + sup] .

Jil-

Aft.r us'fl’ (Se'eS), (Se'es) ‘0 ”t

n . fl
2
(5.l.7) Epiv) .50: rf) - (ii-ti) 5(2 r' + flail—spit

- £- iiiii-i)’ + ‘p ii(ii-l)(li-2)[l2 non” s 6 oz i120 - 6 qiim-t 2p 2-3,].

5.2 Asmtotic Distribution of v
lie observe that,

I" I I * Z '(VJ, u') " 2 ’(uk’ u!)
jol bi

 

4.2-

After defining

i, if u<w and v<w,

.‘W V) V, " ‘7,
f 0, otherwise ,

we can write
_ ”...,“ ”‘4“ N“)?
l; :2: I ‘i' 3; fab-LII) '7" BZ¢%7U£)+Z Y(\j')\1)Ui:>T
J21: i=1 J'fk-‘I

72: Mun Vial/)7 1:14th In UK)

Jf'kr'l k‘fi,

Observing that

n.
:1 #(‘UIUUJ 7' ((7-0)

3'!

and

e. Wt 5» mun = (emf-2) )

I
j:,*...(:‘

we can write from (5.l.li)

/ an
_'i I L’IJ#k:,
“W, N'“ 2".
+ £2 \ 14‘5” U ,g-U)+ 7 7 5/ ’7 V + 413—1.: 712.511.
L'iku; ”9 l” /

iiowdefine three functions ii, K and L as
(i, if X,-0,xj-l and Y,<YJ

”(1's 21) ' j A
g 0, otherwise.

‘J

4.3-

x-o,x-Isndv<v v<v

i if X a 0
idlp zjszk)'{.’ " ,1 k- 7' k’j
0, otherwise.
i, if XI-O,Xj-I,Xk-l anle<Yk,YJ<Y
L(Z,,Z , 2k) . _
J 0, otherwise.
Clearly,
(M) .n r3;
T“:- <.‘ , ___ 7‘ l '
L A: ¢Yj; it) ‘ .../i. H [Z] 9 Z!) )
It; (:5: L)?" I
y, N-n I\/
'\')‘ 7" . . . ‘ Z / ..
;.2_Y(%Jvk7u,)‘ . K\Zj,ak,[))
“"7 1% (MM '
’r _ if N
r- r ‘; 5i“- ‘4" 1/“?
Z A f/C b ‘76) -.. /..— L \L-J ‘2]: Z, / 3
:' 373w: J21? :" 1/4"." fr-’
and hence,
DC __(1. ....(3)‘:
..Vi. _N(N___~:~i) O__ U )_ :5. U ‘
N N1— UN N, ..g
,. .1..__[2~4<n+ MIND: *3N(N+|)-£(N+l)~n(n+12:
iLN3
where 5:.) , 9:2) , 533) are ii-statistics defined by,

k,

/
f“) 1-, 4.....- j I" . j
ANN-”‘1: ’ , '
i N
...iy-j M .’ ‘,’ Z \ .
I. [ r‘\i'l {I .i'!’ . fl.“ .
C ,!?K:']
.43.“ ,
u 2: 4.--,“ --,“ )2: MIL:
/ U
N (..., ‘N' ‘2( #xrc‘l

Theoren 5.24. Let 1' - wit3 . ‘i‘he asmtotic distribution of
(T - film/«Pm is %(o, l) both under the hypothesis as usii as the

alternative.

2291' Observe that the second tern of (5.2.0 converges in probability
to “lipa— 6p -i- 3)]l2 . Dy iloeffding's theore- U3, Theore- 7.2] it
follows that the asyqtotic joint distribution of 6:" , 5:2) , 5:3)
is trivariate normal. The required result follows by an application of

a theoren of truer [9, 9. 251i]. M

5.3 Case then I is‘ (him

More we sstinsts p by 3- nlil and consider the test based on
[T - Egan/tag?) , ...... £300 and can”) are obtained fron (54.2)
and (5.i.3) ”replacing s by S and q .9 a. it is interesting

to note that this test is asmtoticaiiy distribution-free.

Theore- §.}.l. tinder thehypothesis ll: F' - F0 , the asyqtotic dis-
tribution of [1' - Egan/a3") is mio, his) .

A»

_P___roof. Since piio’) - p , the iiaiting distribution of [1 - aim/o3")
is the sense as that of ['r - tum/«(1). Also

I

T—E"(T— ”El/'7- :rvr-‘ ,--.' ‘7'}
(5.3.!) ______b_-._) : _:._;t;:.;- .. :1.— .1:M;_z-____‘/__ :_ a. — L-
“;(T) O‘TfiW 5pm)

Note that after using expressions for E3") , 6'") and 09(1) , b in

(5.3.!) can be written as

b . fiff’flL'x/t}, ’- r Er’; + C 7-:- l.) '1. C 0
CL ’7’)
where c converges in probability to zero. Note that a and bI are
jointly asyqtoticaliy nor-ally distributed with nean vector zero and co-
variance ntrix I; s. (0”) with a" o l , an - °2l - ‘22 . 5/9 .

lience the theor. follows. H

6. We Run fest ‘

As before let 2. - (X', Y.) , i . l, 2, ...,ii be ll independent
observations free a bivarlate population there X asst-es only two values
i and zero with probabilities p and i - p a q respectively; and let
r{v_<_y | x-fi orjm (j .0, i) . Let u], 02, ..., on (n>0) be
those Y observations fro which the corresponding x observations are
one, and V', ..., vii-n be the remaining Y observations. For testing
the hypothesis H: F' (y) . Fo(y) , contine the two saeples of 0's and
V's and arrange than in _the order of negnitude. here we consider the

test based on d , the total umber of runs of 0's and V's . The

4.5-

hypothesis ii is rejected if d is too snail. flood [i9] has given the
exact sewpling distribution of d under the hypothesis H when p is
known and further proved that under the hypothesis Ii , the asmtotic
distribution of [d - zupqlllzhquI - 3»)! is ”Me, i) . These
results are obtained by other authors, see, for example, wishart and

 

Hirshfeld [20], lyer [2i].

Here we consider the case than p is unknown. Estinte p by its
usual estiieetor ’5 - hill and consider the test based on
[d - mfifi/[zf‘mfil . it is proved in the following theoren
that this test is not asynptotically distribution-free in that the
limiting distribution of the statistic depends on p .

Theoreie 6.l. Under the hypothesis ii: Fl :- F the‘asyqtotic distribu-

.—'—-—-—. ...-...!“

tion of (d - Mini/7:30 433) l is none! with eean zero and
variance l - (i - 2p)2/(l - 3pq) .

Proof. As in Theoree 3.6.] the asymptotic distribution of

.. ...- ‘0 *‘t’v‘ ...-g

(d - mull-3&0 - 3’p2i) is the sees as that of '.,(d - Inga/2] lipq(l - 3N) .
sum “u?- m- «9- pm - 2.) - i3 - n2 ......m.

.A _: ___/. _ ) ..., "7.
JI *- NI; ‘3 OI "' f F“. __ II‘I‘ I5,” (I...I.’- + -.'.f:«'..'_fi:.P:'_--

r..- ”O... In... ...—-....-

 

 

(6.0

 

MWFM NEW 1-3!» 12 We: 5162711 Fi'TI’3FiE
it can be shown that the asymptotic joint distribution of the first two
terns in the above expression is %(O,z) with covariance natrix

2 A
2. (0.1) “r. 0".', 0'2.¢2'.azz. (I ’2') ’(l -3") ° A.”
noting that the 3rd tern in (6.0 converges in probability to zero the

rCquired theoreiii follows. [I

 

{Ill .Il

-u7-

Part ii

ASYlI’TOTiC mm or moirito caNiEn-sniaiiov
TEST STATISTICS

7. introduction

Let X', ..., x“ be n independent observations (randon variables)
fros- a population with continuous distribution function 60:) . For
testing the hypothesis no: 600‘ - flu) where F‘x) is some specified
distribution function the following test was proposed by era-3r II],
Sale [2] and Von llises [3). The test statistic .0: is defined as

.: . n f Unix) - rixn’dtixi .

idiere Fn(x) denotes the empirical distribution function of the sonpio
i.e. Fn(x) - v/n , v being the number of XI (i a i, 2, ..., n) that
are less then x , - oo < x < + m ; and the hypothesis H0 is rejected
for large values of is: . Properties of this test have been studied by
various authors. Creeér in [Is] suggested the idea of. extending the
theory of on: test to the case then the distribution function 7(a) is
not completely specified, but depends on certain parameters that eust be
estiaated fro. the sample. This extension was investigated by Darling
[5] in the case when F(x) depends on one paraneter. he considered the
following problem. Let I.'be an open interval on the‘real line it.

and assume that for every point 0 e I , F(x, 0) is a distribution

4.8-

function. For testing the hypothesis H': 600 - Hat, 0) , share the
functional foria of F is known but the para-liter. O is unknown, the
aodified a: criterion is defined as

+ .-. :1
“L” '7

v' , t " r" "‘
L:;‘x;— Fuji/r 22 (z. '9”)

.4 J

(i
r rm

“d

“h.
’

9

~”Y‘I

1 \

0.

there 8" is an estiisate of 0 obtained froe the ample. The hypothesis
llI is rejected for large values of of“ . Under certain regularity
conditions the asywptotlc distribution of cf" is obtained in [5]. The
limiting distribution depends on tbs properties-of the sstinstor 3“ .
llow assures that I is an open set in 82 , the two dleensional
Euclidean space, and for every point 9 - (0‘, 02) t I , F(x, 0) is a
distribution function. Let .3" - (3h, '62") be an sstinsts of o .
For testing the hypothesis ll: 6(a) - F(x, 0) for some unspecified

0 c I consider the test based on the statistic
+00

’- A
C: :: My [aCX)'F(X)§n2J dFCx) 9”) '

The hypothesis ii is rejected if c: is sufficiently large. liac,

Kiefer and uoifouitz [6] considered the modified Craaer-Snirnov tsst
based on c: ubsn rot, o) is a noraal distribution ii(x,fiu, o2) there
botb tbs aean u and tbs variance o2 are uniutowt. Using tbs sapis
noon and tbs ssnpis variance as sstisstss of u and or2 they derived
the asywtoti'c distribution of the test criterion. The aethods used in
the derivation do not seen to be general enough to obtain the lialting

distribution when F(x, 0) is any arbitrary distribution function.

~49-

The object of this paper is to investigate the liiniting distribution

of (:2 when “x, O) is an arbitrary distribution function involving

n
two unhioiin parameters and satisfying certain regularity conditions. As
in one parameter case it will be seen that the asymptotic distribution

2

of c"

.A A
does depend on the properties of the estinators 0'", O The

2n .
’ limiting distribution of c: is derived by suitably co-bining tbs
techniques of Darling and those of Kac, Kiefer and Holfowitz [6].

we also study the aodiflcation of ii-sample eraser-Smirnow test for
testing the hypothesis of goodness of fit. The k-sample problem is as
follows. Let n10 . i, 2, ..., k) be fixsdpositius integers; and
x1," - l, 2, ..., n :j- i, ..., k) be independent random variables
having unknom continuous distribution functions 610:) . Let I be an
open interval in ii’ so tbst for every 0 c I , F(x, 9) is a distribu-
tion function. For testing tbs hypothesis no: c,(x) - 62fo - -
Gk“) - Hit, 00) , for some specified 90 5'1 , Kiefer [7] has considered

various tests particularly lt-sample Crame’r-Sieirnov test. The test

statistic is defined as

. +90 k . 9_
n: = j ZuJ-[Is;<’>c) - Fosbfldeseo),
-00 J3!
were n stands for the vector n - (n', ..., nk) , and fink) is
the eqirical distribution function of the jth sample, that is
thx) c (ilnj) [ninber of X“ (x , i- l, 2, ..., "jl . The hypoth-
esis is rejected for large values of u'nz. Kiefer has obtained the

liniting distribution of is".2 under the hypothesis "0 and has also

tabulated it.

in this paper we consider the problem of testing the hypothesis
ilk: Gl (x) - .... - Gk(x) - F(x, 0) , when the functional form of F is
loiown but 6 e I is unknown. To test the hypothesis ilk, the k-sample

Crama’r-Smirnov test statistic is modified as
J C‘J k fl.) - 2' ,i ‘

r-— " ' £ i—I / I l,
. 1 “- ’ /./ if”; ;/l {I I.“ j

.1 r A
.‘i
.l | L. L. . )
/~ J ,!e

l I
\
i

\
“s

where ll - E n], and 0' is an estimate of O obtainedbypooling

together all the k saples. Thehypothesis ilk

is sufficiently large. Under certain regularity condition tbs limiting

is rejected if i2".2

distribution of cf is obtained when tbs hypothesis iik is trus. As
in the case of one sample problem the asymptotic distribution depends on
the properties of the estimator 3,, . These results can be extended to
the case when the distribution function F involves two parameters
0', 02 by using methods similar to those employed in one sanle problem.
in Sections 8 and 9 we investigate the limiting distribution of the
modified Crame’r-Smlrnov test statistic c: under the hypothesis ii in
the case of one sample problem. Section 8 gives the asyIptotlc distribu-

A A
tion of c: when the estimators 0', 92 are superefficient. in

A

Section 9 the asymptotic distribution of c: is derived-when 3 O

l’ 2
are jointly efficient in the sense of Cramer [is]. The characteristic
function of the limiting distribution is the Fredholm determinant of a

synetric positive definite kernel of a particular form. Theorems

q.5.l and 9.5.2 give methods of obtaining the Fredholm determinant as-

-5]-

soclated with such kernels. in Section C1.6 we study some properties of
(:3 test and consider some consequences of the theory developed. in .

Section lo we study the k-sample Craér-Smirnov test in parametric case
and investigate its asyiiptotlc distribution when the hypothesis i-lk is

true.

8. The Crame’r-Smirnov Test in the Two-Parameter Case.

8.l Let X', x2, ..., x“ be n independent observations from a con-

tinuous distribution function 6(x) . Assmee that for every point

2

O m (0', Oz) belonging to an open interval I in il F(x, 0) is an

absolutely continuous distribution function. For testing the hypothesis
ll: 6(x) - F(x, 0) where the functional form of l-' is known but 0 is
unspecified, the modified Cramer-Smlrnov test criterion is defined as

400 N 1.. , A
(8.l.l) C: 2:: WI [fiCx)-F(X)9n2j ”IF(XJI9n2
v60

A A
vdiere 3" - (Oln’ 02") is an estimate of 0 obtained from the sample.

The hypothesis H is rejected if c: is sufficiently large.

in the present section we consider the problem of finding the

2

A
asysmtotic distribution of c", when ’5'”, 02“ are superefficient esti-

A

mators and also discuss the case then 3'", 02" are regular estimators.

A
8.2 Case when 9' and 02 are Sugrefficient Estimators.
as 0‘ and O

A
in 2n as 92 .
Suppose that the hypothesis ii is true. Let 0 denote the true unknown

Henceforth for simplicity we write 0

parameter vector, and f(x, 0) be the probability density function cor-
2

 

 

responding to F(x, 0) . (0:0 is defined as
__ x - My '5) chi’xbv) .
(3.2.1) - ”£3 (”I F "I :I
Let XI, x2, ..., x; be a rearrangement of the sample x" X2, ..., x“
sothat XI'<X5<. . .<x; . Then is: .Mfc: canbewritten as,
see [ii]
1 ZL-l)
(8e2e2) can :2. ’7." 'l' Z. i F( X: )9)- (
2*" izi
'fl 0

'2 ~— I A {Li 47
8a2e "‘ .J——- 1; f(x ° C. —- ___..._..._.-
( 3) CH -- ‘1.“ ’I’ g, t) 142 171

A
Theorem 8.2.l. Assume that 0n and F(x, 9) satisfy:

(i) [lim nE(8'-O‘)2-0, i-l,2.
n->a>

(ii) for 0,0'eI ,

|F(x, e) - F(x, o')| < not) 5(0, 9') ,

there 6(0, 0') s- [(p' - 9" )2 .., (02 5’21l/2

for some A0 < o, where probability is according to the true distribution

, and r (A200 > no) - 0

fix, 0) . Then 62-w2-l-6 , where plim b -0.
n n n n
n->oo

-53 -

Loaf. This theorem is a direct analogue of Theorem 2.l of [S] and can

be proved in a simller manner. H

Remerk. bhen conditions (i) and (ii) of Theorem 8.2.l are satisfied,

2 2

the asymptotic distribution of C" and bin era the same.

A A
8.3 Case when 0', 02 are Regular Estimators.
in general, condition (i) of Theorem 8.2.l is not setisfied, so now

we consider the case of regular estimation, Cram‘r (ii, p. ”Sis where

A
Var(0‘) Z A‘ln , (i - l, 2) for some positive A, . in many cases the

estimates 3‘ ere such that plim n"2 - 5(3‘ - 0') - O for some

n->m

8 such that '§>8 > O . The following lane td'iich is a direct extension

of Leena 3.i of [5] treats such cases.

Leimea8.2.l. If
(i) for %>8>0 plim n"2-b(8‘-8')-0, i-i,2;

n->a>

. for almost all x

 

III) )5 PU 9)/ < me)

(iii) if.-. ..._r( '1 22/
< ”mnbc)

where the functions m'(x), m2(x), m'2(x), h'(x), h2(x) are square

integrable, independent of O and, do not depend on the exceptional set.
Then,

(3.3-0 C: '2 C“ -r 8,,

where

and plim 8n-0.
n—>@

Proof. Expand l-‘(x, O) and f(x, 0) in a Taylor's series about the

true value 0:

* 7.
F013“): F0992I'2. C§2”9£)% FIX)”
L3, 0 a

l

)I.

9" A A ’\ ’
+ J; Z é9.-wbwwu%-&/)€9.“9._>2,m.€‘

d /
' L31

wisrs lq,i < l . Iqul <2" 3

- A ‘5’ “L .A ‘ . ~ I

i :— +m9> +'Z_(9;“92)45Im IA :41 . ~ u -

’5‘?

Substitution of tbsss expressions in (8.l.l) yields

(8.3.3) C::n n; [FOO- F(x 9)- (2(5 mag. F(x)9jf(xe):1x
" I- A A A , :
+14ij [ ( 9; -9; ) ~71 MCCX HMS, *6,)( 9; 9;) 77:21 4127;429:961 2c

400

l A
-715[QX2*F099)'L(9£~QZ)LF(X)9) 1X
”‘0 2:: 38;
2- A 2. A A- ‘ , ‘
[gag—9.?)Z.M1.Lic)+2(9’—9,)(9L 91)? 3"": 269504,,

+90

.. _ 9- A
+ to; g[fh(x)-FL299)~gag-903%,:(IJ9) _

l

O' l

L(Z(9‘”3)qhomLx)-ra£9, ~9, 2(991)g/m§}).-

’«Or'.

[2L9 9 )A; A inj flick:

-55-

Using tha asst-lotions (l) - (iv) and that sup nIFnbt) - Hat, 0)] is
x

mm in probability, Kolaogorov [8], no find that aach tam axcapt
tho first one in (8.3.3) tands in probability to zaro as n -—> o) .
iionca tha lama follows. ||

Thus by Lana 8.3.l tha probiaa of finding tha asmtotic distribu-
tion of c: is aquivaiant to finding that of (2:2 . '

iiou considar sou-a transformations fluid: are basic in the following

mrka “t

(8.3.15) u - F(x, 0) , u

J-F("°) j-l,2,...,n.

By this transformation 3: is dafinad iqiicitiy as a function of u and
O , axcopt possibly at a dam-arabia sat of valuas of u , at which at
can ha dafinad arbitrarily so as to nab tha function aonotona non-

dacraasing. Dafina
'8 ‘- ‘ -... r i f f. R 7' " 1,. C. ' '2.
(8.3.5) .2 ~22 —< x ’ ”<2- / ~2 ~ ' -- ’ 2 ‘ b

and tha function ”(x) as

i if x<t
”(3‘) - 1< " ,

0’ if XZt a

Then as can write
Ex 2 “'1

whara u's are defined by (8.3.1.) .

if 6n(u) - (iln)(nu¢er of uj's iessthan u) and

F 1 iii" .. /i. .9
(8.3.6) Zn”? -" f7: . 44“) “:1 {7‘ LL 7;” 4:: FAMJ) Ll
then using (8.3.i), c: can be written as
(8-3-7) v” :: 9 7,104) H + Y) J

there

(8.3.8) ‘ V r ("”5” " 25(97‘95 5;“) :

. Y) ‘ i

and piin an-O.
n->m

The liaiting forn of the stochastic process Yum) defined by
(8.3.8), required to obtain the asymtotic distribution: of c" ’ is given
by the Lama 8.3.2 below. This is an extension of La-a 3.2 of [5] to .
the present case. Also note that La-Ia 3.2 of .[5] is proved under sona-
‘ that different conditions than those of the following lama. For the
tine being consider the one paraneter case studied by Darling. After
writing E Zn(u) (“an - O) in a suitable for. Darling arrived at the
following two conditions. (Conditions (to) and (6) of Lenn- 3.2 of [5]).

l) lia n “6;. - O) :- 0 , i.e. 3 is "weakly unbiased”.
n --> o n

‘51

-53-

r‘ A 2 ‘
2) iim nu£fi>(9n-o)r(x',e)<u,¢ -h(u),0<u<i, and
n—>oo - - 2

h(0) . h(i) II 0 .
instead of assuming the above two conditions for each ofthe estimators
3‘ we make assumtion (iv) of the following lea-e. There is an exaapie
of a distribution function F(x, 9) for which 3 is not weakly un-

biased but at the same tint lim E Zn(u) r5 (5 - o) - h(u) exists
n -> m

and has the required properties. it will be seen in Section 9.6 that

for the normal distribution ii(x, u, 02) the estimate

n

2 -2 2 ' ‘

s - (l/n) 2 (xi - x) for o is not weakly unbiased but at the
i-i

same time iim E Zn(u) in (s2 - 02) exists.
n -> m

Lennie 8.3.2. if

i .
(5., '2. ..
(i) L.» - £1900“ +8“ , where “:ng an - 0 , (i.e. we make
the assunptions (i) - (iv) of Lena 8.3.l)

(ii) in (6‘, - 9,) is a sum of independently and identically distributed

random variables,

(iii) the asmtotic joint distribution of (in (6' - 0') , n (82 - 02))

is normal with mean zero and nonsinguiar covariance matrix 2 - (on) ,

-59..

(W) "Li-ngZnhfrn (8‘ - 0') - h‘(u),0<u<l,h‘(0)-hi(l)-O,

i I i, 2 ,
Then Y"(u) converges in distribution to a Gaussian process Y(u) , with

mean zero and covariance function [(u, v) given by

(8.3.9) 9(u, V) - mini". V) i av ° 91(U)h,(V) - s,(V)h,(u) - 92(U)h2(V)
2

' ”(052(0) * Z oUg,(u)gJ. (v) , 0 g u, v S i .
I: 1"

Proof. The stochastic process Zn(u) converges in distribution to a

Gaussian process which has mean zero and covariance function

(8.3.i0) K(u, v) - min(u, v) - uv, 0 S u, v Si ,

see for example [9]. Under the assulption (iii) the asymptotic distribu-
2

A
tion of Z In (GI - 0,)gI (u) is normal with mean zero and variance
8" r

2 .

X i1"'g'(u)gj (v) . By multidimensional central limit theorem it foi-
‘: 1"
lows that Yn(u) given by (8.3.8) converges in distribution to a
Gaussian process with mean zero. To find the covariance. function we

have A“, V) " “Vnhlnnh”

-50-

2 .
.. £(Zn(u)2n(v)) -F<Zn(u) 2,75 (5} - 0,)9,M>
hi
2

- e (znm I; in (GI - spam)

2 [I 2 \
+ E Z“ (3. - 9.)g'(u) \( £5 (6‘ - O‘)g,(v) 2: .
i-i /\ i-i /

Under the I$fl~tim3 '(i) - (iv) as n —> m , I000" v) tends to
,0“, V) given by (8.3.9) and the lone follows. ||

9. Limiting Distribution of c: - Case of Efficient Estimators,

9.l in this Section we obtain the limiting distributions of c: defined

A

A
by (8.i.i) than the estimators O 92 are regular, jointly efficient

'1
(or asymptotically jointly efficient) in the sense defined by Cramir
[‘i, pp. 180-1095]. it will be seen in Section 9.3 that the asymptotic
- distribution of c: .is the distribution of the random variable
i
(:2 - f Y2(u) du , where Y(u) is a Gaussian process with mean zero
0

and covariance function f (u, v). defined by (9.3.i). Section 9.5 gives
two methods of finding the Fredholm determinant (F.D.) of the kernel

590:, y) which is required to obtain the characteristic function of the

-6i-‘

limiting distribution. Lastly Section 9.6 deals with some properties of

c: test and derives the results of [6] as a special case of the results

given in this section

9.2 Case 2f Efficient Estimators

Following Cram‘r if we make a transformation from (“P 32, ..=.°, x")

—'> (0', 02, €" eee, g [1-2) H h.”

\
k

‘ A
99;. 9.1...- gun-,2;

A A I“ A
, '5
if 9', 92 are regular efficient estimators than Mi“, ..., ‘J-n-Z'Oi’ 02)

is independent of 0', 02 and g is such that

,. . A e ‘ '
(902°I) 2.)... [OJ (2 L {9 ’ ' , /' '3’" L'll(6L—91)

(9.2.2)

,1 (o 7’ 18.15.42) '9’ {in-(87:91)
.. .29

A A
where it” may depend on 0', 02 but are independent of 0', 02 .

From (9.2.l), (9.2.2) differentiating each of them w.r.t. 9', 92 and

taking expectations we obtain

2.

(9.2.3) twang p23 9’29992; i “fag/999,9
’2.

k’L-i a: r; :nE(/::.9 ”1031608 9/ SE (534CX)LC.)) .

0‘
'-

.62-

iiuitiplY (9-2.” and (9.2.2) by (6' - 0') , (32 - 92) respectively

and (take expectations to obtain
A A A
(9.2.li) kHVar(O‘) + k'zcovh', 62) - l
A A ’\
(902e5) kz‘COV(°2’ 9') + k22VIr(Oz) . i e

. A A
The covariance matrix Z. - (on) of (9', 02) is nonsinguiar if and
only if

2 [E (35’ “'“LWJiQ‘”? _[’fj'fL‘x,£-2)7

Ffi: (.127: I { 71E (£((71€{X§)L V

if r2 - l , covariance matrix of 0', 02 is singular and 9' and 32

 

2+2-

ara linearly dependent. As they are unbiased estimates of 0' and 92
it follows that 9‘ is a linear function of 92 i and then we are es-
snetiaiiy in a single parameter case. So henceforth we assum that

rzsil . Nowdefine
.9. "x 9).?» [L29 '(XG
f(wijcyh ’ ’05.. Jf ))
i;- (9 (xi/x Li“'[‘*’ 2 (vaX 9)”- V2:- 9
Se ‘2’ “2' L21...- 2 2
.' 'd.“2—

(9-2-7) 0" 2 51.:

(9e2e6) Y" :

 

2.. ]
<5 "‘ '1‘ .o T: (in - (1“. U‘ (z
I C! ~—- ————'——---—. U L - I a

2’1. 1 (/_)e1) E(—3; £(""F(XJ§I.))T / [2 2.]
:2 lull a

lilth this notation from (9.2.l), (9.2.2) we have

m
'n
.A .2 s ,3 ‘2 ‘
(9.2-8) H): 97 Z-fi-«iocvfluwh 913—23—‘3-190339) )
I I "‘ y " ' . . LL
Y.) J..] (6’ y) 3:,- "‘2‘
7" m
“ 2’ Lawne>+ 37.2 2 a. m... ,)
(929) (927523 “7:1" :- ~’ ‘n J J
. e i J3, L J"! '

For efficient estieetors conditions (i) and (iv) of Lea-ea 8.3.l are
satisfied by assmptions of Cranér and we further assume that (ii) and

(iii) hold. Now let
(9.2.l0) ’37" (u) - 0.9, (u) there g'(u) is defined by (8.3.5) .

The limiting form of the process Yn(u') given by (8.3.8) is obtained in

A
the following lease when 3', 92 are efficient estieetors.

A n

Lame 2.2.l. If 0', 02 are regular, unbiased, jointly efficient esti-
netors of 0', 02 , then the process Yn(u) given by (8.3.8) has mean

zero and covariance function

(9.2.ll) [(u, v) a min(u, v) - uv - Lflhl) (f'(v) - W2“) zf2")
- riffluflsz - “30.009200,

where 901(0) are defined by (9.2.l0) and have the following preperties.
’ 2 1
U) "I j - 2 f I , I
_ [(51.0.1) Au :. I/Q»)~ ) (2.)) <9 (.,).3,~_,.,)_.(u... _,/(,_r2.).
0 0 i "I,

25291. From (9.2.8) and (9.2.9) it is seen that condition (ii) of Len-Ia
8.3.2 is satisfied. Since the asymptotic joint distribution of

JR (8' g 9') ,' {E (32 - 92) is normal "(0, Z) where the covariance
matrix 2- (an) is given by (9.2.7), the condition (iii) of Lane 8.3.2

is satisfied. Let hm(u) - E(Zn(u)fi (6“ - 9.)) . Then proceeding as

in Lemea 3.3 of [5] we can show that h;n(u) - n “do", - 9‘)|r(x', o) n a}...
- n E (3, - 0‘) . As ’3' is an unbiased estimator of 9' ,

n E (3' - 0') - 0 and hence in the present case using (9.2.8) h.'"(u).

can be written as

410,0: 71E{(_

’Yi

’Yi
2 Z (’5..-7-f{’)(,‘£ +22: :29. (“7E (X, )9)’F(X09 9): Li ,
KY)

5"...
. .‘ c J 4 .. .
j: ) 0 u- i J ’ I R J...

.3 {“01

Since x', x2, ..., x“ are independently and identically distributed,

5(3)- (oqf{X;)9)/F:’XU9): a): E- __a___./f_.,g{ >335) )
J9; ‘ J‘ {33%. v

for 1-2, 3,...,n and i-l,2. Aisoas F(X', 9)-u isacon-

dltion on xI ,

~65-

Hence
\ p ,
I . l ’ J I " C’ '.’i.‘--(,X, Q
4W2475{m7%®+qlxflw L
I” (39‘ 4 (.1 ‘2'".
Similarly,

p

halal“) - 6-13,, log f(x, 0) + (’72. ,0... log f(x, 9) .
'2. '59,). 2.5)

As h"n(ll) (i - l, 2) is independent of n we omit the subscript n .

From (8-3-5)
,;(u) . ————L—-—: .3. f(x) 9) ' ...)..- log f(x, 0) , I I I, 2, chh
givas '£(1,9j 69; as;

/ (Q I / -. 1""). I“ - . I
(9.2.12) k/WZ‘ , S'WH (“72.91:”); hgu)... ,2” gym) f U’lg‘m).

integrating (9.l.l2) and noting g,(i) . g‘(0) . 0 we get

I‘

(9'2'l3) ’i‘fl”); 5:3, “0+5; i‘fiaz‘; {in}. L5“ alufir (r "22“,.) .
I p 2 J. l

' f3'2.

l
0

Thus, condition (iv) of Lenin 8.3.2 is satisfied. Substitution of (9.2.l3)
in (8.3.9) yields (9.2.ll), which proves first part of the lame. Now

(i) and (2) follow as

1" 7
I i 7 i [alum] .1,
fig 4 jf‘l-‘J ” “p ' , --.- .. — .1”...
‘ ‘ x - 7
O / 1') ... / 2’2 / ‘ ‘6 ( lay-L
LI 2‘ , EL £5.23, (Xx-'7)

 

... . - «.....‘p‘ o

 

~‘__.‘—-—- “no—~-

'b

[g ( 7222.)) t (23.339229)

Viva—+1) , ' ll

2
9.3 Limiting Distribution of C"

l/ / E(‘::: O'LTQf/K /C[°*ip(xry
j @{U ';(’Ul)-{L( "...: _L__,,,, rm)
0

H

The following theorem proves that c: converges in distribution to

i
i:2 . f Y2(u)du , where Y(u) , O 5 u gl is a Gaussian process
0

with mean zero and covariance function f(u, v) defined by (9.2.ll).

Also note that we have not made any auxiliary assueptions on the function
(9,“) used by Darling [5, p. 9].

A
Theorem 2.3.l. If ’5', 92 are regular, unbiased jointly efficient

estimators, then

- 2 0 2
12'»: p) C: 4 X {I :: Pi [IY‘ELUOW 4x)

71-916.; J

where Y(u) is a Gaussian process with mean zero and covariance function

-67-

93-0 [(0. V) - slain. V) —- uv - £f,(U) '3'“) - (2,020!) (92M

- rif'iuilfzhi - rgfi'hflfzh) , 05.. , v51,

2:201: Note that the functions C3,“) defined by (9.2.l0) are continu-

ous and Lgl(u) c L2(O, l) , l- l, 2 . By Leena 9.2.1 the process

Yn(u) given by (8.3.8) converges in distribution to a Gaussian process

Y(u) which has mean zero and covariance function f(u, v) defined by

(9.3.l). write f(u, v) as ’P (u, v) - min(u, v) - uv - ”(u)”(fl -
-’z(u)'2(v) , where

mu) - V’LI - 22) Wu) , .... ’2‘") - 29.02) + L9,“) .

By a method similar to that used in [6, pp. l9S-l97) we can get a Kac-
Siegert representation, [l0] for Gaussian process Y(u) with mean zero
and covariance function f(u, v) and show that the sample functions of
the process Y(u) are continuous with probability one. Hence an applica-
tion of Donsker's Theorem [ll] gives the required result. H

The characteristic function of the random variable
I

2 2
C 2.: ferju

. l5 9"." bY: ”0 [9]:
o

-/2_

it") ' WJ’ )

Jill

where {“j j are the eigen values of the kernel ,0 (u, v) defined by

(9.3.l) i.e. roots of the integral equation

i
sin) - it}; fits. V) 90!) dv .

The expression on the right hand side of (9.3.2) is nothing but

[0 (2 it)]-'/2 , where Mp) denotes the Fredholm determinant (F.D.)
associated with the kernel [(u, v) . Thus to obtain the characteristic
function of the limiting distribution we have to find the F.D. of the

kernel f(u, v) . lie find this characteristic function in Section 9.5.

9.“, Case of Maximum-likelihood Estimators ‘

Assume that all the conditions 'of Cramgr (ii, pp. SOO-SOlil' are satis-
fied. These conditions imply those of the Lei-as 8.3.l and 8.3.2 except
possibly condition (iv) of the latter. lie assume that condition. Then
by argiaaents similar to those used by Darling [5, Section 5] in the case
then 3', 82 are maximum likelihood estimators, the asymptotic distribu-

tion of c: is given by Theorem 9.3.i.

9.5 Fredhoim Determinant of the Kernel f(x, 1)
This section gives two methods of finding the F.D. of positive
definite kernels of special form which enable us to get the characteristic

function of the limiting distribution of c: .

Theorem 2.5.i. Let
(9-5-1) )0 (x. y) - Kim v) - 0,00 01M - Ozix) tziy) ’. 0 5x, y 5 l ,

be a positive definite kernel, where K(x, y) is a bounded synetric,

positive definite kernel over the unit square 0 g x , y S' and

-69-

"(x) e LZ(O, i) , i - i, 2 . Let the kernel K(x, y) have sinpie
eigen values 0 <).' <x2 < and f'(x) , f2(x) 'be the corres-
ponding normalized eigen functions of K(x, y) , also let d'Ot) be
the Fredholm determinant (F.D.) associated with K(x, y) . Define

I I ‘
___ ’3 \A l ‘r (A - \ I "’ ... a‘
(9e5e2) «J. "" £ f:(x)‘€'(X)V[)(o ) [(3). "J0 f;(%1-6'/X/LI)’)J'IJZ

I
(9.5.3) Ci (3) 3. J; ‘1é',’x7:?g'/.C(X ,)

(9.5-4) [9,09 )0 - K(X. v) - ”(X) ”(1) . i - l. 2 -

16 i ~K+ i
Letfiafj } ({Xfl f) and {fjbd} (Afj (’03-) denote respectively

the eigen values (in the order of magnitude) and the corresponding nor-

malized eigen functions of the kernel f'bt, y) (fzbt, y)) ; 0:01;) ,

B] (3] 3 be defined as in (9 5. 2) with fj “plum, by fat “3»)

Also define

0(1- ‘9 1‘.
(9.5.5) 'P(A).-:. Ii): Pt’UzHRZL k?)-
j: i‘Jz/Aj ) szi’A/} J.

 

P2 *0.)(P 0.)) are obtained by replacing 51(0)) by a) *(ajf) in
92006 (1.)). Then the r. o. no.) associated with the kernel f(x, y)

is given by

(9.5.6) 00.) - mm 9.0.) rfm - MA) r20.) 92"?» .

£5991. lie prove that DO.) - d'O.) P'Ot) P: (x) . Since f(x, y) is
a positive definite‘kernel, ['(x, y) being the sum of two positive

definite kernels is also positive definite. by theorem 6.2 of [5), the
F.D. 0.0.) of the kernel f'bi, y) is 0'0.) - d‘O.) P10.) . liowwe
proceed to show that the F.D. associated with f(x, y) is 00.) - 0'0.) P: (A) .

The integral equation
i
(9.2.7) 3m) : A a) [K(x,y)"nfi(7()nfil‘1)'"f’l(‘x)"-rz(\.’)]9l\I)dy
can be written as

l
(9-5.8) 3(1) :- ->\ Cl‘fi)“i'z(9)+7‘§ fi(x,y)3(y)dy

Then we have

i-

oo . if. *
(9,5,9) 3(1) 1::- -- AC1(‘9)Z L'fiflidj 3 #2:).

Fl "V15“

)

see, [i2, p. 228]. As 9 appears on both sides of (9.5.9) it is not a
solution of (9.5.8). iluitiplying both sides of (9.5.9) by ”(ad and

integrating we obtain

00 pi"; ‘ ‘k
61(9) [H- AZ .4...— ] == 0 J i. e. Cityflifl) =0 .
J"

.L./

x31:

-71-

This implies that either 9 is such that c2(g) - 0 or >. is a zero

of P:()\) . ll'ien it If); cz(g) 4 0 , because if c2(g) - 0 , (9.5.8)
is a homogeneous equation with a non zero solution for x d i: . There-

fore, only for those values of ). , which are either zeros of 5:0.) or
are eigen valuesof the kernel ,0'(x, y) , the equation (9.5.8) can

have a solution i.e. it is a zero of 0,0.) 5:0.) . 3:0.) is

5’:- Al so

analytic except for possible simple poles at A. - j .
mo) 5;“) . l .

To prove that 0'0.) 7:0.) is the no. of the kernel pot, y)
we have to show that for any zero ). - i of 01008;“) there exists
a solution §(x) of the integral equation (9.5.8) such that

l
f§2(x) dx - i. In the course of the proof of Theorem 6.2 of [5] we

0

observe that the zeros of 0,0.) are either simple or double. Let i
be a zero of 0,0.) 2:0.) . lie have to consider the following three

(ii) i - xj , there A} is a simple zero of 0,0.) ; 37- o .
- ‘li‘ 1t-

(lli) x - x] ,. there A] is a double root of 0.0.) say
4: -*- 4' .x.
A I 0 .

1+1 "

*' 1’
Note that in case (ii) it is necessary that 51- 0 , because if 314 O ,

-72-

i cannot be a zero of DO.) . Similarly in case (iii) it is necessary
at 'K"

in case (i) since i is not a zero of 0'0.) , it is such that

or-
P200 - 0 . Then

(9.5.10) 5(a) -

 

 

is the solution of (9.5.8). As f(x, y) is synetric i is real.

Also since
*’ 0° *
&(A)=Z '55") >0, forreaix,>:
1:: I-A/A.

is a simple zero of 9:0.) . Thus for any )2 under case (i) §(x)
given by (9.5.l0) satisfies (9.5.8).

in case (ii) we have two subcases. (a) i is such that 0'02) . o ,
lid.) 4 0 . in this case )2 is a simple zero of 0,0.) 5:00 and
fr“) satisfies (9.5.8). (o) if mi) .. o , 3:02) - o , 0,0.) 920.)
has a double root at i uni: . in this case fjbi) and §(x) given
by (9.5.i0) are solutions of (9.5.8).

in case (iii) if i is such that 0,02) - o , and 9:0?) 4 o , i
is a double root of 0'0.) 5:0.) and fI-(x) , fit, (x) satisfy (9.5.8).
if i is a zero of 5:0.) and also 0'02) -0 then i is a triple
zero of 0'0.) 9:0.) . f:_(x) , ffi'h) and §(x) given by (9.5.l0)

are the solutions of (9.5.8).

-73-

Thus for each zero of blot) 3:0.) we obtain solutions of appro-
priate multiplicity to the equation (9.5.8). Hence 00.) - d'()t) P10») 5:0.)
is the F.D. associated with F(x, y) . liriting the equation (9.5.7) as

I
an) s -/\C,(3)1v’(x) 2. AID €(x,y)9(y)oiy )

and proceeding in the same manner as above we can show that
00.) - d'O.) P20.) info.) . This proves the theorem. ||

Even if Theorem 9.5.l gives a method of obtaining the F.D. of
f(x, y) , the method requires the laborious task of finding the eigen
values and eigen functions of two kernels, namely K(x,‘y) and f, (x, y)
or [02(x, y) . The following theorem which is a generalization of
Theorem 6.2 of Darling [5], avoids the above mentioned difficulty by
giving an expression for the F.D. of f(x, y) for which only the eigen
values and the eigen functions of. the kernel K(x, y) are needed. The

proof of the theorem was suggested by Professor Gopinath Kallianpur.

Theorem 2.5.2. Let
[0" Y) ' ‘0‘: Y) " .10" .1‘7) ' ’20‘) .2(Y)

by a positive definite kernel as described in Theorem 9.5.l. Then the
F.D. of the kernel [(x, y) is given by

(9.5.ii) viii-ammo.) .

where

(9.5.l2) A (A) -

 

 

ah) izm

 

00 00/3.
(9543) QM) = ,A Z J ,J A74: AJ'
J2! I’)‘/)\J' 2

Proof. Hrite the integral equation (9.5.7) as
i

(9.5.110 9M '3:- ‘3[Ci(9’)1"(x)+C2(9)Y{1)]+7\)thflgtpdy .
0

Then

00 00
(9.5.5) 3(1) :: -)\C'(3)ZJ~7}\J_€(x)- )t[1(3)2 i {jot}.
J's, i-A J j:

llultiply (9.5.l5) by .'(x) and 82(x) respectively and integrateto
obtain
(9.5.l6) ~c‘(g) P10.) + czigi QN - 0.

c,(9) to.) + czisl P20.) - o.

(9.5.l6) is a system of homogeneous equations in c‘(g) , c2(g) and has

-75-

a non-zero solution if and only if A0.) - 0 . if i. d A]

and cz(g) cannot be zero, because cl(g) . c2(g) - 0 implies that the_

both c. (9)

equation (9.5.lii) is homogeneous which cannot have non-trivial solution
unless d'().) - o . Therefore the equation (9.5.lli) has a solution only
when either i. is such that A0.) - 0 or k is a zero of d'().) .

To prove that 00.) - d'O.) A0.) is the F.D. of the kernel ,o(x, y)
we show that

(9.5m 4.0040.) - no.) no.) rim - i.e.) mu m.) .

it is sufficient to prove that zeros of d'().)A().) and d'Ot) P'Ot) 2:0.) -
d'().) P20.) PTXPO.) are the same. If i is a zero of d‘O.) than it is

a zero d'().)A().) , no.) no.) 3:0.) and also of no.) r20.) rife.) .
Suppose that i: is a zero of Ab.) and d'OI) d 0 . Since ADI) - 0
there exists a solution (c'(g) , c2(g)) (of (9.5.l6) so that at least

one of cI (g) l - l, 2, is not zero. without any loss of generality

assume c.(9) I 0 . From the integral equation

 

.. .. I '
an) 2:. -)C,(C})“fl~’1)+ )é/ihfimtpdy rwf Am:
00 if)!
._ ifT
9(X) :11. —}\C'(3)Z o(J. fJX) ,
J'm / " 5/27 J

Multiply this by ”(x) and integrate to obtain cl(g)l?;+(i) - O .
Since c'(g) f o , Pym . o , which implies that d'():)l’2(i)l’:*(i) . o .

Thus we have proved that if i is a zero of d'0.)A().) it is a zero of

fl
«(who)», on .
Now we prove that a zero of the right hand side of (9.5.l?) is a

zero of dl(9.)A0.) . Here the following three cases arise.

Case (i) i 4 )‘j

By Schwarz's inequality (10:) - O and hence A03 . 0 .

and i is such that P‘OI) - o , 926) . o .

Case (ii) 5.24).] , P,0:) d O , 920:) c 0 . in this case also
Schwerz's inequality yields ‘00:) - o , hence Act) - o . Similarly

when “03-0, P203910, Aha-0.

Case (iii) in. , 9,02) d o , 9201’) 4 o . Since ism] both
c'(g) , c2(g) cannot be zero. Because if c‘(g) - c2(g) - 0 equation
(9.5.lll) is homogeneous which cannot have a solution unless i - k1 .

without any loss of generality assume c2(g) d 0 . Then'from (9.5.l6),
2 - - - A
57(9)“ ' Q (M/PflAN’ZOJI - 0 . As czig) i 0 :
Q26) - P'02)P20:) , and i is a zero of AG.) .

Hence a zero of d‘0.)P20.)P‘,:f0.) is a zero of d'0.)A0.) . This

comletes the proof. H
Corollary l. lf "(3‘) . f.(x) If)?" , 02(x) - fn(x) [fin , then

00
DO) :JIII (1" ”/Aj)
:l

Hwy"
Proof. in this case Q0.) - O and P,(}.) - A. I 0.“ - k) ,

92(3) .- A" I 0.“ ch) ; hence the result follows. H

Corollag 2. The F.D. of the kernel f(x, y) defined by (9.3.l) is

 

 

 

 

:DO‘) :1. “...—”mfg A”) ) winch/C
WA"
00
00 I2.. 2. .
H'Ml—YWZWJ ‘2': ) /\ irr‘ raJ-«rajh
AU):- ’ J3] PVT” J2; "Viral.
o. ,1). ._
I; >\ i’YQ‘Z Ta). *0"); I+A (Taj'rbj)
3 ': i-A ...; . _- .
J i /TfJ F, I )i/n-iz.

l
with aj - {if (91(x)sin(1ijx)dx,
O

j‘l’2,eee

Proof. Hrite 0‘(x) :- if'bt) l - r2

then (9.3.l) reduces to

 

i
.i as I?! L? 2(it)sin(njii)dx ,

, 025x) - "3M + this) .

fix, y) - minixs Y) jxv - “Milli/l - .2005“) .

Also for the kernel K(x, y) - min(x,
xj - 11212 , fj(x) - f2 sin(l‘ij

Substitution of mi . (l - r2) aJ ,

v) -xy

’0 a 4,00 " (“it“) 15:

Bj-raj+b ,and

d'0.) - (sinri ) If): in (9.5.ll) yields the required result. ||

 

9.6 gage Properties of c: Test and Applications.
Cumulants of the limiting distributions' As in [9] it follows

that the culiulants Kj of the asymptotic distribution of c: are given
by

(9-50') itj - 2] '(J— I), ”22"!"— j.= I) 2).. r a

)
[NJ
where {u}? are eigen values of [(u, v) . On account of Mercer's

theorem [l3] it). can aisobe obtained from
J l i '
itj-Z (j—iflofilumnlu ,

where fj(u, v) is the jth interate of the kernel f(u, v) i.e.
I

F'(“s V) ' Ph’s V) : fjhl, V) ' I I? (ll, S)/(S, V)dS .
o J“!
Hence the mean and the variance of the limiting distribution are obtained

as
l
in, : f f/u/umlu
0

I 2. l i
I __ 7— __
—. Z. jaundu—j‘; (fauna/Ll if!) (flaiflfzimduj

b

-79-

I I
0 O ‘ I a.
.2.
—Hgl(v)(gIU)_:] 61“”
z.

I I
__ 2' Z
_ 41; + 1(fl—t)+lf(l-—r 4! (f’(v)@£v)(f’(u)lfl(u) dualv

z I l1 2.. ,I V
.. (HI—r )4 f (fifvflfl(u)o(udv-8’£(Iov}tg(v)jugh/I)0(qu
i) I o I

i V i i
.. slim-WSQCWJD Ugguwuolv— "Y£('“V)Sof"’g“ gyms/«0N .

when 0', 92 are both known, the Crame’r-Smirnov test based on a:
is used for testing the hypothesis 6(x) . F(x, 0) . The limiting dis-

tribution of a: is the distribution of the random variable

I
.2 . f V2(u)du , where H(u) is a Gaussian process with mean zero
0 _ .

and covariance function min(u, v) - uv . Using Kac-Siegert representa-

tion [l0] for the process H(u) , .2 can be written as

an
2 ,.
’07—; 2161/”? 2' , where 6', 62, are independently normally dis-
1-"

tributed with mean zero and variance l .

wlon o 92 are unknown the limiting distribution of c: ”the

'I

l
distribution of the random variable C2 - f Y2(u)du , lid'iere Wu)
0

is a Gaussian process with mean zero and covariance function f(u, v)

on
given by (9.3.1). c2 can be expressed as c2 - 2“} I uj) , where
1-1 -
{“13 are eigen values of /0(u, v) and 6', 62 , are independently

normally distributed with mean zero and variance i .
Note! that (fiix) 9.01) + Wzix)‘f2(y) + “9,00 £920!) + r9,(y) (hill)

is a positive definite kernel. Hence by maxim-minim property of
eigen values, [l3, ill] it follows that the weights ”"1 in C2 are not

greater than the weights il-n'zj2 in '2

. In the case when “(3‘) and
’2“) are functions of the special form as described in Corollary l of
Theorem 9.3.2, the number of terms- in the infinite product for 00.) is
reduced by 2 . This is analogous to reduction of degrees of freedom in
the usual 7&2 theory.

The cumuiants of the distribution of .2 are
a) .
1
31(0) " 214(14): :6 /n2r2) , while those of i:2 are given by
r-l

(9e6e')e 5'0“ 'I'Trzjz Z 'I'lj , ‘(0) 2 nj e

Scale and Location Parameters: A test is said to be asymptotically

parametenfree if its limiting distribution under the hypothesis is in-

 

..I it! I..tlJl

-8l-

dependent of the unknown parameters. The c: test under investigation
will be asymptotically parameter-free if f(u, v) , the covariance
function involved in the asmtotic distribution of c: , does not de-

pend on the unknown parameters 0', O in F(x, 9', 92) . The follow-

2
ing theorem shows that when OI is a location parameter and 02 a
scale parameter, [(u, v) is independent of 9', 92 and hence the c:
test is asymptotically parameter-free. ln the case when the distribution
depends on only one unknown parameter 0 , Darling has shown that if 0
is the scale parameter or the location parameter the modified Cram‘r-

Smirnov test is asymptotically parameter-free.

Theorem 2.6.l. if the distribution function F is such that
flat, 0) - "((x - 99/92) , - m < 0‘ < +m , 02 > 0 , then f (u, v)
defined by (9.3.l) is independent of 0', 92 .

Proof.
{(2,9) -.—. Mtg) ; #42214.) . Heme
+00
EGZ (harm/9)) :. j0L; [4iC/)j/A(y)}oly
”i 2, ’00

and

-82-

450
\ i 2.
E(;Qg'(cjf(x}9).§gIqu-fWXfl): jif{y[h(y)]/In(7)}dy ,
Av (”'00

j 17[A(v>)/Ny)}dy

”Cl ‘

{El {[WwJ/hlvlfdijp; “Zilk‘m/hm} djfvz.

Using these res: its

 

“is independent of 9‘, 02 .

+ co
Q'Wflg‘iv)z:{h(H.iu)Jk[Hpév)]j/("TVJ‘ ([ h’flflVMQ 0’7} ,
(fluflflhl) ={H’('u)ri"(v) l’l LHJNYJ kLH’ (“JVZO- '7‘ )j([\1hiy)j/lliu)d7’ j

rh (vihlh '(uilhih ’(vfl

rigiwqtv)‘ - (l r1 ){[Iw 62[h'(Y2]/N”}dfl[f {[thfl/hi‘ifil‘i )f/

Wf‘ihgw— 7'” (“)MH M] “H m]
i 2 (l— ~r1){[f: {FL/Wf/N‘ifidflU‘ —;{[yk(y)]7kqu) hi7! jg“, ,

 

 

 

are independent of 0‘, 92 and hence the result. ||

-83-

Case of Normal Distribution: In case F(x,.9) is a normal distri-

bution Mat, u, 02) with unknown mean u and variance oz , we estimate

is by x- (l/n) Xx. , and oz by $2:- (i/n) z (x‘ -§)2 . in
I" III

this case r , a? , a; defined by (9.l.6) and (9.l.7) are found to be

r-O, aft-oz, 0:320“. Nowwecompute (9'(u), g2“) required

to obtain f(u, v) given by (9.3.l). Let VJ II (XJ - u)lo , j .- I,

2, ..., n . Y ., Y" is a sawle of n independent observations

', 0.
from ll(y,0,l). Let Ogugl and
.4 1

"2" V , . , , _ ‘
4,”): -:-r7:— 9. ) ENFJWW’H";f‘“”47'“‘§("fl5'

Now we find h (u) , hz(u) as

hm: hm E(r7iw)7);i.:n «nuEéy’yLTMomEij

m{mut[\/IY,4J”]}¥- 44W“)-

swam“

hm)- i... E m‘ziul(s..i>]

747”

:im [mu E{(51’i>/7’£j(u)}-’ 7w 61554)]

04’9”

____' [pm $1" \l'éTll/lfl :; jCH)‘P(-j(u)) o

64"?00

4%;
hence Using of - l , oi:- 2 , (flu) - h'(u)lai , i- l, 2 we obtain
(3, (u) - mini) and igziu) - 37;, J(u) mini) , and

Fill, v) - nlniu. v) - w - MJM) iialull - Millie» NJM) .

This was obtained in [6] by quite a different method.

i0. k-Samgle Cram‘r-Smirnov Test in the Parametric Case

l0.l in Sections 8 and 9 we considered CramKr-Smirnov test for one
samle problem when the functional form of the underlying distribution
as knosel but the parameters on silich it depended were unknolel. in
this Section we propose to study the modification of k-saple Cramir-
Smirnov test in parametric case.

Let Xj‘(i - l, 2, ..., n}, j - l, 2, ..., k) be independent random
variables with continuous distribution function 610‘) . For every
0 e I , an open interval in It. . let F(x, 0) be an absolutely continuous
distribution function. For testing the hypothesis lik: 6'00 - 62(x) .
- - 0 :- Gk(x) . F(x, 0) , \dlen the functional form of F is know!
but 0 is unknown, consider the test based on the statistic

+00 k

. ' 7_ -
, (J) A A
(l0.l.i) C 2— : 5 2'5]: if”) " F”; 9N)de(1;5N))

~00 jzi

k

where li - Z llj , n- (n', n2, ..., nk) , Pg)“, is the eqiricai.
l-i

-85-

A
distribution function of Jth sample, and O is an estimate of 0

ll
obtained from the pooled sample. The hypothesis ilk is rejected if

2 is too large. The aim of this section is to find the asymptotic

0
en
distribution of cf under the hypothesis iik . l-lere also, methods
used in [5] and [6] are employed. Throughout this chapter it is asslmled
tmt m I ->m “Ch “1-9” 0 - l, 2’ eee, k) .M

iim (nJ/li) - aJ exists.
ii -> 0

lie note that the asymptotic distribution depends on the properties
A
of the estimator O" and the characteristic function of the limiting

distribution given in Section l0.ll involves aj's .

lo.2 easowlon 3.

Suppose the hypothesis ii" is true. Let 0 be the true unknown

is a Superefficient Estimator

value of the parameter and f(x, 0) be the probability density function

corresponding to F(x, 9) . k-sampie Crame/r-Smirnov test statistic

a": is defined as ,

(J') 2.
2f izmfl Fjix) - FKXfl 8)] {(1,5)41.

Let X1", x32 , ..., jxj'n be a rearrangement of the jth sample
J
sothat x' <x' <°-'<x' . Then «a;

2
X)" X12, ..., X1"J jl12< In.)

and c".2 can be written as

 

7. k k 3' , { . 2—
.i y l — '. _ 11.“!
)2, J j" {1’ J
i fin.
L: K .i , 7.
2 ..-
I "‘ I r , _ 2' _
J?" J Jfilc'ti J

A .
Theorem l0.2.l. Let 9N and F(x, 9) satisfy

(i) lim was” -o)2-o,
u«-+> oo
(ii) For a , o' e‘1;

|F(x, a) - F(x, o')| e A(x)|9 - o'| ,

where A is such that P{A2(x)>Ao} .0 forsome Ao<m , where

the probability is according to true distribution F(x, 0) .

(3") "’ "1/"..1, 1", 2, eee, k e
li—>o
Then Cézcwt'fdibu where lim bun-O.
ii—>a)

Proof. This theorem can be proved in a manner analogous to that of

Theorem 2.l of‘oarling [5]. ||

.Remark. Under the conditions of Theorem 3.2.! the limiting distribution

.2

of 0'12 is’the same as that of a"

which is given in [7].

-87-

A
l0.3 Case Hhen O" is a Regular Estimator.

A
Let 9N be a regular estimator in the sense of Cramér (to, p. 1079].

in this case Var(6~) 2 Alli for some positive A . in general even if
assumption (i) of Theorem l0.2.i may not be true, in many cases we shall
”2 ' 5(A

have forsome 6 such that l/2>6>0 lim ii Ou-9)-0.

ll -—> a)
The following lemaa enables us to write (2.": in a suitable form that

will be useful in obtaining the limiting distribution.

Lem l0.2.l. Let

(i) m- (nj/u) -aJ , j- l, 2, ..., k .
u—>ao

(ii) For l/2>8>0 lim N'lz-5(;"-O)-O.
ii—>m

For almost all x and all 9:1"

 

(iii). .9. F(7,&)l 4 (j (at)
be" 0
B 4. (x
(w) HUM“! 9' >

where 900‘) , g'(x) are integrable functions and also independent of

the exceptional set. Then

.2 g 2
('OeBa') c" - c" + 5" ,
More plim 6" - 0 , and
ii --> m

*1 400k _ (J')
I 'J’ ' a
(l0.3.2) Cfi ”—w JZWJ' LFJOU- F(I)[)— —(€=~ 559 {‘(x 5]}{ 753/} )(
'l

A /\
Proof. Expand F(x, 9") and f(x, 9“) in a Taylor's series around

the true value 9 :

A

A A ’L
FOB/5N) = F029,)?“(9N'6):§§F(729)+{wg’GMocjolfl9 ”30’“;

ll

Wxg'5)+(9~’9)’1l9,{"); MAIL, '

Substitution of these espressions in (l0.i.l) yields,

2 .
(no.3.3) Cl: ’“jffiFU ”O'H’W- (SN 6);}. Fir 9)],€(y,(9>dx
n le Jc-a 7i) 39

Ufl/flf: (SN 6) A: 9 (2)—f(7,c9)clx

.):I
k -+00 49’ A
’wa [51.(’)'F/’99)’(9N (9):: For 93(88)1A05j(x)je/10M)L
j=l J*a) .J
L flioo (j)
’ I M
+27} wa[F{’*F”>5> (6»0)31—(2,a](e &)A9 at x.
77'
J

J“

.+

-89-

1. W. -
T: (fly/[0f (DN'O)SA013:(X)3’H ) 01L

3!

.. ij’j [a'(x),lr(y)5)-{Qv»§)bBFf/Jfifl(GA/'0) o ‘90 709' L 0
J5! "a

By an application of Kolmogorov's theorem, [8] under the assumptions (i) -
(iv) of the I... it follows that all the terms except the rm: onein
(l0.3.3) converge in probability to zero and hence the result follows. I]
Lena l0.3.l reduces the problem of finding the asymptotic distri-
bution of cf to obtaining an: of cfz given by (10.3.2).
The following transformations will be used in the sections to fol-
low. _As in Section 8.3 let u :- F(x, 0) , u]. - ”x1" 0) . Define
as before 0.00 .1 if x<t , ’z‘" -o, if th . Thenwlth
probability one we have

m. ”’3‘
(J) ' J .1. " um
51.01) 2 a]: . #335,) : ’Yl‘ 2_ “fix JL)
J J Uzi J 6:,
Also set
(no.3.u) Z,n .(u): \‘71J'[F;1.(u}—M]:fil3[%o Z: Yaw}? ) —- a] j
J j J L"!
(10.3.5) 31a) -.—. Arm/,5),

00

Observe that g(u) is in general a function of o . Employing these

transformations, 6".2 can be written as

I 2” ~é / I ”' J ‘ ‘2 i S
e a :- 2. I j ‘X "3' I I: -( Ll 'i ~
(lo 3 6) Ch _- .0 _ /7. J A.
J" J
where plim bu - 0 , and

"-9.0
( l ( l "
00.3.7) v": (u).- 1n: (u) - In] (0,,~ - a) .

The following lama gives the limiting form of the stochastic process

Yin“) . He note that this iemaa is analogous to Lemaa 3.2 of [5]
J
which is proved under somewhat different assumption. For comparison of

these one may refer back to consents before Lea-a 8.3.62.

Lemaa l0.3. . Assine that

(i) cg: can be written as in 00.3.6) ,

(ii) {“3" - o) is ssy-ptoticsiiy normally distributed with
mean zero and variance 62 > o .
(m) lim E (29)“) xi." (6". - 0)) - aJh(u) , where h(u) is
. N -> oo 1
such that h(l) - h(0) - 0 , and a - iim (n Iii) .
J ii—>tn 1

Then the stochastic process 1.9)“) given by 00.3.7) converges in

 

iii-11.; ill Ill I‘f giii‘l‘ ii. I:

 

 

ai-

distribution to a Gaussian process Yj (u) which has mean zero and co-

variance function
00.3.8) f] (u, v) . min(u, v) - uv - ajg(v)h(u) - ajg(u)h(v) + aJozg(u)g(v) . '

Proof. It is known that the process 2.9)“) converges in distribution
to a Gaussian process 21 (u) idiich has mean zero and covariance function
K(u, v) - min(u, v) - uv , see [9]. From the assimtion that

lim (n Iii) -

j sj and (ii) it follows that in] (3" - a) is ssynp-
N ->O

toticaiiy 91 (0,a 12.0) Hence the process Yfijhu) converges in dis-

tribution to a Gaussian process '1‘“) which has mean zero and the as-

sumtion (iii) yields that

”Lima/(1)“) - "1- lim mEEU)“, YU)(vfl- [01(u, v)

given by (l0.3.l) and hence the result. H

To find the limiting distribution of c: the estimator 3‘ is

ii
specialized further in the next section.

l0.“ Case of Efficient Estimator

Suppose that 6‘" is an unbiased,lregular efficient estimator in
the sense of [ti]. Further we assume that Gramér's‘condltions [‘i, pp. ‘07-
1.89] are satisfied and also conditions (i) and (iii) of Lei—a l0.3.l
are fulfilled. Let

-92-

L ’“j 1’"-
7. __ '\
1511—11. {(61/61) and 0‘ ~.-. [tgggolvaffmml’]

J
(l0.li.i) ..g.1(9~'§) : 2 2 log f(xj‘, O) , and variance of

(J76; - oi) - a2 , is independent of u .
To obtain g(u)(J defined by (l0 3. 5) we proceed as in [5). Hrite

haw-5‘ z:{)u>r(9 m-tfleramw )]-— waif/o”- 6)

Since under the hypothesis iik , u“ (i - l, 2, ..., j“ , j- i, 2,

.., k) are independently identically distributed each having uniform

distribution on unit interval, hn(u) can be written es

J

A A
hn}u) snjil E {(O" - O)|uH <u} - nju E (0" - O) .

By Leama 3.3 of [5] and due to unbiesedness of 3 using (i0.li.l) um

ii
have

liars) m n] E{(€" - OHM"- U} 2'.

= i: 23.37 «mm/mi

—'
'

Since $5 log f(xj', 0) are independently identically distributed vd'ien
the hypothesis ilk is true and “ii - u is a condition on X” ,

d ... 1M ( i9 .
0mm” ‘&§)-'°'_L.jabjf7))
From 00.3.5) and 00.4.2)

9'“) - 4L— .3 f(x, 0)- 39 log f(x, 9) . ...- 5 2A "2(a)
-f/7750 ‘33: ”D /
Hilch gives after integration end noting 9(0) . g(l) . 0 , that

N 11; ohnfU) - o zajghi).

Hence

(1)
, I H
’01“ v) “j ; fnj (u, v)

.- lim [E(ZU)(u)ZU)(v) - g(v)i'ia (u) - g(u)hn (v)

iiJ --->0 "1

+ 9(u)9(V) E njibN - Olzl
- min(u, v) - uv - ozajg(u)g(v) .

Thus in the case idien ’0‘" is on efficient estimator Lei-ea l0.3.2 yields,

.91.-

.A
LullOAJ. If 0

ii is an efficient estimator, the stochastic process

Yin“) given by (i0.3.7) converges in distribution to a Gaussian
J .

process VJ (u) with mean zero and covariance function 4“, v) de-

fined by

(l0.li.3) @(u, v) - min(u, v) - uv - eJLf(u)<}(v) , were
00.5.16) @(u) :- og(u) .

Now we are in a position to find the asyntotic distribution of 0;: .

it is interesting to note that the characteristic function of the limit-
ing distribution of of involves the proportions (aj's) intuition the
jth population Gj is sampled. Further it might be observed thet the

2

. limiting distribution of m' obtained by Kiefer is independent of

aj's .

A
more. 'Oehe I e If on

"2;;{c'f a}- Pgi flvfmdoq} .

Jill 0

is an unbiased efficient estimator .

where VJ (u)(j . I, 2, ..., k) are mutually independent Gaussian pro-
cesses with zero means and covariance function ’0] (u, v) givenby

(l0.li.3).

Proof. Observe that (f(u) defined by (i0.li.li) is a continuous function

-95-

and (5e L2(0, I) . Let {xk} be the eigen values and §fk(u)§
the corresponding normalized eigen functions of the kernel

K(u, v) i- iain(u, v) - uv . Let

I on
2 2
(lo.li.5) aj . {gin} fJ(u)du , a 1-12. x1 :11 .

By Lena l of Kac, Kiefer, Holfowitz, as f] (u, v) is e covariance
function aj (:2 5 l . Let H(u) denote Kac-Siegert representation of e
Gaussian process with mean zero and covarience function K(u, v) . Then

proceeding as in [6], it can be verified wot

 

f W
.U- (i-aa’n '
(l0.16.6) Y1 (u) - li(u) - J a2 J (3(a) X )‘k akfli(u)fk(u)du ,
ic-l o

 

is e representation of a Gaus'sien process Yj (u) which has mean zero and
covariance function ’01“, v) given by (l0.li.3). Since the sample
functions of the process li(u) ere continuous with probability one, end
. on
by an application of Lemma 2 of [6], X akfk(u) converges uniformly to
lull
(flu) , the sample functions of Y1 (u) ere also continuous with proba-

bility one. By Donsker's theorem [ii] the required result follows. H

.95-

Now we obtain the characteristic function of the limiting distri-

bution of 032 . Let iij (t) denote the characteristic function of
i

f Yf(u)du , then the characteristic function Mt) of the esymptotic
0

distribution of 6:2 is given by

k
(lo.li.7) nit) - H iij(t) .
i"

-Let gulf} denote the eigen values of the kernel flj (u, v) . Then

Hj(t) is given by ,

' oo -l/2
2i: -l/2
H. . ’ C — -
Jit) 11' ( “If iojizmi ,

where 010.) is the F.0. essocieted with the positive definite kernel
[0} (u, v) . The F.0. d'().) of the kernel K(u, v) - min(u, v) - uv
is d.().) - (sinfi) NT. end its eigen values )‘r and eigen functions
fr(x) are i... - 11er , fr(x) - fisinfirrx) . Then by Theorem 6.2

of [5] we have

0
.2 ....
Dj(x)-M Lid-eJJLZT-Tixaj, Xihr

i. rel

end or - (2' fight) sinfnrx)dx, r- l, 2, ... Putting ). -2it

the characteristic function of the. limiting distribution is obtained
from (l0.li.7) .. The cherecteristic function depends on aj , i.e.,

the proportion in which jth population is sempled.

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T T

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3 129