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ABSTRACT

ASYMPTOTIC NORMALITY OF SIMPLE LINEAR RANDOM
RANK STATISTICS UNDER THE ALTERNATIVES

BY

Janet Tolson Eyster

In this paper we study the problem of the asymptotic normality
of random signed rank statistics under the alternatives when the score
functions, m, are bounded. When the random variables are independent
and identically distributed, W is assumed to be square integrable.
This extends the work of Koul (1970) and (1972), Sen-Ghosh (1971) and
Ghosh-Sen (1972) to m which may be discontinuous. To relax the
assumptions of differentiability on m, restrictions are placed on
the distribution functions of the random variables similar to those
used by DupaE-Héjek (1969). This work is useful in generating

bounded length confidence intervals for the regression problems.

ASYMPTOTIC NORMALITY OF SIMPLE LINEAR RANDOM
RANK STATISTICS UNDER THE ALTERNATIVES

BY

Janet Tolson Eyster

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Department of Statistics and Probability

1977

TO

George, Mark and Sandy

ii

ACKNOWLEDGEMENTS

I wish to express my sincere thanks to Professor Hira Koul
for his guidance and patience during the preparation of this dis-
sertation. His advice and encouragement given during my study at
Michigan State University are greatly appreciated.

I also wish to thank Professors D. Gilliland, R. Phillips
and J. Stapleton for reading this thesis and for their helpful
comments. Special thanks are due to Mrs. Noralee Burkhardt for her
typing of the manuscript and the patience with which she did it.

Finally, I wish to express my gratitude to the National
Institutes of Health and to the Department of Statistics and
Probability, Michigan State University for their financial support

during my stay at Michigan State University.

iii

TABLE OF CONTENTS

Chapter Page
I INTRODUCTION . . . . . . . . . . . . . . . . . . . 1

II MAIN THEOREMS . . . . . . . . . . . . . . . . . . 6
III APPLICATIONS . . . . . . . . . . . . . . . . . . . 13
IV RANDOM WEIGHTED EMPIRICAL CUMULATIVES . . . . . . 18

V PROOF OF THEOREMS 2.1 AND 2.2 . . . . . . . . . . 36

VI PROOF OF THEOREM 2.3 . . . . . . . . . . . . . . . 42
APPENDIX . . . . . . . . . . . . . . . . . . . . . 45

REFERENCES . . . . . . . . . . . . . . . . . . . . 58

iv

CHAPTER I

INTRODUCTION

We consider the problem of the asymptotic normality of simple
linear signed rank statistics S+(¢) under the alternatives based on
a random number of observations. They are called random signed rank
statistics. Corresponding theorems are also presented for random rank
statistics. In particular suppose {Xi}' i :_l, is a sequence of in-
dependent random variables with continuous distribution functions
{Fi}, i :_1, and {Nr}, r :_1, is a sequence of positive integer
valued random variables. All random variables are assumed to be de-
fined on the same probability space. Let {Ci}, i :_1, be a sequence
of real numbers. We will be investigating the asymptotic distribution
of random signed rank and random rank statistics which correspond to

a :p and are defined as

n +
Z c.a (R. )s(X ) and
1 n 1n 1

S+
n(cp)

l-‘

:5

Sn(m) El Cian(Rin)

where
(1.1) s(x) =I(x:0) -I(x:0), -°°<x<°°
+ n
.2 o = . n <
(1 ) Rm 23:1 1(lle _ [xi|),

n
(1.3) Rin - Zj=1 I(Xj :_Xi), and

(1.4) an(i) m(i/(n + 1)) or

(1.5) an(i) E<¢<uni)), i = 1.....n

where Un1 :_... i-Unn is the order statistic of a sample of size

n from a uniform distribution on (0,1). When Xi' i :_l, have continuous
distribution functions Fi, i :_l, m is assumed to be bounded and

‘flq and to satisfy some other conditions presented later. These W

may be discontinuous. When Xi’ i :_l, are independent and identically
distributed (i.i.d.), m is assumed to be square integrable and .1!

Also presented are theorems related to the asymptotic normality
of 5(m) for non—random sample sizes and an application of.the theorems
for random sample size which are of interest by themselves.

This work is motivated by a desire to use m which are dis-
continuous to generate bounded length confidence intervals for the
regression problem as presented by Koul (1970) and (1972) and Ghosh-

Sen (1972). In particular the median test can be used since
¢(t) = 1(5 < t :_1) is bounded and discontinuous at t = 5.

This work should be compared to the papers by Pyke—Shorack
(1968a) and (1968b), Koul (1970) and (1972), Sen-Ghosh (1971) and
(1972) and Ghosh-Sen (1972).

Pyke-Shorack (1968a) and (1968b) presented a Chernoff-Savage
theorem on the asymptotic normality of the two sample linear rank
statistic for non-random and random sample sizes with a broad class

of m which include W with jumps. With slightly more restrictive

score functions, Braun (1976) shows that the two sample linear rank
statistic with random sample sizes converges weakly to a Wiener pro—
cess. Koul (1970) proved the asymptotic normality of the random signed
rank statistics when {an(i)} satisfy (1.4), {ci} satisfy Noether's
condition, {Xi} are i.i.d. and m is square integrable and absolutely
continuous with derivative T. which exists a.e. and has bounded
variation on [0,1]. Koul (1972) proved the asymptotic normality of
the random rank and signed rank statistics under the alternatives

when an satisfies (1.4) and m satisfies the conditions in Koul
(1970) and has a continuous derivative m'. These m are bounded

and uniformly continuous.

Sen-Ghosh (1971) presented the asymptotic normality of func-
tions of random signed rank statistics when ci 5 l, {an(i)} satisfy
either (1.4) or (1.5), m satisfy various smoothness conditions and
{Xi}' 1.: l, is a sequence of i.i.d. random variables having an
absolutely continuous distribution function F symmetric about zero
for which both the density function f and its first derivative f'
exist and are bounded for almost all x. These W include the normal'
scores and Wilcoxon, but not the median tests.

With essentially the same assumptions on m, {an(i)} and F,
except symmetry which is not assumed, Ghosh-Sen (1972) generalized to
the linear regressionrmxhalwith {ci} satisfies the boundedness con-
dition (2.2).

The results in Sen-Ghosh (1971) are used in obtaining bounded
length confidence intervals for the median while the results in Koul

(1972) and Ghosh-Sen (1972) are useful in obtaining sequential bounded

length confidence intervals for the regression coefficient 8.

In two of our main theorems we establish the asymptotic
normality of random signed rank and rank statistics under the alter-
natives for a class of bounded m which may be discontinuous.

In particular, in Theorems 2.1 and 2.2 we assume that m is
the difference of two non—decreasing bounded functions. To relax the
usual assumptions of differentiability oncp, we need to place restrictions
on Fi similar to those used by DupaE-Hajek (1969). In view of the
example in Hajek (1968), these conditions are close to being necessary.
As an application of these theorems, we present the regression problem.

To complete the proof of the main theorems, the appendix con-
tains the fixed sample size theorems. They are presented in-a form
which relaxes the boundedness condition on m by assuming m
satisfies Hoeffding's (1973) condition. By imposing Hoeffding's con-
dition on m we are able to eliminate condition D—H(2.12) and relax
condition D-H(2.l3) on Fi in DupaE-Héjek (1969).

Theorem 2.3 presents the weak convergence of random signed rank
statistics when xi, 1 :_l, are i.i.d., an is defined by (1.5) and
w is square integrable. The proof uses techniques similar to Sen-
Ghosh (1972). The essential step in proving the uniform continuity
in probability needed to apply the Anscombe theorem (1952) is the proof
that if Fn is the o-field generated by {(s(Xi), Ri)' 1 :_i :_n}
and the distribution F of {Xi} is symmetric about zero, then
{82, Fn} is a martingale sequence and the Kolmogorov inequality for
uartingales holds. Sen-Ghosh (1971) and Ghosh-Sen (1972) used this

technique to prove their results.

In general this technique cannot be used. If F is not
symmetric about 0 or if an is defined by (1.4) instead of (1.5),
the martingale property need not hold for 8;. Similarly if an is
defined by (1.4), the martingale property need not hold for Sn.
Since 5; and Sn are sums of dependent random variables, the uni-
form continuity in probability needed to apply the Anscombe theorem
is in general difficult to prove.

To prove the main theorems for an defined by (1.4), we
utilize the so called weak convergence technique as used by earlier
authors, Pyke-Shorack (1968a) and (1968b) and Koul (1970) and (1972).
We decompose the normalized random signed rank or rank statistic into
two parts and approximate each part by a fixed sample size sum of
independent random variables.

Chapter 2 contains the main theorems. In Chapter 3 Theorems
2.1 and 2.2 are specialized to the regression problem. Also some
sufficient conditions for Theorems 2.1 and 2.2 are presented.

Chapter 4 contains preliminary results for the empirical process
which are necessary for the proofs that follow. Chapters 5 and 6
contain the proofs of the main theorems. The asymptotic normality
of the non-random signed rank and rank statistics is presented in

the Appendix.

CHAPTER 2

MAIN THEOREMS

Suppose {Xi}, {Fi}' {Nr} and {ci} are defined as before.
About {Nr} we assume the following:
(2.1) there exist sequences of positive integers {ar} and {br}
such that ar + w, br + m, ar/br + l and if

A = [a :_N

r r :.b 1: then P[Ar] + l; or equivalently there

r r
exists an increasing sequence of positive integers {nr} such
that Nr/nr + l in probability as r tends to infinity.

For Theorems 2.1 and 2.2 we will assume that {Ci} satisfy

the boundedness condition for all n

(2.2) k < k < m where 02 = Zn c? and o-lng max lc.| = k .
n-— c n l i n . n
ljiin
. 2 -2 2 -2 .
Slnce 1 > o o > 1 - (b - a ) max |c Io , (2.1) and (2.2) imply
- a b -— r . b .
r r l<1<b r
--r
(2.3) 0 /o + l .
a b
r r
(2.1) and (2.3) imply
(2.4) ONr/Obr + l w.p. l and

0a /0N + l w.p. l .

For Theorem 2.3 we will only assume that {Ci} satisfy (2.3)

and Noether's condition

(2.5) max Ic.|o + 0 as n + w.
. 1 n
liiin

Define for any n

+ +
S (t) Zn c,I(R, < (n + l)t)s(X.), 0 < t < l,
n l 1 1‘- 1 -— -

and

| A

S (t) Zn c.I(R. (n+l)t), O<t< 1,
n 1 1 1 -— —

+
where s(xi), Ri and Ri are defined by (1.1), (1.2) and (1.3)

respectively. One observes that

+ 1 +
sn(¢) IO ¢(t)dsn(t)

and

l
sn(¢> f0 w<t>dsn(t) .

This representation makes it clear that one needs to study the
. .. +
standardized weighted empirical processes {SN (t), 0 :.t :_l} and

r

{SN (t), O :_t 5.1} in the cases in which m is bounded. This is
r

done in Chapter 4.
In the remainder of this chapter we will state our main
theorems for random signed rank and random rank statistics.

In Theorems 2.1 and 2.2 we assume

(am v=¢1-%

where (Pm is \ bounded on (0,1), m = 1,2.

For signed rank statistics we define for 0 :_x < +m and

0 i.t.i l,
-l n
K (x) = n z I{[x.| < x},
l 1 '—
-1 n
K (x) = n X [F.(X) - F.(-X)],
l 1 1
+ _ -l
(2.7) Linl(t) — Fi(§n (t)) - Fi(0)’
+ _ _ _ -1
Lin2(t) — Fi(0) Fi( 55 (t)).
+ + +
Lin(t) — Linl(t) + Lin2(t)'
+ + +
pin(t) — Linl(t) - Lin2(t)'
E+(t) = Zn c.uf (t). and
n l 1 1n
-+ n +
un(¢) — £1 cifm(t)dpin(t).

Obviously for any n

-1 n +
. Z L E .
(2 8) n 1 in(t) t

Theorem 2.1. Let {an(1)}, {Nr}, {Ci}, m and qh, m = 1,2, sat1sfy

 

 

+
(1.4), (2.1), (2.2) and (2.6). Assume there exist functions Rijk(s),

0 < s < l, l :_i :_j, a §_j :_b , k = 1,2, and sets Er such that

r r

vd,0<d<oo,and Vn>0
(2.9) lim mh(Er) = O, m = 1,2,
r—wo

he e E = E E
XL_E;_ r rl U r2

(2.10) E = {s; max max max sup

rl (t)
k=l,2 aréjfibr 121:) It-slidj

.5 +
—%3 lLijk

+ +
- Lijk(8) - (t-S)£ijk(5)l > n}.

 

—1 .+ .+
(2.11) E = {8; 0 a% max lc.(S) - c (S)| > n},
r2 a r . a
r a <j<b r
r—- r
and for any integer n
“+- _ -1 n + +
(2.12) cn(t) — n Zlci(2inl(t) Rin2(t)), O < t < 1.

Finally for any n define

 

 

2 _ n ‘ _ 1+ -1 _
Tn+(o) - Zi=1Var{f[(ci cn)[I(O :_Xi ju§n (t)) Linl(t)]
2 l3 - ( + A+) I -K-1 t) < X < O) - L t))]d (t)

( ° ) Ci Cn ‘ ( —n ‘ —- i in2( v

- [s(X) - + 1)] (1)}

Ci i “1‘ W "
Then
(2 14) lim inf 12 ( )o'2 > o
' aJi-cP a
r r r
implies
-1 + -+

(2.15) Ta +(cp)[SN (m) - UN (¢)] + N(0,1) r.v.

r r r D

Proof: See Chapter 5.

For linear rank statistics we define for -w < x < w,

o :_t §_1

10

H (x) = n-1 2? I{Xi :_x},
_ -l n
H (x) - n 21 Fi(x),
_l .
(2.16) Lin(t) - Fi(§n (t)), l :_1 :_n and

n
un(¢) = Zlcifw(t)dLin(t).

 

Theorem 2.2. Assume {an(i)}, {Nr}, {Ci}, m and qh, m = 1,2

satisfy (1.4), (2.1), (2.2) and (2.6) respectively. Assume there

 

 

exist functions Rij(s), 0 < s < l, l :-i :_j, a :_j i-br' and sets

 

r

Dr such that 'v d, 0 < d < w, and 'v n > 0

(2.17) lim (pm(Dr) = o, m = 1,2,
M

h =
w ere Dr Dr1 U Dr2'

D = {s; max max sup

r1 12|Li.(t) - Li.(s)
a <j<b l<i<j |t-s|<dj 3 3
r—-r __ _

-5j

- (t-S)£ij(S)I > n}.

 

(2.18) D = {5; o-lag max Ic.(s) - a (5)] > n}
r2 a r . a
r a <j<b r
r—-— r
and for any n
. _ -l n

Define for any n

 

11

2 n - -l
(2.l9) Tn(m) — Zi=1Var[f(ci - Cn)(I(Xi : an (t))

-Lin(t))d¢(t)].

2 -2 .
Then lim inf Ta (cp)oa > 0 implies
r r r

-1
Ta (<p)(SN (o) - uN (o)) + N(0,1) r.v.
r r r D

Proof: See Chapter 5.

 

Remark 1. Without loss of generality we may assume that

-l 2 n +
(2.20) n Zk=lzi=l Kink(t) — 1, 0 < t < l

+
and that gin (t) are measurable functions on (0,1). These follow

k

4.
from the fact that if there exist £in satisfying (2.9) then it is

k

- - + 8 + -% +
satisfied b 2 = -
y ink(t) n [Link(t + n ) Link(t) 1.

Remark 2. When {Xi} are i.i.d. and F symmetric about zero,
2Ijk(t) E 5 (in i,j,k and 0 :.t :_1) and for any n, én(t) E 0.
Therefore Er = ¢ and (2.9) is automatically satisfied.
Remark 3. The theorems above allow a broader class of m than Koul
(1972) by removing the absolute continuity and uniform continuity
conditions on m, but we impose the conditions (2.9) through (2.11)
on the Fi'
Remark 4. In the fixed sample size problem when T is an indicator
function (2.17) and (2.9) reduce to D-H (2.13) in DupaE-Hajek (1969)
and a similar condition in Koul-Staudte (1972a).

The following corollary allows the application of Theorem

2.2 to the two sample Chernoff-Savage problem.

Define A0 such that O < A i-AO :_1 - A < l.

12

Corollary 2.1. Theorems 2.1 and 2.2 hold when (2.2) i§_rep1aced by

 

(2.21) ci = I(i :_nr)

where nr i§_a_random variable and there exist positive integers
' and b' ch th t ' + w b' + m a' ' + l and if
{ar} { r} su a ar ) r I r/br ,__

I = I I I ° I =
Ar [ar :_nr :_br], then PEAr] + 1 and 1:m ar/ar A

0.

Proof: See Chapter 5.

We now present a theorem for random signed rank statistics
with a different centering constant when T is square integrable and

{Xi} are independent and identically distributed.

Theorem 2.3. Assume X1'°°°'xn have the continuous distribution

 

function F satisfying F(x) = 1 - F(-x). Assume W i§_square

 

integrable in_ (0,1). Assume {an(i)}, {Nr} 229_ {Ci} satisfy
(1.5), (2.1), (2.3) 111d(2.5). Then (s; - Es; )(AUN )'1 4 N(0,1)
r r

r.v. where A2 = f3¢2(t)dt.

Proof: See Chapter 6.

CHAPTER 3

APPLICATIONS

The following lemma replaces (2.11) in Theorem 2.1 by a

stronger condition on Fi but perhaps an easily verifiable one. De-

fine
+ +
E 3 = {0 < s < 1; max max max I2..k(s) - 2i k(s)| > n} .
r k=1,2 a <j<b 1<i<a 3 a
r—--r -—-r
Lemma 3.1. Define Er4 = Er1 U Er3° Condition (2.9) £§ implied by
(3.1) lim ¢m(Er4) = 0, m = 1,2,

r
whe e he 2:. grg,defined in Theorem 2.1.

Prggf: Unless otherwise specified ar, Nr and br will be
denoted by r, N and b. To prove the lemma it is sufficient to
prove E D E

r3 r2'
In the following we use the fact that for all j and r
.-l -l . . -l
(3.2) 3 = r + (j - r)(rj) .

Since 5 will be fixed in (0,1) we will simplify notation by not

revealing the s. For fixed 5 6 (0,1) and r < j :_b we have

13

14

-1 B
or r lg ql
0-1 r5 -1 + + + +
5-0 r Izr icir (Elijl “ijzj [zirl £1r23)
j . _ . -1 + _ +
+ Zlci(3 r)(rj) (Zijl Eijz)
+ zj c r-l(2+ - 2+ )l b (3 2)
r+l i ijl ij2 Y '
- + +
:_o lrk max IciIEZ max max max I2"k - lirk|
r 111:1) k=1,2 r:j_<_b 1:i_<_r 13
— . .-1 j + +
+ - +
r (J r)J 21(2ij1 £1j2)
-l r + +
' — E +
r [J l‘gijl Rij2)]]
+
< 8kC max max max (1+ - l. I + o(l) by (2.2) and (2.20).
- ijk irk

Ck: l, 2 r<j<b 1<i<r

This completes the proof that Er3 D Er2°

Remark 5. Results similar to Lemma 3.1 hold for rank statistics. In

particular (2.18) may be replaced by

D = {o < s < 1 max max |2..(s) - 2. (s)| > n} and (2.17)
r3 . . 1] 1a

a <j<b l<j<a r

r—--r --r

by lim qh(Drl U Dr3) = 0, m = 1,2.
m

Theorems 2.1 and 2.2 can be applied to the regression problem.
The proof will be carried out for Theorem 2.1 although similar results
hold for Theorem 2.2.
_ 2
Assume for all n, Xin - 80 + CinB + Ei' cin =c2/Zn c21, ci

satisfy (2.5) and 8i have a distribution function F which has a

uniformly continuous derivative f which is positive on (~w,m).

+

—+
' ' . . F ' , L.
Also assume Nr satisfies (2 l) or any n define un(¢) ink'

15

2
k = 1,2, Kh' Tn+(¢) and kn(t) by (2.7), (2.13) and
-1 n -l -l
k (t) = n 2[f(1< (t) - c. B) + f(-K (t) - c. 8)]. t e (0.1).
n l —n in —n in

Also define F*(x) = F(x) - F(-x), 0.: x < w .

2 -2 . . -l . + _+
Then Tr+(¢)or > 0 implies Ir+(¢)(SN(m) - uN(m)) + N(0,1).

The proof follows from Theorem 2.1 and Lemma 3.1 by showing
that (3.1) holds.

Fix t0 6 (0,1). For every h such that

(t - h, t + h) c (0,1) and n there exist en,gn (dependent on h,

0 O

O -1
t0 and n), < n < 9n < such that en K (tO h)

-l . _ . -1 -l ’
max CinB <<§n (tO + h) - min CinB -.gn. Since En (t) + F* (t) and

max Ici I + 0 as n + w, there exists e and g, -m < e < g < m
lfijn
such that for all n, e :_en < gn‘: g.

Since f is positive on (-m,m) and e and g are finite

there exists €2>0 dependent on t0 and h such that for all n.: 1

inf kn(t) > e.
It-t0|<h

Noting that [kn(t)]u1 represents the derivative of §;1(t), we obtain

for all n.: l

sup |d§;1(t)/dt| < 5'1 .

It-t0|<h

Therefore §;l(t), n': l, is uniformly continuous in a neighborhood

of to. The same is true of

16

+ -1
dLinl(t)/dt f(§n (t) - cin8)/kn(t)

and
dL+ (t)/dt - f( K’l(t) s)/k (t)
in2 - —n Cin n

since f, kn and §;l(t) are uniformly continuous in a neighborhood
of t0 for all n :_1.

N f 2+ — dL+ /dt k — 1 2 ' < ' < ' < b h

ow or ijk — ijk , — , , i __j, ar __j __ r' we ave
(3 3) max max max su 'IIL+ (t) - L+ (t )

' p -53 ijk ijk o

k=1,2 arjjibr 1:15j lt—tolidj

+
- (t - to)£ijk(t0)l + 0.
Since (3.3) holds for each t0 6 (0,1),
I(Erl) + 0.

Next we will show ¢(Er3) + 0 which completes the proof of

(3.1). For t0 6 (0,1) and n > 0 choose

-1 _
e > 0-3- 25[f(F* (t0)) - e] l < 5n. Then there exists rl-Bor > rl

implies

f(F;l(tO)) - 5

IA

min min f(§:l(t0) - ci.B)
rfjjb ljijr J

IA

max max f(§:l(to) - Ci'B)
rjjjb 151:; 3

f(F:l(tO)) + e.

|A

Then for r > rl

max max [2

+
(t) - £.1(t0)|
1<i<r r<j<b 1r

(3.4) i ICf(F:1(tO)) + €][f(F:l(tO)) - eJ-l

- [f(F;1(tO)) - e][f(F:1(tO)) + sJ'll < n .

+
Since similar results hold for 2. (t ), i < r, r > r

ir2 0 —- 1
and (3.4) holds for each t0 6 (0,1), ¢(Er3) + 0 and (3.1) holds
Remark 6. By Corollary 2.1 the regression problem can be specialized
to the two sample Chernoff-Savage problem with random sample size.

Pyke-Shorack (1968b) obtain similar results for a broader class of m.

CHAPTER 4

RANDOM WEIGHTED EMPIRICAL CUMULATIVES

In this chapter we prove the relative compactness of the
weighted empirical cumulatives based on independent random variables
with improper but finite distribution functions. This extends the
results of Shorack (1973) and Koul (1974).

This result is used to establish the asymptotic normality
of random weighted empirical cumulatives based on signed ranks or
ranks by establishing that they have the same asymptotic distribu-
tion as non-random weighted empirical cumulatives. The asymptotic
normality of the latter is established through approximation by
their projection on independent random variables.

Define {Yi' 1 §_i :_n} to be a sequence of random vari-

ables on [0,1] with improper but finite and continuous distribution

functions {6. ,1 < i < n}. Also define 02 = in d? and
in - —- d l 1
(4.1) w (t) = o-lznd {I(Y < t) - G (t)} O < t < 1 .
d d l i i - in ' —- -—

In Proposition 4.1 and the remainder of this work, “f“

will be the sup norm for any function f on [0,1].

 

Proposition 4.1. Assume {di} satisfy a_condition similar tg_(2.2).

 

 

 

Assume for the Gin defined above that

n
(4.2) 21 Gin(t) - nt

18

l9

i§_a_non-increasing function on_ [0,1]. Then (is > 0

 

(4.3) lim lim sup p[ sup lwd(t) - wd(s)| 3 a] = o .
6+0 n+00 It-sl:§

Proof: The proof is presented in a lemma and remark.

Lemma 4.1. Assume the conditions g£_Proposition 4.1. Then

 

(4.4) Elwd(t) - Wd(s)|4 : k:{3(t - s)2 + n'llt - sl}.

Prgof: Without loss of generality assume 0 :_s :_t < 1.
Let Ei = I(s 5-Yi :_t) - pi where pi = Gi(t) - Gi(s), i = 1,...,n.

Then using the independence of Y1 and the fact that Egi

ll
0
t
(D

have by (4.1)

4 -4 4
Elwd(t) - wd(s)| — 0d Elzi: digil
-4 4 4 2 2 2 2
= +
0d E? diEgi 3? § didngiEgj]

1 17‘]

-4 2 2 2 4 4 2 2
— 0d [301: diEii) + £1 di(E£:i — 3E (aiHJ

-4 2 2 4 2
od [3(2 dipi(1 - pi)) + 2 di(pi(l - pi)(1 - 6pi + 6pi))]

i 1
-4 4 2 4
< o [3 max Id.| (X p.) + max Idl Z P.]
- d . i . i . i . i
1§}§P i 1§}§P i
-4 4 2 — 2 -
< o max (d.|n [3(n 12 p.) + n 2 2 p.]
- d . i . i . i
liiin 1 i
4 _
§_kd[3(t - s)2 + n llt - SI]

which is (4.4). The last inequality follows because (4.2) implies

20

Z(G.(t) - G (s)) < nt - ns.
1 i -—

This completes the proof of Lemma 4.1.
Remark 7. In Koul (1974), Proposition 4.1 was proved with (4.2) re-
placed with a stronger condition. Since (4.4), which was basic to the
proof in Koul (1974), is also true under the weaker condition, Proposi-
tion 4.1 also holds under the weaker condition.

We now present an application of Proposition 4.1 which is used

in proving Theorem 4.1. Whenever possible we suppress the subscript

n used in the notation of Chapter 2 and denote En be K.

Define for 0 :_t :_l

_ -l n -l +
Vn(t) - on 21 ci[I(IXi| :_x (t))s(xi) - ui(t)],

_ 5 -1 _ _ -% n -l +
Wn(t) — n [Kn(K (t)) t]— n zl{1(|xi| i K (t)) - Li(t)},

(4.5) an(t)

0'12“ c.[I(0 < x. < K'1(t)) - LI (t))
n 1 i - i-— 11

and

-1 n —l +
Vn2(t) on 21 ci[I(-K (t) fi-Xi :_0) - Li2(t)].

Note that for 0 :_t §_l

(4.6) Vn(t) = an(t) - vn (t)

2

and when ci 5 l

(4.7) Wn(t) = Vn1(t) + Vn (t) w.p. l.

2

21

Corollary 4.1. Assume {Ci} satisfy (2.2). Then v e > 0

 

(4.8) lim lim sup pt sup |Vn(t) - Vn(s)| 3_€] = 0

5+0 n It-slfifi
and
(4.9) lim lim sup P[ sup |Wn(t) - Wn(s)| :_e] = 0 .
6+0 n It-s|:§
Proof: In Proposition 4.1 take Yi E K(Xi)I(0 :_Xi) and
d = c Then G (t) = L+ (t) nd f o (2 8) an+ (t) t -
i ‘ 1' in ‘ inl a r m ° 1 inl “ ‘
+
-2n L. (t). Hence (4.2) is satisfied for these G. '5. Moreover
l in2 in

Vn1(t) = wd(t). Hence an satisfies (4.3). Similar application of

Proposition 4.1 to Yi E K(-Xi)I(Xi < 0) and di 5 ci will yield (4.8)
in view of (4.6). Similarly by (4.7), (4.9) follows when di 5 l.
The following results which appear in Koul-Staudte (1972b)

and Koul (1972) are presented here without proof.

Lemma 4.2. The random variable { sup IW (t)I} §§_bounded in_
O<t<l

probability as_ n + m. Also

 

 

sup Ix (K;l(t)) - tl = o (1) and
OEtEI r r p
(4.10)
sup Iga (Eighth - tl = o (1).
Ofitjl r r p
In view of (4.10) and the fact that K (x) = H (x) - H (-x)
-n —n -n
we have
(4.11) sup lga (15;;th — t) = o (1) .
Ojtfil r r p

22

We now state the main theorem of the chapter. Its proof
will be presented in a series of lemmas which establish the asymptotic
normality of the random weighted empirical cumulatives based on
signed ranks by first establishing the asymptotic normality of non-
random weighted empirical cumulatives and then establishing that the
random and non-random weighted empirical cumulatives have the same
asymptotic distribution. Similar results are then presented for
random weighted empirical cumulatives based on ranks.

For signed rank statistics introduce for a fixed point

V 6 [0:1]

(4.12) T+(v) = 0'12“ c.{I(lx.| :K’1(v))s(x.) - LCM}
n n l i i n 1 i

where Kn is the empirical cumulative of {lxi|, 1 __i :_n} de-

fined by (2.7). Also define for v 6 (0,1)

-1 _
(4.13) 23w) on {2" ci[I(|Xi| 5. K 1(v))s(Xi) - 11:0»)

1
- nsc+(v)W (v)}
n n

+
where cm is defined by (2.12).

 

Theorem 4.1. Assume {an(i)}, {Nr} and {ci} satisfy (1.4), (2.1)

and (2.2) and for fixed v, 0 < v < 1, there exists numbers

 

 

+ . . .
Qijk(v), 1 :_i :_3, ar :_3 : br' k = 1,2, such that for any d,

 

0<d<°° and k=l,2,

(4.14) max sup -5|Link(t) - Link(v) - (t-v)£:nk(v)l = o(n-§)

1:i_<_n It-v lidn

and

23

(4.15) o-lafi max |&[(v) - 6+ (v)| + 0 as r + m.
a r a
r a <j<b r
r
§2£.32X n define
12 (v) = “(a 2[L:n (v) - (pf (v))2] + (6+(v))2Lf (v)[1 — LT (v)]
n+ in n in in
(4. 16)
.+ + +
— 2cicn(v)[uin(v)(l - Lin(V))]] .
Then
(4.17) lim inf T: +(V)/O: > 0
r+® ar+ ar
implies
(4 18) ‘1 T+ 4 o 1
. TN +(v)ON N (v) D N( , ) r.v.

r r r
The proof of Theorem 4.1 utilizes Lemmas 4.3 through 4.6 and

is completed after them.

+ .
Lemma 4.3. Assume {Ci} satisfy (2.2) and Zirk(v), l :_1 :_r,

k = 1,2, satisfy (4.10). Then (4.17) implies

(4.19) l(\))0r T: (V) '* N(OI1)-
D
Proof: (4.19) will follow when we show
+ +
(4.20) IT (v) - z (v)! = o (1)
r r P

-1 +
and Tr+OrZr(v) D N(0,1).

(4.21) T:(v) = vr(K(K;1(v))) + a; lEu: (K(K; 1(v))) - u (v)].

(4.8) and (4.10) imply

24

-1 _
(4.22) sup IVr(K(Kr (t))) - Vr(t)| _ op(l) .
03:31

Now since IK (K—1(t)) - t] < r-l, we have by (4.5) and (4.9)
r r -

(4.23) r5[K(K'1(t)) - t] = r5[K(K’1(t)) - K (K"1(t))] + 0(r‘5)
r r r r

-1 -a _ _
=’wr‘K‘Kr (t))) + 0(r ) - Wr(t) + op(1).

Since the sequence file‘} is bounded in probability in the

limit a cha-

(4.24) P[Br] + l

where

—1 -%

B = {sup|K(K (t)) - tl :_dr }
r t r

Combining Assumption (4.14) and the above observations yield

-1 -+ -1 -+
(4.25) or lur(K(Kr (v))) - ur(v) + rgc:(v)wr(v)l = op(1).

Finally, combining (4.21), (4.22) and (4.25), we have shown

+ - -
(Tr(v) - Vr(v) + o 1r56:(v)wr(v)l = 0p(l).

Combining (4.22) and (4.13), we note that
(4.26) v (v) - o'lr*e+(v)w (v) = Z+(v)
r r r r

which completes the proof of (4.20).

To complete the lemma, write

25
Z+(v) = o'lzr{(c - 6+)[1(o < x < K'1(v)) - L+ (v)]
r r l i r - i —- irl

.+ -1 +
- (ci + cr)[I(-K (v) fi-Xi < 0) - Lir2(v)]}

r +
leri(v) (say).

+
Then since Iér(v)| :_max|cil follows from (2.20), we have

max (2+.(v)l j'4 maxIc.|0.-l = 0(1).
ljifir 1 1 r

+
Therefore I! e > 0 and r sufficiently large, max (Z .(v)l < e
1<i<r

and the Lindeberg-Feller theorem plus (4.17) yield that T-lorz:(v)

r+
is asymptotically normal with parameters (0,1). Therefore (4.19)
follows from (4.20).
Before proceeding with Lemma 4.4, we state an inequality

which is given in Fernandez (1970).

Inequality. Let D[0,l] pg_the space p£_functions pp_ [0,1], the

 

elements pf_which are right continuous and have left limits. Let

 

{Yi} p§_seqpence p§_independent random variables ip_ D[0,l] and

c = 2?

, >
j 1:1 Yi Then ‘v s O

1

V

(4.27) p[ max [lgjl

-2€] < PU g H > e](l - n )-
1:153) _ l n n

where

D
II

max PHICP - cl|> e] .

 

Lemma 4.4. Assume {Nr} and {ci} satisfy (2.1) and (2.2). Then

Y e > O

26

(4.28) HVN - va H = op(1).
r r
(4.29) “WN - wa H = op(l)
r r
229.
30 4 *
(4. ) .thr“ — op(l)

 

where for any, n, V*(t) = V (K (K-l(t))) - V (t).
n n -n n n

Finally the sequence (“WN L} i§_bounded in probability 12_
r

 

 

Ppppf: Unless otherwise specified ar, Nr and br will be
denoted by r, N and b. First note that (4.29) and the fact that
{hwrh} is bounded in probability imply that the sequence (“WNh]
is bounded in probability. Since (4.29) can be proved by the same
methodology used to show (4.28), only the proof of (4.28) and (4.30)

will be presented here.

Define for 1 5.1 :_b
-1 -1 +
= I . . - .
Yi(t) or ciE (lxll 5-52 (t))s(x1) u1r(t)]

and

V'(t) = 2r Y.(t)

r l i

To prove (4.28) we will first show that

(4.31) “v;q - Vr“ = op(l) .

27

Note {Yi} are independent random variables and Yi E D[O,l]. Then
by (4.27), we have
. c
PLHVfi - Vr“ > zej]§_P[ max “v3 - vr“ > 2g] + P(Ar)
riJEP

(4.32)
I ‘ I I — l
< P[“Vb - Vrh > e](l - max P[[[Vb - Vj“ > 5]) + op(1) + O

r:1'_<.b
as r + m as the following argument shows. For any t and any j,

r :_j i'b, we have

I I 2
E(|Vb(t) - Vj(t>| )

-2 b 2 -1 + 2
(4.33) — or £j+l ciE[I{|Xi| i-Er (t)}s(Xi) - uir(t)]
< o-'(o2 - 02) +>O as r + w b (2 3)
- r b r y ' °

Noting that

sup [K.(K;1(t)) - tl = 0(1)

Oifiil

and
-1

V'.(t) = 0, 0V.(
‘ J J

K.(
r J ‘3

K’1(t))).r < j < b.
_.r _ _

we have by (4. 8) that

(434) PE sup I(vyt) -.v5(t)) - (vys) - vJ!(s))| > e] + o
It-s|<6

as r + w and then 6 + 0.

Combining (4. 33) and (4. 34) , we have

28
max P[“V' - V!“ > e] + 0 as r + w.
. b j
rjjjb
Therefore (4.32) and consequently (4.31) hold. To complete the proof
of (4.28)) we start by combining (4.31) with (4.8) to get

(4.35) p[ sup IV&(t) - V§(s)| > e] + o .
It-slsS

. _ -l , -1
Noting that VN(t) - CrON VN(_K_r(KN (t))), we have

(4.36) “VN - vr“

:_| sup OrON

1 -l
(v'(K (K (t))) -v'(t))l
Qifiil N —r -N N

+

or0;1“V§ - VI“ + “Vrhlorogl - 1]

+ .
R1 + R2 R3 (say)

(4.37) R3 = op(l) by (2.4) and the fact that {“vrn} are bounded

in probability.

(4.38) R2 = op(l) by (4.31) and (2.4).

+ —
Finally define D = [ sup (K (K 1(t)) - t] < 6] and note from (4.11)
r 0<t<1 -r -N

(4.39) P(D:) + 1.

Then by (4.35) and (4.39)

'1
(4.4(» P[Rl > 5].: P[ sup OrON

|v'(t) - v'(s)| > e] + p(DC ) + 0.
lt-S|<5 N N r+

Combining (4.36) through (4.4C» yields (4.28).

29

Finally, the proof of the lemma is complete when we show that
(4.30) holds. Since the sequence {thh} is bounded in probability,

there exists d such that

_ -1 -%
cr — {sip IKN(KN (t)) - tl :_dN }

and

(4.41) P[Cr] + 1 .
Then (4.30) follows by combining (4.8), (4.28) and (4.41)

lim p{nv;“ > 5}
pm

§_1im lim (P{ sup IVN(t) - VN(s)| > e} + P[CC]) = 0.
6+0 r+m It-sl<6 r

Lemma 4.5. Assume the conditions 9£_Theorem 4.1. Then aE_ v

 

 

I -1 -* (v) - 0-1 -* (v)| = o (l)
(4’42) ON uN a pa P
r r r r

where for any n
~* -+ -1 -+
= K - v).
(4.43) un(v) un(§h( n (v))) un(
Proof: Again we will write r, N, and b for ar, N and

r

+
br. We will repeatedly use the fact that since [irk 3.0, i :_r,

k = 1,2, (2.20) implies for any r 3_1

- +. - - +
(4.44) O lrkué‘: “ < o lrs max Ic.(r lhz Z 2. H < k .
r r -— r . i . irk — c
liiir i k
To prove (4.42) we will first show that
-1 5 A4' -1 g, A+ _
(4.45) loN Nr 6N (v)WN (v) - 0a ar ca (v)wa (v)! — op(1)

r r r r r r

30

In particular we will show that

-1 B + -1 5 +
.46 = “ - “ =
(4 ) R1 ION N cN(v) or r cr(v)| op(l)
which by (4.44) also implies
(4.47) lim P[o;lN5|c;(v)| i K < w] = l .

r

The proof of (4.45) will then be complete by the triangle in—

equality, (4.29) and the fact that the sequence {hWa H} is bounded
r
in probability. We now prove (4.46). By (2.1), (4.15) and (4.44)

—1 5 .+ .+
|R1| :_ON N lcN(v) - cr(v)l + op(l)

-1 5

+ + ‘

_<_orb max Ic.(v)-c(v)|+o(l)=o(1) .
a <j<b r P P
r—-— r

To prove (4.42) we first observe that (4.14) implies that
8

V's > 0 there exists r-.9. if r > rE and It - VI fi_dj— o
e

r §_j §.b, then we have

'1 j + + -l r +
lo. 2 c.[u..(t) - u.. v — o z - +
J l 1 11 13( )3 r 1 Ci[“ir(t) “ir(“)]l
(4.48)
-l..+ -1 +

< o. c. v t - v - “ _
__l J 1 J( )( ) or rcr(v)(t 6)] + 4ekc .

. -1 + +

+ 430. max Icil max max sup _ lLi'k(t) - L..k(v)
1515; k=1,2 15153 It-vljdj 3 13

' +
- (t - v)£ijk(v)l

< o'lje+(v)(t-v) - 6‘1 6+(v)(t )| +- 4th ‘
j j r r r . v c.

To proceed let Cr be defined by (4.41). Note on A H C . we have
r r
for r :.j :_b
-1 _
sup (K.(K. (t)) - tl < dj 1’
t J J '—

31

We complete the proof of (4.42) by partitioning 9 based on {Nr}, apply-

ing (4.5), (4.23) and (4.48) and recombining. Let 60 > 4kce > 0. Then

-1

P[|0N

_* _1_*
uN(v) - or ur(v)| > 360]
< b [I -l'* 'l'* . n . — 'J ]
__P[jgr( oj uj(v) - or ur(v)l > 3601;)[cr1 .[Kr — 3 )

+ P(AC) + P(Cc)
r r
b

-1.5 .+ -l 5 .+ _= --
PEjgr([loj 3 wj1v)cj(v) - or r wr(v)cr(v)| > eO]n[cr]n [Nr 31)]

| A

+ o (1)
P

. -1 8 .+ -1 %.+
PL [ION N cN(V)WN(V) - or r cr(v)wr(v)| > 60] n [ArJJ + op(1)

I A

= 1 b 4.4 .
op( ) y ( 5)

Lemma 4.6. Assume the conditions of Theorem 4.1. Then

 

 

 

-l
TN+(v)Tr+(V) + 1 w.p. 1 as r + m.

ngpf; As usual we will use r, N and b instead of ar,
N and br' Since v is fixed we will simplify notation by not
revealing v when possible.

We will repeatedly use the observations that (4.17) insures

that r;20: is bounded, that (2.8) and (4.11) yield

-1 r + + _ -1 _
(4.49) r (21(LiN(v) - Lir(v))l — | (KN (v)) - VI - op(1)

K
-r

and that

32

-1zr + +
l u1N uir

-1 r + +
(4.50) | :_r [21(LiN - Lir)|

To show that (4.50) holds assume K;l

easily shown that

L+ L+ < + + < L+
iN ir —-“iN uir —- ir
. . -l -1
Similarly when K < K
r - N
L+ _ L+ < + + < +
ir iN —-“ir “1N —- iN
Therefore
r + + r + + r
2 . - . < Z L - L, = L
lluiN “irl -' 1l ir 1Nl lzl(
Now by (2.1) and (4.16)
(451) 2()’2()—1+): 11+ (1)
. TN+ v Tr+ v — m=1 m op
where
-2 r 2 + +
R = L _ L
1 Tr+ £1Ci( iN ir)
-2 r 2 + 2 + 2
R = -
2 Tr+ Zlci((uir) (uiN) )
-2r2+ +
R: C _ _
3 Tr+ 21E N+LiN(l LiN)
-2r A++ +
= l- _"
R4 2Tr+ l 1 ruir( Lir) C
-2 N 2 + + 2
P: _
5 ”2+ Zr+1EC ( iN ( iN) ) +
+ + +
- .é - L .
2C NuiN(l iN)]

= o (l).
P
:_K; , then it is
iN'
+
L. 0
II
+ +
. — L. )|.
1r 1N
2 + +
L ..
r+ ir(l Lir)J
+ +
1 -
u N( LiN)]
2 + +
. L _
N+ iN(l LiN)

33

In View of (4.49) and (4.50)

(4.52) IR1I :_T;i mix CIIZI(L:N - L:r)I = op(l)
and
(4.53) IRZI 5,1}: max six: “Ir + “INIIuir - “:NI = o (1)
Next
|R3I :1; ((22+)?- (6:)2I|z§L:N(1 — L:N)|
+ Tri(e 6+ )2I21L+ LiN(l - L:N) - Z:L1r(l - Lir)l
= IR31I+ IR32I (say).
In view of (4.15), (4.44) and (4.49)
IR31I _ 1;: riggb Ioj - C:IIC;+¢:II21L+1NI = op(l)
and
IR32l :_3Tri(e:)2I21(L:N - Lir)l = °p(l)°
Therefore
(4.54) IR3I = o (1)

Similar techniques yield
(4.55) IR = o (1).
P

Finally in view of (4.47)

34

— + +
22“ (a? + (e )2 + 2c.|e I) + o (1)
N 1 N p

(4.56) IRSI :_rr+ r+1 1

T—2(N — r)[ max Icgl + (6+)2 + 2Ié+I max Ic.I]
r+ . N . i
1:1_<_b 1:130

IA

+0 (1) =0 (l).
p P

Combining (4.51) through (4.56) completes the proof of the lemma.
Proof 9£_Theorem 4.1. As usual we will use r, N and b
instead of a , N and b .
r r r

We prove (4.18) by using the decomposition
+ —1 -1-*
T (v) = V (K(K (v))) + 0 u (v)
n n n n n

-‘k
where un(v) is defined by (4.43). Note that from Lemmas 4.4 and

4.5 we have
+ + * *
(4.57) ITN(V) - Tr(v)I :_IVN(v) + VN(v) - Vr(v) - Vr(v)I

_ -* _ _*
+ I0 lu (v) - 0 In (v)I = o (l) . .
r r p

N N

Combining (2.4) and (4.57) with the results of Lemmas 4.3 and 4.6

completes the proof of (4.18).

35

For rank statistics we present Theorem 4.2 which is the
analog of Theorem 4.1. The proof of Theorem 4.2 is similar to the

proof of Theorem 4.1 with appropriate changes in notation. Define

-1 n -1
Tn(v) on 2:1ci{1(xi i Hn (0)) - Lin(\))}

and

T2(v) z“[(c. - e (v))2(1 - L.(v))L.(v)].
n l i n 1 i

 

Theorem 4.2. Assume {an(i)}) {Nr} apd_ {Ci} satisfy (1.4), (2.1)

 

 

 

and (2.2) and there exist numbers £ij(v), l :_i :_j, ar §_j fi-br
such that, ap_§_fixed point v, for any d, 0 < d < w
max sup _5ILin(t) - Lin(v) - (t - v)£.n(v)I = o(n“%)’
lfiifin It-indn l
and
-1 A A '-
o aL1 max Ic.(v) - c (v)I = o(a 5) as r + w .
a r . a r ——
r a <j<b r
r-- r

Then 1im inf 12 (V)/O: > 0 implies
r+°° ar

 

T;1(V)UN TN (v) D N(0,1) r.v.

r r r

CHAPTER-5

PROOF OF THEOREMS 2.1 AND 2.2

Proof of Theorem 2.1. As before we will use r, N and b

 

to represent ar, Nr and br' BY (2.6) we may assume o is \IJ-

Define for any n

+ -1 + -+ \ -
(5.1) Tn(m) = on [sn(¢) - un(¢)]_.

Since the assumptions of Theorem 2.1 are stronger than the assumptions

of Proposition A.1, we have by (A.28) and Remark 9

-1 +
rr+(v)orTr(m) B N(0,1) .

Thus to prove (2.15) it suffices, in view of (2.14), to prove
+ +
(5.2) ITN(@) - Tr(o)I — op(l) .

+
To prove (5.2) note that for all r.: l, Tr(o) =

fr/r+l

+ -
0 o(t)dT (r 1(r + l)t) w.p. l where T+(t) is defined by (4.12).

+
Since T (0) = 0 and m is bounded and \L, upon integration by

parts one has

36

37

+ e -f:/r+1T:(r-l(r + l)t)de(t) + T:(1)6(1)

(5.3)

1 + . +
—fo T (t)dm(t) + Tr(1)¢(1) + op(1) -

The latter follows from the boundedness of m and the fact that

- +
sup IT:(r 1(r + l)t) - Tr(t)I = op(l). Similar results hold for

T;(o) by (4.57).

Now by (4.28), the boundedness of m, and the fact that

+
for any n, Tn(l) = Vn(l)' we have
+ +

6(1)ITr(1) - TN(1)I — op(l).

Thus to prove (5.2) it remains to show
1 + +
- T = .
IronN r)dcpl op(1)

* .*
Defining for any n, Vn and “n by (4.30) and (4.43)

respectively and using the decomposition presented at (4.57) we have

1 + + 1 * *
IfO(TN - Tr)deI : fOIVN+(t) + VN(t) - Vr+(t) - vr(t)|dm(t)

1 -1-* —1-*
(5.4) + IIO{6N uN(t) - or ur(t)}do(t)I
= R1 + R2 (say).
(5.5) IRiI = op(l)

by (4.28), (4.30) and the boundedness of m. Next we proceed to

prove R is op(l). Since (4.44) holds for any j 3_1

2

38

-l *2..+

P[II0N N CNII _<_ kc]

PE u ([N = j] 0 [IIOleIc‘szI : x 1)] = 1.
j=l 1 J c

(5.6)

For any n, 0 < n < w, r sufficiently large and Er defined by

(2.9) we have by (2.4) and (4.44)

-1 + - +
sup ION N8&N(t) - Orlr%&r(t)l
tEE
r
- + +
< sup 0 lNIIE‘: (t) - & (t)I + o (l)
—- c N N r p
t€E
r
-1 5 .+ .+
(5.7) :_0 b max sup Ic (t) - cr(t)I + op(l) §_2n + op(l).

r<j<b tEEC

-- r

Define A and C as in (2.1) and (4.41). Let G = A n C .

r r r r r

Note

(5.8) P(G:) +0 as r+°° .

For an arbitrary e > 0, n > 0 (a function of e), we have for Er

l -l-* -l-*
IIOwN uN Or urhml

5 9 f { '1‘* ’1‘*}d + If { '1'* "1" d
< — -
( . ) __I Er ON uN or ur wI EC 0N UN or ur} ml
r
= + .
R21 R22 .(say)
We first show R21 = op(l). For any j, r :_j :_b and t 6 (0,1)

we have on Gr

39
Ifif(t)|
J

j + 1 _ + _ + -l +
Izlci(Lijl(Kj(KJ (t))) Lijl(t) Lij2(§j(Kj (t))) + Lij2(t))l

I A

Izic.(LI.(K.(K71(t))) - LI.(t))I
1 11 -3 J 11

I A

I + -1 +
max Ic.I|£3(L..(K.(K. (t))) — L..(t))|
lfiijj i l ij —j j 1]

| A

d max Icins by (2.8) and (4.41).
lfifiij

Consequently on Gr we have

R21 :_0-1 max Ic.I(db11 + dr%)m(E )
ljiib r
:_3dkcm(Er),
Therefore by (2.9) and (5.8)
(5.10) R21 = 0p(l).
Next we show R = o (1). On G n [N = j]
22 p r r
-1.5 .+ -1 t .+ ,
IRZZI : sup I0. 3 Wj(t)cj(t) - or r wr(t)cr(tflno“ + 4nkchm“
tEE
r
-1 5 .+ -1 k .+ _
i.izg l0N N WN(t)CN(t) - or r wr(t)cr(tnnmu + as .
r

The first ineqUality follows from (2.10) by an argument similar to the

proof of (4.48). Therefore since “W“ is bounded we have on G .
- r

4O

-1 H
I :_sup IO N
22 tEE: N

.+ -1 5 .+ .
WN(t)cN(t) - or r wr(t)cr(t)I“mh + 56

o;1N*ne;nan - w.“ )4“ + 4.

I A

-1 s + +
+ o N. IIW IIsup If: (t) " f: (t) I II‘PII
N r tEE: N r

= o (l).
P

- +
The first term on the right side is op(l) since oNlNkhéN“ is
bounded w.p. l and “W - W H = o (l). The second tenm is o (l)
N r p P
because “WI“ is bounded in probability in the limit and

+
sup I&;(t) - ér(t)I < n for n arbitrarily small. Consequently

c
t E
6 r

by (5.9)

(5.11) R22 = op(l).

Combining (5.5) and (5.9) through (5.11) completes the proof

1 + +
Of T - T - =
If0( N r)d4>I Op(l) and the theorem

Proof 9£_Theorem 2.2. With appropriate changes in notation

 

the proof of Theorem 2.2 is similar to that of Theorem 2.1. The

details are left to the reader.

Proof p£_Corollary 2.1. First note that {ci} defined by

 

(2.21) satisfy conditions similar to (2.2) and (2.3) since

-%

O b

r r
Since the proof of Corollary 2.1 parallels the proof of Theorem 2.1

5 .
(a/n) =1 +o(l) ando/o =a'/b'+o(l)=l+o(l).
r r p a r r P P
only the modifications will be discussed. Each place (2.2) or (2.3)
are used in the proof of Theorem 2.1 the conditions above may be used.
Similarly Corollary 4.1 and Lemma 4.6 hold as before for these {Ci}°

Therefore (5.5) holds. Finally (5.6) and (5.7) hold since for all r

41

-B

-l H. 5 -1
No l 0

r .
r arcrI _<_ (ar/nr) r IIzliirII : 1 + op(1) .

The remainder of the proof follows without change.

Similar arguments show that Theorem 2.2 also holds for these

CHAPTER 6

PROOF OF THEOREM 2.3

This proof is developed in Lemmas 6.1
through 6.3. Koul-Staudte (1972a) presents the asymptotic normality
of the nonrandom linear sign-rank statistics. The specialization of
their Theorem 2.2 to {Xi} which are iid with symmetric distribution
F is presented in Lemma 6.1. Lemmas 6.2 and 6.3 verify the meth-
odology developed by Anscombe (1952), thereby completing the proof.

Lemma 6.1. Let Xl"°"Xn b§_iid random variables with a_continuous

 

 

 

symmetric distribution function P. Let o(t) = ¢l(t) - ¢2(t),

 

 

t E (0,1), where Ii are nondecreasing, square integrable. Let an(i)

and {Ci} satisfy (1.5) 229 (2.5) and A2 = I; 42(t)dt. Then

- + +
A10 nl(Sn - ESn) + N(0,1).

Lemma 6.2. Define {an(i)} py (1.5). Let Fn pp the o-field generated

 

+ , + , _
py {(s(xi), Rin)' l :_1 :_n}. Then {sn'Fn} i§_§_martingale.

 

Proof: Using the density of the ith order statistic of a
set of independent observations each distributed uniformly on (0,1)

one can show that

l

'1 )) + (n - i+1)(n+1)' E(q(u

i(n+l)

E(W(U i)) = E(w(Uni)).

n+1 i+1 n+1

Therefore

i) .

(6.1) an(i) = [i/(n+l)]an+ l(1+1) + [(n-i+l)/(n+1)]a n+1(

42

43

. . . +
Since F is symmetric, {s(xi), l :_1 :_n} and {Rni'

+
l :_i :_n} are independent. Therefore given {(s(Xi),Rni), l i'i :_n}

(6.2) s(Xn+l) = :_l with probability 5

and

6 3 R+ — R+ + 1 with b b'l't R+ /(n+l)

( ° ) n+1i _ ni pro a l l y ni
— R+ with b bil't ( +1 R+ )/(n+l)
— ni pro a 1 y n ni .

Combining (6.1) and (6.3) we have for i = 1,...,n

+ + +
EEan+l(Rn+1i)an:I = (Rni/(n+l))an+l(Rni+l)
(6.4)
+ + +
+ ((n+l - Rni)/(n+l))an+l(Rni) - an(Rni).
Now by (6.2) and (6.4)
+ _ n+1a +
EESn+lanJ ‘ ED:1 biS(Xi)an+1(Rn+li)anJ
= c E[s(x )lF JEEa (R+ )IF 1 + ch s(X )EEa (R+ )IF 1
n+1 n+1 n n+1 n+li n l i i n+1 n+1i n
n + +
— Zlcis(Xi)an(Rni) - Sn .

+
Therefore {Sn'Fn} is a martingale. This completes the proof.

Lemma 6.3. Assume the conditions of Theorem 2.3. For any n let

 

+ .
Tn = Sn/Aon. Then.‘v e > O and n > 0, there ex1sts 6 > 0 such

 

 

 

that as r + w

 

PE max_ IT‘ - T. | > n] < e .
lr-j|:§r r 3

Proof: Denote r - [6r] by 2 and r + [6r] by u where
[r] is the largest integer smaller than r. Now since 5 can be

arbitrarily small, we have by (2.3)

44

(6.5) max IO 2(02. - O )I :_Io-2(o - 02)] + O as r + m .
lj -r|<6r j u

+
Since S = A0 T
r r

-1 -1 -1 + +
(6.6) ITr - Tj | i [(0]. - arm]. llTrl + A 03. sr - sj .

Combining (6.5) and the fact that Tr is bounded in probability in

the limit, the first term on the right hand side is op(l).

- + +
Define 12 = r l Zr a2(i). Note that ES = 0, V(S ) =
r 1 r r r
12 fire? = 0212 and 32 = A2 + o(l). Since 0-202 + 1 as r + m,
r 1 1 r l u
(6.6) implies for a suitably chosen 6 and n1 < 5An
-l + +
P[ sup ITr - T, I > n] :_P[ sup 0. ISr - S_ I > n1] + o (1)
lr‘j |<6r ] Ir- jl<6r J 3 p

-2 +2 +2 -2 2 2 2 2

< - + < — < .
__(nlog) E(Su Sfi ) op(1) —-(n10£) [cu]u ORJRJ e

The third inequality follows by the Kolmogorov inequality for martin-

gales (Loeve, 1963, pg. 386). This completes the proof of Anscombe's

condition and therefore the proof of Theorem 2.3.

APPENDIX

APPENDIX

Proposition A.l presents the asymptotic normality for signed
rank statistics for :p satisfying Hoeffding's condition. These p
may be discontinuous. This proposition retains a simpler centering
constant similar to the centering constants used for rank statistics
presented in Dupac (1969), Hoeffding (1973) and Koul (1974). However,
it eliminates the condition D—H (2.12) which DupaE-Héjek (1969) and
Dupac (1969) used and relaxes the boundedness condition which Koul
(1974) used. No conditions are required on the alternatives for the
absolutely continuous part of :p as was shown by Héjek (1968) for

rank statistics and Huskova (1970) for signed rank statistics.

To prove Proposition A.l we first present a lemma for signed
rank statistics. This lemma is an analog of a similar lemma for rank

statistics in Hoeffding (1973).

Define
l5 1:
(A.l) 31?) e fit 41 - t) dm(t)

Lemma A.l. 2:; m i§_non-decreasing, then

 

(A.2) Z$|EP(R:/(n+l))s(xi) — fép(t)du:(t)| :_8n5Jcp) .

Proof: For x 6 I-m,w) define

45

46
wi(x) = I(Ixil > lxl) + nKn(|x|).
Then
(A.3) Etp(R:/(n+1))||xi| = x} = Eh(wi<x)/(n+1)).
Consequently for 1 :_i :_n

IEp(R:/(n+l))S(Xi) - fip(t)dui(t)|

+ +
lEp(Ri/(n+l))I(Xi > 0) - f.p(t)dLin1(t)

+ +
(A.4) - (Ep(Ri/(n+l))I(Xi < 0) -.flp(t)dLin2(t))|

|A

f2Lp<wi<K'1<t>)/<n+1>) — w(t)|dL:n(t)
Since wi(x) :_nKn(x) + l and wi(x) §_n, we have

(A.5) @(wi(x)/(n+1)) _<_9(nKn(x)) 19(wi(x))
where

g(i) = min{;p((i+1)/(n+l)), '9 (n/(n+1))}, o i i 1 n.
Now by (A.4)
(A.6) ZilEp (RI/(n+l))S(Xi) - fép(t)du:(t)|

g_z$ fELp<wi<K’1(t))/(n+1)> - g<nxn(x‘1(t)))|dL:n(t)
+ z? fElg(nKn(K-l(t))) -2p(t)IdL:n(t)

= R1 + R2 (say)

But by (A.3) and (A.5)

47

73
A

n -1 . -1 +
_ £1 fECgW]. (K (t))) - p (Wi (K (t))/(n+l))]dLin(t)

n + _ +
(A.7) £1 E[g(Ri) - p (Ri/(n+l))]

Z:{g(i) -:p(i/(n+l))}

:p(n/(n+l)) - ~p(l/(n+l)) .

Now the remainder of Hoeffding's proof of Proposition 2 holds with-
out change by considering K and Kn to be the distribution and
empirical distribution of the random variables IXll, |X2|'-°°'|Xn|

respectively. In particular by (2.8)

R +:p(n/(n+l)) -:p(l/(n+l))

(A.8)
= nfElg(nKn(K l(t))) - :p(t)|dt +cp(n/(n+l)) - <p(l/(n+1))

8nL'jJ (:p ) .

IA

The last inequality follows from Hoeffding's proof. Combining (A.6)
through (A.8) yields (A.2), which completes the proof.

The asymptotic normality of signed rank statistics will
be established by approximating them by asymptotically equivalent sums

of independent random variables. For this purpose define

+. _ -l n _ -l _ +
zncp) — on [Zlciflp(t)d[l(lxil 5.x (t))s<xi) ui(t)]

(A.9) 1’ +
+ n fen(t)wn(t)dp(t)].

+ +
where CD is defined by (2.12). Obviously ZHCp) is a sum of in-

dependent random variables.

48

Lemma A.2. Let {an(i)} and {Ci} satisfy (1.4) and (2.2). Let :p

 

.bg \* and bounded. Assume there exist measurable functions on (0,1),

 

{if

Jmk,k=1,2,1_<_i:n} suchthat vd.0<d<°°.andv n>0.

lim;p(E ) = 0 where
n+w n

E { max sup n5IL+ (t) L+ (s)
= s; max _ . - .
n k=1,2 ljiin It-slidn 5 Ink Ink
4.
—(t—s)£. (t)I > n}.
. ink
, +
Define TnCP) .by (5.1).
+ +
A.10 Th T - - Z ‘ = l .
( ) en I n((0) n(WI op()
Proof: As in (5.3) we have

+. _ _ 1 + + _
Tn(~P) - f0 Tn(t)dp(t) + Tn(1)p(1) + 0p(l)

+
where Tn(t) is defined by (4.12). By integration by parts

+ + -1 n +
Zn(p) — -f anp + on {zlci[s(xi) - ui(1)]kp(1)

+
where Zn(t) is defined by (4-13) on (0,1). Then

T+cp) - z+cp) = fé(z+(t) - T+(t))dp(t) + op(1) .

-*
Define un(t) by (4.43). By the decompositions (4.21) and (4.26)

T+Cp) - Z+Cp) f[-Vh(K(K;1(t))) + vn(t)]dh<t)

’11['*(t) — “+(t)(x(x’1 ]
on un Incn n (t)) - t) dp(t)

‘5 3:

(A.ll) - o'lfn 6+(t){w (t) + n (K(K-1(t)) - t)}dp(t)
n n n n

+ o (l)
P

Rl - R2 - R3 + op(l) (say).

49

By (4.22) and the boundedness of m it follows that

. 2 R = 1 .
(A 1 ) | °p‘ )

By (4.23), (4.44), and the boundedness of :p,
(A.13) |R3| = op(1) .

+ —
Now to show [R = o (1) define C. = C.I(C. > O) and C. =
p 1 1 1 - 1

2|
CiI(Ci < 0). Also define Bn and d by (4.24). Observe on Bn

that

1|K(K;1(t)) - ql :dn-11

and

0:11 5:6)” °

- n + -l +
unfilzlciminmmn (t)) - uinm)”

-1
I

I A

[zicftLT (K(K-l(t))) - LT (t)1|
1 1n n 1n

+

- n — + -1 +
onH121c1[Lin(K(Kn (t))) - Lin(t)1|

-l 5

+
20 max [c.ldn .
n . 1
1<1<n

IA

The second inequality follows from (2.8) and the first from the fact

+ +
that for t > s, L, (t) -_L. (s) > O, k = 1,2, i < n. Therefore on
ink ink - -

B
n

- +
IR l < f c 1[2 max Ic.ldn% + dngé (t)]dp + f ndp
2 —' n . 1 n c
En 1§}EP En

The first term on the right hand side is o(l) since lim:p(E ) = O.
n+00 n
Since :p is bounded the second term can be made arbitrarily small by

50

Finally since P(Bn) + l as n + m

making n arbitrarily small.

IR = o (l) .
P

(A.14) 2|

Combining (A.11) through (A.14) yields (A.10).

Lemma A.3. Let {an(1)} and {ci} sat1sfy (1.4) and (2.2). Let

@ EEE\L , absolutely continuous and satisfy the Hoeffding's condition

pr(t)l(t(l - t))_%dt < m ,

(A.15) Then E(T;kp) - zch))2 + O as_ n + w .

Proof:
By Lemma 1 of Hoeffding (1973),<p can be decomposed into

:9 (t) = Yl(t) + y2(t) - 73(t)

and Y are \L, and

where Y1 is a polynomial and Y2 3

J(Y2) + J(y3) < a

(A.16)
where J(Ym), m = 2,3, is defined by (A.l).
NOW
E(T+Cp) - Z+Cp))2 :_2E(0;1(S+«p) - Es+«p)) - Z+«p))2
(A.17) + 20;2(ES+(;p) - n+(n))2
= 2R1 + 2R2 (say).

-1 + + + 2
IRll :_3E(on (s (wl) - ES (w1)) - z (w1))

3 -2 + + 2 + 2
[0 ms Wm) - ES WmH + M2 Wm” ]

+ 62m=2

51
Since W1 has a bounded second derivative the first term on the right
hand side goes to zero as n + m by applying Lemma 2 of Huskové (1970).
The second term goes to zero by applying Theorem 4 and Lemma 5 of

Huskova (1970). Therefore
(A-18) lRll +0 as n+co

(A.19) IES+¢p) — u+¢p)l

| A

3 + +
zllhs (ym> - u (ym)|
= R21 + R22 + R23 (say).

For m = 22,23

(A.20) R
m

n + +
Izlci{Eym(Ri/(n+1))s<xi) - me(t)dui(t)}|

I A

n + +
max IciIXllEYm(Ri/(n+l))s(Xi) — fym(t)dui(t)l .
15}£P

Combining (A.2), (A.16) and (A.20) yields
(A.2l) R + R < 8 max Ic Inga
22 23 . i °
1:}EP
Since 71 has a bounded second derivative, it follows from
Lemma 2 of Huskova (1970) that there exists K dependent only on

Y1 such that
-%
(A.22) R < Kn O .
Combining (A.19), (A.21), and (A.22), yields
+ + -5
IES (m) - u (WI/0 _<_n K + 8kca .

Finally for any a > O, a is selected so that l6kca < 6. Now K

is fixed. Then for n 3_(2K/e)2, we have

52

+ +
IES (:p) - u (WI/o < 5.
Therefore
(A.23) |R2| + o as n + m .

Combining (A.17), (A.18) and (A.23) yields (A.15). This completes the
proof of Lemma A.3.

In the following propositions :p is assumed to satisfy
Hoeffding's condition on (0,1). To simplify the notation we will
assume that .p is non-decreasing on (0,1). This can be done without

loss of generality since :p may be written

:p =cpl -cp2 where '91 and :p2 are N,

and satisfying Hoeffding's condition on (0,1).

Decompose :p into
(A.24) m = ¢ + é

where (D1 and <1>2 are ‘2 and satisfy Hoeffding's condition on (0,1);

¢1 is the absolutely continuous part of m and satisfies

fcd¢1 = 0; ¢2 is the singular part of' m and satisfies

B

fdd>2 = O; and BC is acp-measurable set containing the singular set
E

of' p.

Proposition A.l. Let {an(i)} and {Ci} satisfy (1.4) and (2.2)

 

and :p be \L' and satisfy Hoeffding's condition

 

(A.25) félm(t)|(t(l - t))-%dt < m .

Let :p bg_decompgsed into ¢m' m = 1,2. Assume that there exist:

 

 

measurable functions

 

53

}, i = 1,...,n, n :_1, k = 1,2, on (0,1) such that ‘V d,

 

 

0 < d < w, and. V n > O and k = 1,2,

 

 

, 5 + +
(A.26) 11m ¢ {5; max sup _ n ILink(t) - Link(s)
n ljifin It-slidn
+
- (t-sm. (s)| >n}=0.
1nk
Define
+ -l n + + c
A = — C
cn(t) n Zlci(£inl(t) lin2(t)), t._ B
and
.+ - +'
c (t) = l nc.L. (t), t e B
n l 1 1
+I
when the derivative Li (t) exists.
Then
(A 27) 1im inf 12 . )0-2 > o
' n+(P n
n
implies
-l + -+
T . - - .
(A.28) n+(p)[Sn(p) un(p)] 3 N(0,1) r.v.

4..
Remark 8. Since the Lni are absolutely continuous and (A.26)
+ -
holds, én(t) is well defined almost everywhere with respect to .9.

Consequently Tn+«p) is well defined. Also (2.8) implies

_ I
n lanf (t) = 1.
l 1

Proof of Proposition A.l. Given a > 0, there exists a

 

decomposition

54

(A.29) w = ¢1 + $1 + $2
where
<I>2(b) O<t<b
¢l(t) = ¢2(t) b §_t §_1 - b

¢2(l-b) l - b < t < 1,

$1 and $2 are defined by (A.24) and

b is chosen so that fétfi(l — t)1§dw2 < a and

l 2 2
d < .
f0 $2 t a
The choice of b is possible because :p satisfies Hoeffding's con-
dition and is therefore square integrable. Now ml is bounded on
(0,1) and satisfies the conditions of Lemma A.2. $1 is absolutely
continuous and satisfies the conditions of Lemma A.3.

, + + + + ,
Define T ($1), Z (¢l), T (mm) and 2 (mm), m = 1,2, as 1n

(5.1) and (A.9).

+ + 2 -2 + + 2
(A.30) BIT ($2) - z (w2)| = 40 Els (wz) - ES (wz)l

-2 + -+ 2 + 2
+ 4c lES (wz) -.u (wz)l + 23(z (¢Z)) = 4R1 + 4R2 + 2R3 (say).

By Theorem 4 in Huskova (1970)

- 2
(A.31) R < 800 2n max |c.|2 flw dt < 80k2a2
1 - n . 1 O 2 - c
ljifin

Also by (A.2)

55

2 2 n + 1 + 2
R :_0 max Icil (leEw2(Ri/(n+l))s(xi) - f0w2(t)dui(t)l)

2 ljijp
- 2
(A.32) :_640 2na2 max Ic.|
liifin
2
< 64k a2 .
—- c
Now Lemma 5 of Huskové (1970) implies there exists K such
that
+ 2 2 2
o = < a
(A 33) R3 Var(Z (wz)) __K Rea

Combining (A.30) through (A.33) yields
+ + 2 2 2
(A.34) EIT (wz) - z (w2)| : (320 + 2K + 256)kca .
Since a can be made arbitrarily small, (A.34) implies

+ +
(A.35) IT (wz) - z (w2)| = op(l).

Since Lemmas A.2 and A.3 imply IT+(¢1) — Z+(¢1)l = op(l)

and |T+(¢l) - z+(w1)| = op(1), we have by (A.29) and (A.35)
(A.36) |T+tp) - 2+6)! = 6pm .

4..
Therefore T 69) has the same asymptotic distribution as Z+¢p).

If :p is bounded, write
+. _ _ -1 n _ + -l _
ZnLP) - on Zl{f[(ci 6n) (I(0 _<_ Xi : K (t)) Linl(t))
-( + 2") <1:(-1<’l t) < x < 0) - 1
Ci Cn ( —- i Lin2(t)) dp(t)
- [ (x ) +(1 J- 1.}
ci 5 i - Hi )~p( )

n +
_ ~21 znicp) (say).

56

Since IIGHI :_lma: Ic I and .p is bounded we have
1 n
max |z+i<:p)| 1511;011le max Icil =o<1>-
igign n liiin

Therefore by (A.27), the Lindeberg-Feller theorem yields that
+
T itp)onzntp) 'is asymptotically N(0,1). Then by (A.36) for' m

bounded
(A 37) '1 ) 'T+- ) ' a t t' all N(O 1)
. Tn+(p on (p ‘15 symp 0 1c y , .

If :p is not bounded but is \L_ and satisfies Hoeffding's
condition, decompose cp into (9 =:pl +;p2 where cpl is bounded and

f4):(t)dt < 8. Then by (A.33)

(A.38) E(Z+(CP) - Z+(;p1))2 = Var(Z+(:p2)) _<_Kk:E
and

2 r
(A.39) ((Tn+(spl)/Tn+(:p)) - l) _<_LTn+(1P2)/Tn+(‘.p)]2 _<_ KR: ern ftp) .

Therefore for large n if e is sufficiently small Tn+Cp1)/Tn+cP)
will be as close to l as we want.

Therefore combining (A.38) and (A.39) yields
(A.40) lz Cp)i 1‘?) - 2 (pl )r; l(pl)| = op(l)

Since :pl 'is bounded (A.2?) and (A.40) imply T;:¢p)cnz+¢p) is
asymptotically N(0,1). Then by (A.36), (A.3?) holds for :9 not
bounded which completes the proof of (A.30).

Remark 9 . If m =.pl ~cp2 where W1 and $2 are\L and satisfy
Hoeffding's condition, the same proof whEn applied to T+(wl) and

+ . . .
T (m2) will prove Prop031tion A.l for W which satisfy Hoeffding's

condition.

57

 

Proposition A.2. Let {an(i)} and {Ci} satisfy (1.4) and (2.2) and
cp b_e \ and satisfy Hoeffding's condition (A.25). Let :p be de-

composed into ¢k' k = 1,2, defined_by (A.24). Assume that there

 

 

exist measurable functions {Kin}, i = 1,...,n, n.: l, on (0,1) such

that vd,0<d<°°,and 'vn>0,

 

5
11m ¢ {8; max sup n IL. (t) - L. (s)
n+m 2 lfiifin It-s :dn 5 in in
- (t-s)Rin(s)I > n} = O ,
Define
6(t) = n'lznc IL (t) t6 BC
1 i in ’
and
x -1 n
C(t) =n >3 c.L!(t), t6 B
1 1 1

when the derivative Li(t) exists.

. 2
Define pep) and TnCp) by (2.16) and (2.19) respectively.

Then
2 ..
lim inf T (p)o 2 > O
n n
n
implies
T—ltp)[8 Cp) - U(?)] + N(O l) r.v.
n n D '
The proof is similar to Proposition A.l with appropriate changes and

is not presented here.

REFERENCES

I II l i: I II III III I I l I:

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