WWHHIWIWIWWNHWW1!WIHHHWIHIWHI

113
952
.THS

   

 

LIBRARY
Michigan State
University

 

 

 

This is to certify that the
thesis entitled
A Test For The Change-Point Problem

Based On The Cramer-Von Mises Statistic

presented by

Jae-Sung Kim

has been accepted towards fulfillment
of the requirements for

 

 

Ph.D. degree in StatiSt‘lCS

Jere/~07

Major professor

Date 31 988

0-7639 MS U is an Affirmative Action/Equal Opportunity Institution

 

 

MSU

LIBRARIES
“

 

 

RETURNING MATERIALS:
Place in book drop to
remove this checkout from
your record. FINES will
be charged if book is
returned after the date
stamped below.

 

 

 

 

 

 

A TEST FOR THE CHANGE—POINT PROBLEM
BASED ON THE CRAMER-VON MISES STATISTIC
by

J ae—Sung Kim

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Statistics and Probability
1988

507-0'7/7

ABSTRACT
A TEST FOR THE CHANGE—POINT PROBLEM
BASED ON THE CRAMER—VON MISES STATISTICS
BY
J AE—SUNG KIM

Let X1, X2""’Xn be independent random variables and F, G be
continuous distribution functions on R. We want to test the null hypothesis:
X1, X2,...,Xn are i.i.d. F against the alternative: X1, X2,...,XT are i.i.d.

X

F, X .,XIl are i.i.d. G, F¢G, where r is an unknown positive integer,

7+1’ r+2’"
called a change—point. The Cramer—von Mises statistic is pr0posed to test these
hypotheses. The limiting distributions of the test statistic under the null hypothesis
and particular alternatives are obtained. Under general alternatives , the

estimator in of r is pr0posed by maximizing the proposed test statistic. The
consistency of in is proved and the asymptotic distribution of ATn is expressed in
terms of Kiefer processes. In the case of G( .) = F( o—0), the two sample location
model, the asymptotic normality of 0n: = med {Xj—Xi, 1 S i 5 in, in + 1 g j 5 n}
is also derived.

Finally, distributions of test statistic and estimator in finite samples are

studied and the comparison of the pr0posed test with other tests is carried out via

simulations.

To my wife Min—Soo and my sons

Woo—Jin and Song—Jin

ACKNOWLEDGEMENTS

I wish to express my sincere thanks to Professor Hira L. Koul for suggesting
the problem, for his continuous guidance, encouragement and especially his patience
during the preparation of this dissertation.

I would also like to thank Professors Joseph Gardiner, Dennis Gilliland and
James Stapleton for serving on my committee. My Special thanks go to Professor
James Stapleton for his continuous encouragement, for his constructive comments
on earlier versions of the thesis and for his valuable help with simulations.

I would also like to thank my wife Min—Soo and my sons Woo—J in and
Song—J in for their encouragement and patience during the preparation of this
thesis.

Finally, Special thanks go to Ms. Cathy Sparks for her excellent typing of the

manuscript.

TABLE OF CONTENTS

Chapter Page
1. INTRODUCTION ' 1
2. TESTING A CHANGE—POINT 4
2—1 The proposed test statistic 4
2—2 The asymptotic null distribution of the test
statistic 6
2—3 The asymptotic distribution of the test
statistic under particular alternatives 11
3. ESTIMATING A CHAN GE—POINT AND A LOCATION
PARAMETER 16
3—1 The consistency of the estimator of a
change—point 17
3-2 The asymptotic distribution of the estimator
of a change—point 22
3—3 The asymptotic normality of the estimator
of a location parameter 23
4. FINITE SAMPLES 29
4—1 Distributions of estimators for finite
samples 29
4—2 Comparison of the proposed test with
Schectman's test 30

REFERENCES 32

1. INTRODUCTION

Let X1, X2,...,Xn be a sequence of independent random variables with
distribution functions F1,F2,...,Fn, respectively, Fit? where .7 is a class of
continuous distribution functions on R. The so called change—point problem
concerns the null hypothesis HO: F1 = F2 = = Fn=F(say) against the alternative
hypothesis H1: F1=”'=Fr ,e Fr+1 = = Fn=G(say) for some unknown
7, 1 g r < n. The parameter r is called the change—point. The relevant questions
are the following: How should the change—point be detected? If there is a
change—point, how should it be estimated? If F( -) = G( o—0), and if there is a
change—point, how should 0 be estimated?

In the change—point problem there are two t0pics of interest: i) Testing and
estimating the change—point and ii) Estimating the change under the alternatives.
This thesis deals with both these t0pics.

In detecting a change—point there are two main methods available in the
literature: i) Sequential detection and ii) Quasi—sequential detection. In this
thesis a quasi—sequential procedure is used.

If n is not preassigned, a sequential method is a rather reasonable method
for finding a change—point. About thirty years ago Page (1954, 1955, 1957) A
introduced cumulative sum test to detect a change—point. Pollak (1985) has
derived a st0pping rule which is a limit of Bayes rules to detect the change—point.
Page's (1954) stOpping time was shown to be Optimal for the detection of changes in

distribution by Moustakides (1986). All of these papers have dealt with sequential

methods to detect a change—point.

Most changebpoint papers consider the quasi—sequential case, for which n is
preassigned. We will assume that n is given, that is, X1,X2,...,Xn are given.

This thesis studies the Cramer—von Mises statistic for testing H0 and
estimating a change—point 7' under H1 in a sequence of independent random variable
in the quasi— sequential setting.

For G(-) = F( -—0) the Hodges—Lehmann estimator (1957)
in = med {Xj—Xi’ 1 5 i 5 ;n < j g n } of 0is pr0posed, where in is chosen to
maximize the test statistic in Section 3.1.

The limiting distributions of the test statistic under the null hypothesis and
the particular alternatives are obtained in terms of two independent Kiefer processes
in Section 2—2 and 2—3. The finite sample null distribution of the test statistic is
simulated because it is almost impossible to compute it exactly. The estimator in
of 1- is shown to be consistent in Section 3—1 and the asymptotic distribution of 1- n
under particular alternatives is expressed in terms of the Kiefer processes in Section
3—2. The asymptotic distribution of an is shown to be the same as that of its
analogue in the two sample problems with no change. The prOperties in finite
samples are studied and the comparison of the prOposed test and other tests is
carrried out in Chapter 4.

Other approaches to the change—point in the case of quasi—sequential
procedure include: for testing and estimating the change—point and change in
mean; maximum likelihood estimation of Hinkley, D.V. (1970), Sen and Srivastava
(1975), Hinkley, D.V. and Hinkley, EA. (1970), and Hawkins, D.M. (1977);
cumulative sum test approach of Hinkley, D.V. (1971) and Pettitt (1980); a

non—parametric approach based on the Mann—Whitney statistic studied by Pettitt

(1979), Schectman and Wolfe (1984) and Schectman (1982,1983); the least square
method of Hawkins, D.L. (1985); for estimating a change in the slope; maximum
likelihood estimation approach due to Hinkley, D.V. (1969a, 1969b);
non—parametric approach of Sen (1980, 1982). For more details see the
bibliography of change—point problems of Hindley, D.V. (1980).

Very useful references for this thesis are Hawkins, D.L. ( 1985) and
"Empirical Processes" by Shorack and Wellner (1986) for a discussion on the Kiefer

process.

2. TESTING THE CHANGE—POINT

Recall the hypotheses.
Let X1,X2,...,Xn beindependent with Xi having distribution function Fi’
i = 1,2,...,n.

We want to test

(2.1)

for some unknown 7, 1 g r < n, where F and G are unknown.

2—1. PROPOSED TEST STATISTIC
To introduce the test statistic, proceed as follows: Divide the sequence

X ..,X into two parts of the first j random variables X1,...,X. and the last

J
Xn’ j=1,2,...,n—1. Consider (n—l) Cramer—von

1"

(n—j) random variables Xj+1""’

Mises statistics and take a maximum of them. We do not consider the case of j =

II

11, since j = 11 implies that there is no change—point.

The corresponding Cramer—von Mises statistic of X1,...,X. and Xj+1,...,X

J 11

is

snj=(j<n—j)/n)1 [i,(x)—én_j(x)12dfin(x), lsjsn—l,
R

= 0, j=n, (2.2)

where R is the real line,

 

. _1 ° A — 1
Fj(x) — jifil I[XiSX], Gn_j(x) — Ill—jig:1[xj+i-X]
fink) :le(x) + ill—DGH _j(x )=% E I[Xi SX] xER,

i 1

and where I denotes the indicator function.

The proposed test statistic for H0 against H113 is

Mn: max Sn: max (j(n—j)/n) 1 [F.(x)— an __jx2()] dH n.(x) (2.3)
1<j<n J1<j_ <n—1 R J

For computational purposes let us express S nj explicitly in terms of ranks.

The alternative form of S nj is

Snj = Unj/nj(n—j) — [4j(n—j)]/6n, j=1,2,...,n—1 (2.4)

where

Ill

Rij = Yij's rank in the pooled sample, 1 S i S j,

Tij = Zij's rank in the pooled sample, 1 S i S 11—1.,

o n_j
1:1 1:1

Yij = i th order statistic in X1,...,Xj, 1 S i S j,

Z.. - ——i th order observation 1n X.

1j j+1’" “’Xn’ 1 S1 gn—j.

To compute Ri j and Ti j easily proceed as follows:

i) Divide into two groups X1,...,Xj and Xj+1""’Xn’

ii) Determine the order statistics Yij and Zi

iii) Find ranks Rij and Ti

j in the pooled saIanle.
This alternative form is due to Anderson (1962).

We reject the null hypothesis when Mn is too large. We note that Mn is
taken to be maximum of a sequence of random variables {Snj; j=1,2,...,n—1} which
are not independent. For this reason it is not easy to compute the exact

distribution of Mn. Hence we study the large sample approximations to this

distribution.

2.2. THE ASYMPTOTIC NULL DISTRIBUTION OF THE TEST
STATISTICS Mn'

Let c be a small positive number and r = [nto] for t0 6 [6,1—6] = I (say).
We exclude the cases of t0 = 0, t0 = 1 since if t0 = 0,1, then there is no
change—point. Let N(e) = {[nc], [me] + 1,...,n—[nc]} — {0} and

M6: max S .. (2.5)
1‘ jeN(c) I”

Note that M 1i 3 Mn, since the maximum in MIC1 is taken over a smaller set
than in Mn‘ Observe that M; and Mn are very close if c is small.

We will try to find the asymptotic null distribution of MI: instead of Mn
as n -—1 00. We use M :1 rather than Mn because it is very hard to detect a
change—point when it happens near 1 or n. Assume that

T=[nt0]—-Ioo as n——»oo.

Define the empirical stochastic process Sn by

Sn:{Sn(t);teI},Sn(t)=Sn[nt], n21, tEI. (2.6)
Then
s .=s (j/n), jeN(c) and M‘ = max 5 .=supS (1). (2.7)
“J n '1 jeN(c) “1 th ’1

Note that SIl has its paths in D(I), where D(I) is the Space of functions on I

that are right—continuous and have left—limits.

Let {K(t,x); 0 S t, 0 5 x S 1} be the Kiefer process with EK(t,x) = 0 and the
covariance function

EK(t1,x1)K(t2,x2) = min(t1,t2){min(x1,x2) — X1X2}'

Note that a Kiefer process {K(t,x)} can be expressed in terms of the two time
parameter Wiener process {W(t,x); 0 5 t, 0 s x g 1} as follows: {K(t,x)} 2
{W(t,x) — xW(t,1)}.

A visual way of seeing a Kiefer process is that it is a Brownian bridge in x
and a Wiener process in t. For more details on the Kiefer process, see Csorgo
(1983, Chapter 3), and Shorack and Wellner (1986, Sec2—2 and p131).

We need the following lemma to study the asymptotic null distribution of

6
Mn.

Lemma 1. Let Yl’Y2""’Yn be independent uniform random variables. Define the

partial empirical process

Kn(taY) :(1/@[_§_:]{I[Yis Y] ‘3’}, O -<- t, y 51'

Then Kn-vl—iK in (D([0,1]x[0,1]),|| H) as 11-400 where

K is a Kiefer process. L means the weak convergence, (see Billingsley (1968)) and

II II denotes the supremum norm.
Proof. See Shorack and Wellner (1986, p131). u

The asymptotic null distribution of M; is given by the following theorem. In

order to state it we need to define
A(t,x) = J(I—t)7t K1(t,x) —Jt7(1'—t) K2(1—t,x) (2.8)

~ 1
and S(t) = j A2(t,x)dx where K1,K2 are two independent Kiefer processes.
0

Note that EA(t,x) = 0 since EK(t,x) = 0, and
EA(t1,x1)A(t2,x2)

= E(J(—)7'_l—tl t1 K1(t1,x1) —Jt17l1—tll K2(1-t1.x1))'

(Vi 1-t27t2 K1(t2,x2) — i132“ 1—t2) K2(l—t2,x2))

 

= ill-t1” t1 “(HQ/t2 t1("1‘x1x2) _

 

Jt1/(1-t1) Jib/(14'?) (1_t2)(X1_X1X2)

 

= Jt1(1—t1)F2(1—t2) (1—t1+t2)(x1—x1x2) when t1 < t2, x1 < x2.

 

We see that A(t,x) is a Brownian bridge in x for fixed t.
Theorem 1. Under H0 (F1 = F2 = = FB = F and F is continous).
(1) sum in (D0).” 1) as n—m.

(2) M; —D—-+ sup S(t) = S6 (say) as n —-1 00.
1361

Proof. Without loss of generality we may assume that the Xi's are i.i.d. Uniform

(0,1). Otherwise, work with F(Xi)'s instead of Xi's. Since Snj
is a rank statistic, this transformation does not change Snj’ 1 g j<n. Moreover, F

is continuous implies F(Xi)'s are Uniform (0,1) random variables. Thus

snj = {i(n—j)/n} 11 113p) — (imam? dfine)
0

= {1(n—1)/n} 1(1) 11rj<x>—x} — {én_j(x>—x}12 dams)

° n 2 .
= I;[JFK-WE10:1(I[Xinl-x)—1/(n—j) 2; 1(11xisx1—x11] dHnm.

1=j+

Set j: [nt]. Let

_ n— nt 1/2 1 [nt]
An(t.X) [ nt] .3131 (mesa—x)
—[ “t ]1/2._1_. 121 (I[X.<x]—x). (2.9)
“'1“ m i=[nt]+1 1‘

10

Then

§n(t) = SI'I[Ilt] = I; A121(t,X) dfin(X)

= 1(1) A§(t,1i;1(x)) dx. (2.10)

From Lemma 1, the independence of the Xi's and (2.9),
An _W__. A m (D(Ix[0,1], u u) (2.11)
where A is as in (2.8).

Moreover, by the Glivenko—Cantelli Lemma,sup | H;1 (x) — x | @413 0.
xER

Hence by (2.10) and (2.11),

sup 1 sun) — )1 A121(t,x) dx 1 = o (1). (2.12)
tel 0 1’

Now, for any g E D(Ix[0,l]), [1 g2(t,x) dx is a continuous function
0

of g with respect to the supremum norm. Hence (2.11) and (2.12) imply
~ W 1 2 .
sum ——» 10 A 0.x) dxm (D0). 1 II)-
This proves (1) of Theorem 1 where (2) follows from the invariance

principle. 1:1

In order to implement the M :1 test, Table 1 below gives simulated critical
values of M; and 3f for n = 40,100,200 and a: .10, .05, .01. The simulation

is based on 1000 repetitions.

 

11

TABLE 1

l—a quantiles of M; a and 8; (c = 1/12)

 

 

6 6 6 C
0’ M40 or M100 a M200 0 Sa
.10 . 7425 . 8324 9082 9196
.05 .8371 1.015 1.059 1 119
.01 1.155 1 273 1.380 1 444
2.3. ASYMPTOTIC DISTRIBUTION OF THE TEST STATISTIC UNDER
PARTICULAR ALTERNATIVES.
Let an(x) = Fj(x) — Gn_j(x). (2.13)
Recall that
. . 2 ‘
M€= max {1(n-1)/n}l V - (30dH (X)-
‘1 jEN(c) R ”J n

For jS 7(t S to):

an(x) = an(x) — Ean(x) + Ean(x)

=(1/1),>5 (leisxl—Rxn—(l/(n—j»,5, (I[XiSXl-F(x))
1:1 l=j+1

- (1/(n—J')), £1; (I[Xi 5X] — G(X)) + ((H-T)/(n-i))(F(X) - G(X))-

I=T+

12

Hence

Air—1775 vnj(x> = «(3:973 (WIT) $1099 5 x1 — 1‘00)
471—71 n—1 (1/719i=.j€+1<11xi 3 x1 — F00)

— 075:9 (1/75)i=§+l(11xis x1 — G(x»

-~/Jln—Jl7n ((n-T)/(n-J°))(F(X) —G(X)))

= Ln10,F<x).G(x)) + an1(t.F(x).G(x)) (say) (214)
where

an1(t.F(x).G(x)) = W n—1 n - (<n—r)/(n—1)) (F(x) — 900).

For j> r(t > to),

tin—5775 vnj<x> = W (WIT) ii (lei 5 x1— ax»

+ «(IT—Tm (we i$10199 s x1 — G(x»

1=j+

471—71 n—1 (we 31; 1(IIXiSXl-G(x)))

+ «(Jill-15711 (T/J') (F(X) — G(X))

= Ln2(t,F(X),G(X)) + 011203.17 (X),G(X)) (SW) (215)
where 01112 = W - T/j (F(X) — G(x)).

13

The last terms of (2.14) and (2.15) tend to 00 as n —-> 00 for any
x e R. So in order to have meaningful results we need some conditions on F(x) and
G(x).
We consider the following alternatives:
Hm: G(x) — F(x) = n—1/2(F(x) — F(x)k), for some positive integer k.
Note that
:13; |G(X)-F(X)| = 0(1).
an1(t,F(x),G(x)) ——+ 1 1—1 (1—t0) (F(x) — F(x)k) = al(t,F), (say)
and (2.16)
an2(t.F(x).G(x)) ——» W (to) (F(x) — F00“) = c1203). (say)
uniformly in x E R and t E I as n ——1 00. Note that under Hln’ L ni(t,F(x),G(x))
are functions of t and F(x) only. Thus write Lni(t,F(x),G(x)) = L ni(t,F(x)), i = 1,2.

The limiting distribution of the L ni(t,F(x)) are given by
Lemma 2. Under Hln’
Ln1(t,F(x))1V—-1 L1(t,F(x)) in (D(IxR),|| H) as n —-+ 00,
where, for OSySI, tStO,
L103) = W K103) — W K100—ty) — 7771—175 K2(1-t0.y), (2.17)
and
Ln2(t.F<x)) Vi» Laure» in (130an ll 1) as n ——» e,
where, for ogy51, t>t0,
L2(t,Y) = (IT-WT K1003) + W K2(t-t0,y) - (WIT-ti K2(1-t,Y), (2-18)
where

Ki's are independent Kiefer processes, i=1,2.

 

14

Proof. For t5 t0,

Ln1(t F(X))= (n-lntllllntl (1/1/5) [Et 1](IIXi S X] - F(Kl)

 

 

—./[mj/(n—[m])(1(/,/fi) [I120] 1i(I[x <x]— F(x)) (2.19)

i=nt+l]

—nJ| t|7(n—|nt|(1/J—) 2 ] l(I[Xi5x]—G(x)).

i=nt0+

The first term of (2.19) converges weakly, by Lemma 1 applied to {F(Xi)},
to K1(t,F(x)) in uniform metric. The second term of (2.19) converges weakly, by
Lemma 1 applied to {F(Xi)}, to K1(t0—t,F(x)) in uniform metric. The third term
of (2.19) converges, by Lemma 1 applied to {G(Xi)}, to K(1—t,F(x)) in uniform
metric. Hence (2.17) holds. Similarly (2.18) can be proved. a

The asymptotic distribution of MICl under H1n is given by
Theorem 2. Under H

1n’

n (1301111) as a...
st (0 = 100.103) + «may»? dy lit 5 to]

+ 10(L2(s.y) + a2(s.y))2dy11t > so],

15

and where Li(t,y) and ai(t,y) are in (2.16), (2.17), and (2.18), i=1,2.

(2) M:1 24 sup{St (t)} = S: (say), as n —-a co.
tEI 0 0

Proof. For t g to,

1 (Ln1(t.F(x))+ an1(t.F(x)))2 dame)
R

_ 1 “—1 “—1 2
-— l0 (LnlfiaPTHIl (y)) + 0311(13,F(Hn (y)))) dy-
Let
Hn(x) = EHn(x) = 1 /n [TF(X) + (n—T)G(x)]. (2.20)
Note that sup |Hn(x) — F(x)| S supl Hn(x) — Hn(x) | + supl Hn(x) — F(x)|
xER XER xER
L 0 (2.21)

as n ——-s 00 because of the Glivenko—Cantelli lemma and because of Hm.
‘ —1 A —1
Next, to see that {Lnl(t,F(Hn (y))) + oznl(t,F(HI1 (y)))} and
{Ln1(t,y) + an1(t,y)} have the same limiting distribution, observe that

:2? IL 0.1103130»)+amsrmgle)»—Ln1(t.y)—an10,y)l

y6{0sll

n1

A —1
S $16111) an1(taF(Hn (y))) — Ln1(t,)’)l

ye [0.1]

“—1 P .
+ $1611; lcvnl(t,F(Hn (y))) - an1(t,y)l —r--+ 0 as n -’ oo
yE[0.ll

 

16

Since su IF(f1;1(Y))-YI
ye 0,1]

A

s surf 1F1figl1y» —f1n(H;1(y))1+1/n s an 1F1s1— fin1s)1+1/n
ye 0.1] ye 0.1]

LOasn—aoo,

because of (2.21) and because Ln1(t,y), an1(t,y) are tight.

By Lemma 2, for t S to,

1111103) + 011103,” L L103,” + 0100’) (D(IX[0,1],“ H) and
1 1

1 1Ln11ay1+ an11ay112 dy l»1 1L1(t,y) + a1(t,y)12dy
0 0

in(D(IMI ll) 38 n—W.

Similarly for t > to,

1 1
I0 [Ln2(t.y) + an2(tay)l2dy L lo [L2(t,y) + a2(t.y)l2dy as n —-+ co.

It follows that (1) and (2) in Theorem 2 hold. 13
3. ESTIMATING A CHANGE—POINT AND LOCATION PARAMETER
An estimator :rn of 1' is given by

; =smallestk such that S = max S .. (3.1)
I] 11k jEN(f) DJ

When H1 is true and G(~) = F(s—0), an estimator Rn of 0 is

0n=med {Xj—Xi:1$i$rn<j§n} (3.2)

 

17

In what follows the consistency of Tn under H1 is proved and the limiting
distribution of TH under H1n is obtained in terms of Kiefer processes. The

asymptotic normality of En is also proved.
3—1. CONSISTENCY OF }n

Recall the definition of Sn from (2.5) and define

2 _1 2 . _1 .
§n(t) : = n Sn(t),tEI and tn = n Tn. (3.3)

Observe that tnEI and it maximizes {n(t), tel since 7 maximizes

n
Sn(t), tel. Theorem 3 below ShOWS that t n 25—» t0 as n -——> 00, that is, (Tn - r)/n
——10 in probability, at the rate O(n—1/2+6) for 6 > 0 where

r = [ntO], tOcI. Note that all expectations in this section are computed under H1.
In what follows use j instead of [nt] for the sake of convenience.

From (2.13),

Evnjtx1=E{<1/1>,>5 I1xisx1—(1/(a—9), ‘25 11x,sx1>}
l=1 1=_]+1

((n—T)/(n-J')) [F(X) -G(X)li j S T

 

= (3.4)
(7/1). [F(x) — G(x)], j > T.
Var an(x)
I12+T.—2Il. _ Il-T .
. . F(x)[1 F(x)] + , 2G(X)[1-G(x)], )3 7-,
2 J (n’l) . (n—j) (3 5)
j}; F(x)11—F(x)l + W G(X)[1—G(x)], j> a

 

18

Use EX2 = {EX}2 + Var X to compute

jg n—j) 2
E n Ean(x)

 

2
i—Lfn—T F—G + —1—le +7 2“ F I-F + n—T2G1—G <7'
n(-)1 [ 12 i(n-i) [ 1 (n -i)2 [ ]j

Let
H(x) = t0F(x) + (1—t0)G(x), xER and
fn(t) = n-1 (R ([nt](n—[nt]) /n) EVI21[nt](x)dHn(x), tel.

From (2. 20),

sup 1 H 00— H(x 1 1= 0,11).
xER

From (3.8) and that the second and third terms of (3.6) converge to O,

and Hn(x) —-) H(x) uniformly in x as n —-+ co, the pointwise limit of fn(t) is

f(t) := lim fn(t)

n-Ioo

lim(tn—r12/n211m1/(n-1nt1 1) IR[F(x)-G(x) 12dH1x),

n-ioo

Iim1r2/n2)<1a-1ns1)/1nt1> 1R1F1x1—G(x>12dH1x) ,

n—im

(HO)2 (t/(I—t» 80,) .t :10.

t3 ((1—t)/t) B(t0 ) , t > tO
where B(t0) = 1R[F(x) — G(x)]2dH(x).

7'2
7—2 [F—G]2 +T—2 F[1—F]+ %1+—T;§31o[1—G], j>r.
J 11"] J

(3.6)

(3.9)

 

19

We need the following Lemma to prove Theorem 3.
Lemma3. Under H1, for any 6 > O
1141/2 suplfn (t)— in(t)|—£—»0 as n—aa (3.10)

‘ . . 2 2 A 2 ”
Proof. 15,10 — 1,101 = J(n—J)/n 11Rv,,- dH, — 1REv,jdH,1

sfl—fi—‘fltling 4211?, )dH, 1+11R EVE, 41H, —H ,)1}

'n—' _ . . —. 2
3 Mn {4IRIan EanldHn+ JRIHn Hn|dEan} (3.11)
Observe that for j S r,
j—(gjl—atU—t) uniformly in tel as n——»oo,
n
.—1 g: __1 7
=1 , {I[X,5x1—F1x)}—1n—1) 2 111xi le—F(X)}
-— i-—1+1
_1 n
-(n-i) 3 {I[XiSXl-G(X)}
i=r+1

= [I Kn1(t,F(x)) -— (31% Kn2110—1.F(x)) - £3,- Kn3(1-tovG(X))

where KI1 i's are as in the proof of Lemma 1. The first term of (3.11) tends to zero

1—/2 —6 1/2— —6

because n sup nItl —-10,n sup ——>0 asn ——ioo and Kn' 'S are

teI tel ‘1 nut
tight, i=1,2,3 which follows from Lemma 1. The second term of (3.11) tends to zero

because 1011—11—1 t( —t) uniformly in tel as n——ioo and

[1-61/2 sup lHn (x)—Hn (x)| -P—-1 0 as n —-1 00 by the Glivenko—Cantelli lemma.
x6

Hence (3.10) holds. 1:1

 

 

 

20

The consistency of Tn is given by

Theorem 3. Under H1, for any 6 > 0,

nl/Q-‘Sltn—tol E540 as n—-)oo.

Proof. Note that
111/ 2‘5 sup |fn(t)—f(t)| ——) 0 as n _.. oo
teR

where fn(t), f(t) are in (3.7) and (3.9), since

ail/2‘5 sup 11,19—1101
tER

s [12/a1/2+51a11/1n—1a101) + n1/2-6/(n_[n,)))11.9.0]

+ (2/n1/2+5(n—[nt0])/[nt01 + n1/2—5/[nt0]).l[t>t01] .B(t0)

Claim: To prove the theorem it suffices to Show that for any 6 > 0

n1/2_6 |f(tn) - f(t0)| _.. 0 as n __) a.
To prove (3.13) first note that for any 0 < u < 1,

l—t t
f1u)=1:g1t)da where g0) = B1t,)[1fi)211tsso1-1,9-)2110101],

For tn 5 to,

t

. t
£110) 411,) = 1,0 g1t)dt = B1t0)-11—to)2-1£° 1/11—s)2dt

II II

3 B(t0)((1—t0)/(l—tn)2)(t0-—tn) ; B(t0) ——§ ~ (to -- in)

(3.12)

(3.13)

 

21

since e<tn <1—e.

Thus for tn 3 t0,

0 g n1/2“5(tO—tn) g n1/2_6[f(t0)—f(tn)]o-(All-—E§s1/B(t0). (3.14)
-t
0

For tn > to,

A

f1£,)—f1t0) = 1tn 3(t)dt = B11343 1tn1—1)/12dt
t0 t0

2
t A A
0 .

Thus we have for t n > to,

1 2— A 1 2— A 2
0 5 n / 5(tn—t0) g n / 511(t0)—i(tn)].e /t§.B(t0) (3.15)
By (3.14) and (3.15), the claim is proved.
To see that 111/2—5lf(tn )—f(t 0)| —-+ 0 as n ——-1 oo, observe that

|f(t n)-f(t0)| 5 WD )-€, (t ,)| + |€,t( ,-) f(t OH

3 sup111t)—&,1t)1 + lsup E,1t)—sup 1101
tel tel tel

(by definition of t and because f(t ) = sup f(t))
n 0 teI

s 2-sup111t)—é,1t)1
tel -

S 2°ftellf|f(t)-f,(t)l + 2°:gllalf,(t)-£,(t)l (3.16)

 

22

The first term of (3.16) tends to zero by (3.12) and the second term of (3.16)

tends to zero by Lemma 3. B

3—2 THE ASYMPTOTIC DISTRIBUTION OF :rn UNDER
PARTICULAR ALTERNATIVES

In this section the limiting distribution of in under H1n is obtained.
Theorem 4. Let k = [ntk], tkcl.
Then PH1n{ Tn>k} —-+ Pr{sup[St0(t): eStStk] < sup[St0(tk):tk<tSI—€]}

where S is as in Theorem 2.

t0

Proof. From the definition of in, (3.2), we have that

{i-n>k} = {max(Snj: [ne]$jgk) < max(Snj: k<j$n—[ne])}.

PHln{max(Snj: [ne]$j$k) < max(Snj: k <j$n—[ne])}

= PH1n{sup[Sn(t): fStStk] < sup[Sn(t): tk<tSl—c]}

——) Pr {sup[St (t): eStStk] < sup[St (t): tk<tSI—e]}
0 0

by Theorem 2. U

23

3—3 THE ASYMPTOTIC NORMALITY or in

Recall 0n 2' med{Xj —Xi: 1 g i S rn< j S II}.

To prove the asymptotic normality of 0n the following Lemma is useful.

Lemma4. Under H1,

A

A

11) 11%,/1a—a,))1/2on‘1/2i=§:1111x,sx1—G1x)}1 1179,1353 0,

A

12) 11%,/1n—r,))1/2sn‘1/2 i5 {leisxl—G1x))111r>‘r,1flao.
i=Tn+l

as n——)oo for any xeR.

Proof. We prove (1) only Since the proof of (2) is Similar.

A

A

(;n/(n—Tn))1/2'n—1/2. E 1|I[Xi5x]—G(x) | I[TSi’n]

1=='r+

s Tn/(n‘f))1/2'n_l/2{Tn—r}

= 1%,on/n1n—r,))1/2.(in—n/nnl/2
=(i’nn1/4/n)1/2o(n/(n_;n))1/2.((;n_r)/n).n3/8fl, 0 as n __) a

by Theorem 3 applied with 6 = 1/4, 1/2, 1/8. 13

The asymptotic normality of 0n is given by

24

Theorem 5. If F has a continuous density h with 0 < j h2(x)dx < 00, then under
R

H1 and G(-) =r(.—3),

712—.(22n—0)——D—aN(0, 1 ) as __.e,
n 12°(l(1)112(y)dy)2 n

 

where kn = rn(n—Tn)/n.

Proof. Let Un(u) = IRGn_;n(x)dF‘Tn(x—u), tcR, then

,. . Tn n
Un(u) = 1/1'n(n—Tn)iE1 2 I[Xj—Xigu].
j= Tn+1

Note that Un(tln—) S 1/2 5 Un(3n) and hence .

{Un(u) > 1/2} 5 {in g u} s {Un(u) 2 1/2}. (3.17)
Consider the event {fit—n (flu—0) 5 x}, xeR. From (3.17) we get the following

relation:

P{Un(x/\/k;+0) > 1/2} 5 P{0 S x/Jk—I;+0} S P{Un(x/t/k;+0) 2 1/2}. (3.18)
the first and third term of (3.18) have the same limit, then we will get the

limit of the distribution of 4k; (Rn—0). Consider the first term.

{Una/7E1; + a) > 1/2}

=17k; 1U,1x/7E, + 1) —1RG1y+x/7E, + o)dF1y)1

> (E [1/2 - IRG(Y+X/~/1'<, + 6’)dF(y)l } (3-19)

 

25

Consider J11; [1 / 2 - G(Y+X/\/k_1; + 0)dF(Y)l

= 711—, [IRF(y)dF(y) - JRG(Y+X/~/1§+0)dF(Y)l

= JR; lR[F(y) - F(y+X/\/1§Q]dF(y)

w

l

r'1—x)1 h1y)dF1y)=1—x)1 1121105 as n—aa, because
R R

__.) .
n (D

'x‘

Consider i/k; [Una/fig + 0) — IRG(y+x/JE'I; +0)dF(y)]

= 91231116211; (y+x/./k;+0)dI3‘;n(y) - JR G(y+X/~/1§+0)dF(y)l

T
n

= 91;; [1R(én_;n(y+x/,/1§1‘+o) — G(y+x/1E,+ 9))d13‘;n(y)

+ 1RG(Y+X/1/I<;+ 9)d[F;n(Y)-F(y)ll
= «R; [1R(én_;n(y+x/,/t;+ a) — G(y+x/./1§;+0))dr(y)
+ 1R1é,_;n1y+x/¢r,‘+o) — Goa/75+ 0))d113“; 1y)—F1y)11

II

+ JR; [1RG(y+x/,/l§+ 0)d[F;n(Y)-F(Y))l

 

26

=(rn/(n—rn))1/2 IRK2,n_:rn(y+x/Jk;+0)dF(y) (3.20)
+ 1‘r,/1“r,))1/2 1RK2,,_;n1y+x/7k—,+o)d11~‘;n1y)—F1y)1 13.21)
_ ((n.?rn)/3rn)1/2 JRK1,;n(y)dG(y+x/Jk; +0) (3.22)

by integrating by parts,

where K2 n—i' (u) = (IA/ID .E (I[Xi S Ill—GUI»,

n ._
I—Tn+1

K1,;n1u) = 1N?) if; 111xisu1—F1u)).

To get the limit of (3.21), note that

(rn/(n—rn))1/2EI—a(tO/(l—t0)1/2 as n—-ioo, and
K2,n—Tn(Y+x/‘/§+ 0)

=11/7m :5 111xjsy+x/./R;+01—G1y+x/7E,+0)
j=Tn+1

= (l/mj_§+1(1[xjsy+x/./1g+111—o(y+x/,/rr;+o))+(rn/(n—rn))—1/2op(1)

by Lemma 4.

 

27

By Lemma 1, (3.20) converges weakly to (to/(l—t0))1/Zj1 K2(1—t0,u)du
0

in uniform metric.

To get the limit of (3.21), note that
1RK2,_:, 1y+x/7k;+o)d1F; 1y)—F1y)1
n n

= 1RK2,,_:,n1y + f + 0d113“;n1y)—F1y)1

= l K2,,_;n(y + 4L1? + 0)dF;n(Y) - IRK2,,_;n(y + fi- + 9)dF(Y)

R n n

= )1K2 ‘ (F71(u) + _x_ +0)du — le _‘ (F—1(u) + i + 0)du
0 ,Il—Tn Tn JR; 0 2,n Tn n
1 . “—1 . —1

= {0(K2,n_Tn(F; (u)+0) — K2,n—rn(F (u)+0))du + op(1)

I1

P—r—i 0 as n ——1 00 since {K2 n_‘T (t)} is tight by its convergence and
. 7 n

sup 1F;1(n)—F‘1(n) 3L0 as n—aa.
OSuSI n

To get the limit of (3.22), use the same method as (3.20) and we get the
1
limit, ((1—t0) /t0)1/ 2 ) K1(t0,u)du where K1(t,u) is the Kiefer process and
0

independent of K2(t,u).

 

28
Therefore the limit of LHS of (3.19) is
1/2 1 1/2 1
(to/(l—t0)) )0K2(1—t0,u)du — ((1—t0)/t0) )0K1(t0,u)du (3.23)

1
Note that } K1(t0,u)du has a normal distribution with mean 0 since for
0

each ue[0,1], EK1(t0,u) = 0, K1(t0,u) is normal and

1 a.s. n 1
j K (t ,u)du = lim 2 (k/n) K (t ,k/n). So is j K (l—t ,u)du. The variance
1 0 _ l 0 2 0
0 Il-ioo k—l 0

of (3.24) is 1/12 since
1 1 2
Var )0K1(t0,u)du = E(jOKl(t0,u)du)
1 1 ' 1 1
:1 I EK1(t0,u)K1(t0,v)dudv=j I t0(uAv—uv)dudv
0 0 0 0
1 1
= 2t0- j I (u—uv)dudv = (1/12)-t0, and
0 0

Var I;K2(I—t0,u)du = (1/12)-(1—t0).

 

29

Hence the first term of (3.19) tends to
1 1
Pr 111M?) 2 > 1—x) 10h21y)dy)) = Pr 111W) 2 < x10h21y)ay1
1
= @1712.» 10h21y)dy)

and the third term of (3.19) has the same limit where Z 2 N(0,1), Q is a

cumulative distribution function of the standard normal. :1
4. FINITE SAMPLES
4—1. DISTRIBUTIONS OF ESTIMATORS IN FINITE SAMPLES

To get some idea of the distributions of estimators in finite samples, TABLE
2 and TABLE 3 give some Simulated results.

For each row in the TABLE 2 and TABLE 3, 1000 samples of size n (n=40,
100) were generated from N(0,1) with a shift 0(0 = 1.0) at rc{n/4, n/2} and e = ‘
.05. The percentiles of rm, and the sample mean i and standard deviation s of

an are given .

 

TABLE 2
Distributions of estimators for n=40

 

Percentiles of in A011
7 0 10 25 50 75 90 E S
10 1.0 5 7 9 13 24 .942 .479
20 1.0 11 16 19 20 25 .959 .376

 

 

 

30

 

TABLE 3
Distirbutions of estimators for n=100
Percentiles of :rn A011
7 0 10 25 50 75 90 SE S
25 1 .0 20 24 25 27 33 . 969 .242
50 1.0 45 48 50 52 56 .972 .238

 

 

 

From the tables, we see that ;n is almost median unbiased in
the sense that the median of the distribution will be equal to the true
value of the parameter. Looking at 3n, 0 is understimated but A011
is getting close to 0 as 11 increases regardless of T. For any case
of n, A011 has smaller variance when r=n/2.

The comparision of our test and other tests is given in the next section.

4—2. COMPARISON

We will compare the test M; with Schectman's test (1983). To compare
them, their empirical powers are computed.

Schectman's test (1983) is pr0posed as follows: let Wi be the
Mann—Whitney

statistic for two samples X1,...,Xi and X

[(W - -1/2)/i(n-i)l
Vi = 1 1/2 , and the test statistic is defined by
[( n +l)/12i(n—i)]

i+1’”"Xn of Sizes i and n—i,

 

Vz= Max IViI.
IS iSIl—I

 

31

For each row of TABLE 4, 1000 samples of size n (n=40) were generated
from normal and double exponential with 0(0=1.0). Let HS and IIk
denote empirical powers at a = .05 of Schectman test and M; test
respectively, that is, the pr0portion of samples of 1000 for which H0 was

rejected.

TABLE 4
Empirical powers 0 f M161 and Schectman for n=4—0

 

 

 

T 0 HS Hk
10 1.0 .301 .303
20 1.0 .405 .405 from NW)
1' 0 HS IIk
10 1.0 .494 .494
20 1.0 .683 .676 from dOUble

exponential

 

To compare M; with Schectman's, see TABLE 4.

For any case of T, Schectman's test and the MI: test have almost the same power.

32

REFERENCES

Anderson, T.W. (1962). On the distribution of the two—sample Cramer—von
Mises criterion. Ann. Math. Stat. 33, pp. 1148—1153.

Billingsley, P. (1968). Convergence of probability measures. Wiley.

Csorgo, M. (1983). Quantile processes with statistical applications. SIAM.

Hawkins, D.L. (1985). A simple least squares method for estimating a
change in mean. Tech. Report, Univ. of Texas at Arlington.

Hawkins, D.M. (1977). Testing a sequence of observations for a Shift in

location. J_AS_A, 72, pp. 180—186.

Hinkley, D.V. (1969a) Inference about the intersection in the two—phase
regression. Biometrika, 56, pp. 495—504.

Hinkley, D.V. (1969b). On the ratio of two correlted normal random
variables. Biometrika, 56, pp. 635—639.

Hinkley, D.V. (1970). Inference about the change—point in a sequence of
random variables. Biometrika, 57, pp. 1—17.

Hinkley, D.V. (1971). Inference about the change—point from cumulative
sum tests. Biometrika, 58, pp. 509—523.

Hinkley, D.V. (1980). Change—point problems. Tech. Report, Univ. of
Minnesota.

Hinkley, D.V. and Hinkley, EA. (1970). Inference about the change—point
in a sequence of binomial variables. Biometrika, 57 , pp. 477—488.

Hodges, J .L. and Lehmann, EL. (1962). Estimates of location based on
ranks tests. Ann. Math. Stat., 34, pp. 598—611.

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

33

Moustakides, G.V. (1986). Optimal stopping times for detecting changes in
distributions. Ann. of Stat., 14, pp. 1379—1387.

Pages, ES. (1954). Continuous inspection schemes. Biometrika, 41, pp.
100—114).

Pages, ES. (1955). A test for a change in parameter occuring at an unknown
point. Biometrika, 42, pp. 523—527.

Pages, ES. (1957). On problems in which a change in a parameter occurs at
an unknown point. Biometrika, 44, pp. 248—252.

Pettitt, A.N. (1979). A non—parametric approach to the change—point
problem. Ann. of Stat., 28, pp. 126—135.

Pettitt, A.N. (1980). A Simple cumulative sum type stastistic for the
change—point problem with zero—one observations. Biometrika, 67,

pp. 79—84.

 

Pollak, M. (1985). Optimal detection of a change in distribution. Ann. of
m 13, pp. 206—227.

Schectman, E. (1982). A non—parametric test for detecting changes in
location. Comm, Stat., Theory, Meth., 11,(13), pp. 1475—1482).

Schectman, E. ( 1983). A conservative non—parametric distribution—free
confidence bound for the shift in the change—point problem. @m_m_.,
Stat., Theory, Meth., 12(21), pp. 2455—2464.

Shorack, and Wellner, (1986). Empirical processes with applications to
statistics. Wiely.

Sen, P.K. (1980). Asymptotic theory of some tests for a possible change in
the regression slope occuring at an unknown time point. Z. Washr.

Verw. Gebiete 52, pp. 203—218.

 

 

34

(24) Sen, P.K. (1982). Tests for change points based on resursive U—statistic.
Seg. Anal., 1, pp. 263—284.

(25) Srivastava, M. and Sen, A. (1975). On tests for detecting change in mean.
Ann. of Stat., 3, pp. 98—108.

(26) Wolfe, DA. and Schectman, E. (1984). Nonparametric statistical procedures
for change—point problem. J_1)_u_r_n_a_l of Statst. Planning and Inference,
9, pp. 389—396.

 

3.“),,‘.,,...._.,.,. . ,.......,._. ,. .. ... - . . ..- W .H .

ATE U

Mllllllllllllll 1| 111111 ||11111111111111111111111111ES
3 1293 03062 2587