BIOMATHEMATICS TRAINING PROGRAM
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•
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f
ASYMPTOTIC THEORY OF SOME TESTS>
FOR A POSSIBLE CHANGE IN THE REGRESS10N
SLOPE OCCURRING AT AN UNKNOWN TIME POINT*
by
Pranab Kumar Sen
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1179
JUNE 1978
Ie
ASYMPTOTIC THEORY OF SOME TESTS FOR A POSSIBLE
CHANGE IN THE REGRESSION SLOPE OCCURRING AT AN UNKNOWN TIME POINT*
by
Pranab Kumar Sen
University of North Carolina, Chapel Hill
Abstract
Based on least squares estimators and aligned linear rank statistics, some
testing procedures for a possible change in the regression slope occurring at an
unknown time point are considered.
..
The asymptotic theory of the proposed tests
rests on certain invariance principles pertaining to least squares estimators and
aligned rank order statistics and these are developed here.
AMS 1970 Classification Nos:
60FOS, 62FOS, 62GlO
Key Words & Phrases: Asympttoic efficiency, asymptotic tests, Brownian bridge,
invariance principles, least squares estimators, rank order statistics, regression coefficient, transition point .
..
*Work partially supported by the National Heart, Lung and Blood Institute, Contract
NIH-NHLBI-7l-2243 from the National Institutes of Health.
-2~
1.
Introduction.
Parametric as well as nonparametric testing procedures for poassible shifts
in the location of a distribution function (df) occurring at unknown time points
between consecutively taken observations have been proposed and studied by Page
(1955), Chernoff and Zacks (1964), Kander and Zacks (1966), Mustafi (1968),
Bhattacharyya and Johnson (1968), Sen and Srivastava (1975) and Sen (1977), among
others.
The object of the present investigation is to consider a related problem
of regression where a change in the regression coefficient may occur at a unknown
time point and to develop suitable testing procedures.
Given the observations on independent random variables (rv)
tl:S ... :s t n with at least
one strict inequality sign), consider the following regression model:
i = 1, ... ,n,
.,
Y.1 = Y(t.),
1
taken at time points
Yet) =
(1.1)
0.+
t l ' ... , t n
I(t<T)a(t-T)
(where
+ I(t~T)y(t-T) +
e(t) ,
...
where
a., a, y
and
Tare unkn01JJn parameters
the indicator function of the set
real
e,
A and e(t)
(tl:S T :S t n ), I (A)
stands for
is a white noise i.e., for every
the df
F(e) = P{e(t) :S e}
(1. 2)
Note that if
= t 1 or
T
t
n
does not depend on
,
t (tl:S t:S t n ) .
then (1.1) reduces to a simple regression model,
while for
t <T<t
and a ~ y, it relates to a segmented regression model with
l
n
a common intercept (a.) at time point T and two different slopes a and y
..
for
t < T and
t
~
T,
respectively;
T is termed a transition point.
We assume
that
< T< t
(1. 3)
t
Then, under
(1.3)~
l
n
while
a
and
y mayor may not be equal .
(1.1) relates to a simple regression model only when
a
= y.
-3-
Such a segmented regression model is not very uncommon in practical problems.
If we let the
t.
1
stand for the
do8~8
Y.1
of a drug and the
for the responses,
such a segmented dose-response regre88ion also arises in some problems, where, at
a higher dose, the regression pattern may differ from the one at a lower dose.
We desire to test for
HO: B =y
(1. 4)
treating
a, Band T
vs.
HI: B > Y or H2 : B;e y ,
as nuisance parameters.
could have considered the two samples
If T were specified, one
{Y.: t. < T}
1
1
and
{Y.: t.
1
1
~ T}
(which
are independent) and, bearing in mind the two simple regression models pertaining to these samples, one might have tested for the identity of the two slopes
.
•
Band y.
Since
T
is not specified, our problem is somewhat more complicated .
Moreover, we do not assume that
F in (1.2) is of a specified form (e.g., nor-
mal), and, for this reason, we take recourse to tests based on rank statistics
and the classical least squares estimators.
In Section 2, along with the preliminary notions, the proposed test statistics are formulated.
Some invariance principles for least squares estimators
(LSE) are considered in Section 3 and also these are incorporated there in the
study of the asumptotic properties of the tests based on the LSE.
Similar
invariance principles are developed for (aligned) linear rank statistics (LRS)
in Section 4 and these are utilized then in the study of the asymptotic properties of the proposed rank tests.
Section 5 deals with the asymptotic comparison
of the procedures based on LSE and LRS.
-4-
2.
Preliminary notions and the proposed tests.
Let us define
-t = Lk -1
k
(2.1)
Note that
the LSE of
k
. It.
1= 1
T~ is ~ in k(~l). Under HO in (1.4), based on Yl, •.. ,Yk ,
B is
for
(2.2)
If we assume that
Ho in (1.4),
cr 2/T 2 ,
..
t mS T
•••
<
A
B2 , ••• ,Bn
,cr 2/Tn ,
t m+ 1
in (1.2) admits of a finite variance
F
A
B
are all unbiased estimators of
respectively.
for some
k = 2, ... ,n; Sl = 0 .
On the other hand, if
m: 1 S mS n-l,
2
cr ,
then under
with variances
O does not hold and
H
then
(2.3)
m+l
where the right hand side (rhs) of (2.3), for
k > m,
k Sn ,
differs from
Thus, the estimators cease to fluctuate around a common
hold.
S
B
when
B (or
HO does not
We consider the residuals
A
A
Y.=Y.-St.,
1
1
n 1
(2.4)
and based on the partial set
A
i=l, ... ,n
A
{Yl, ... ,Yk },
we compute
o.
(2.5)
HO' Sl"",Sn al unbaisedly estimate 0, while under HI or H2 ,
are not so. Our proposed test based on the LSE rests on the statistics
Under
(2.6)
y).
M+ =
n
max
S
Osk:s:n n,k
and Mn
max ISn,k I
= O:s:k:S:n
'
they
-5-
where
Sn,O = Sn,l = 0 and
1 2~
-1 2 ~ ~
n, k = T-n TkSk = Tn Tk (Sk-Sn)' 2:;; k :;; n ,
(2.7)
S
so that
Sn,n
= O.
The test procedure will be formulated in Section 3.
For the rank tests (to follow), we do not need the existence of the second
moment of
that
F.
However, to avoid ties among the observations
F is continuous everywhere.
(Y.),
1
we asswne
Consider the usual LRS
l, ... ,n,
(2.8)
where, for every
the scopes
~(i)
k(~l),
~i
= pa:nk of Yi
are defined by
among
Yl'''''Yk
for
l:;;i:;;k,
(2.9)
e
Ukl < ••• < Ukk are the ordered rv' s of a sample of size k from the rectangular (0,1) df and the scope genepating fUnction ~ = {~(u), 0 < u < I} is
assumed to be square integrable inside
(0,1).
Actually, bearing in mind, the
elimination of the nuisance parameters through estimation, we assume that
~(u)
(2.10)
For every real
b: -
00
<b <
is.?'
00,
in u: O<u<l .
let
(2.11)
where
Rki(b)=rankofYi-bt i
among
Yl-btl""'Yk-btk
[cf. Theorem 6.1 of Sen (1969)], under (2.10), for every
(2.12)
'.
Let then
(2.13)
L (b)
k
is
\r
in b: -
00
< b <
00
•
for
l:;;i:;;k.
k(~l),
Then
-6-
Sk*
(2.14)
Under
H
in (2.14),
O
estimator of
S
=
S~
* + Sk,2),
*
2"1 (Sk,l
k
=
1'00 .,n
is a translation-invariant, robust and consistent
k ~ 1.
[viz., Adichie (1967)], for
As in (2.4), we consider
the residuals
Y~
(2.15)
1
= Y.1
and based on the partial set
L*
(2.16)
n,k
- S*t.
n 1
for
{y~, ... ,Y~},
i
= 1, ... ,R
we define
= Lk(S~)/Tn
=
*
Tn-l~k
l.1= l(t.-tk)ak(R
1
k1.), k
= 1, . oo,n ,
* = rank of y~1 among
for
Conventionally,
Rki
we let L*n,O = 0 . Then, parallel to (2.6), our proposed test statistics are
where
(2.17)
and Dn
-l{O<k<
max
* k I}
__n ILn,
= An
where for n > 2,
(2.18)
and
-
an
=-n1
.
~n
l. la (1) •
1=
n
The test procedure will be formulated in Section 4.
3.
Asymptotic properties of the tests based on M+
n
and
M.
n
For the study of the asymptotic distribution theory of M+n
and Mn
(under the null as well as local alternative hypothesis), we need to study
first some invariance principles relating to the LSE.
study, we consider a triangular array
and
{Y.
1
For this asymptotic
of time-variables
= yetn1.), 1 sis n; n ~ l} are defined accordingly as in
assume that for every
e:
0 < e s 1,
(1.1).
We
-7-
lim = lim [na]~I~~n6]t . = .(8)
n-+oo t n [ne] . n-+oo
L.1.=1 n1
1.1
(3.1)
2
(3.2)
lim
lim { -lr[n8]
n-+oo (Tn [n 6]/n) = n-+oo n L'1= 1 (tn1.-tn [e])
n
and both
1.1(6)
~(e)
and
are continuous inside
exists ,
2} = ~ 2(8)
exists
[0,1].
Note that (3.1) and
as
00
(3.2) insure that
maX{T-llt
nn n1.-tnk l :
(3.3)
Let then
UnO = Unl = 0
and for
(3.4)
Unk
and define a stochastic process
(3.5)
n -..
•
k~2
~k
-
= L..1= l(t n1.-t nk)Y.1
w(l) = {w(l) (u), 0 ~ u ~ I}
n
n
be letting
W~l)(U) = Uk (u/(IilO"~(I)), kn(u) = max{k: T~k~UT~n}, O~u~l .
n
Note that
k (u)
n
of u € [0, 1]
is a nondecreasing, right-continuous and integer-valued function
W(l)
and
belongs to the
n
0[0,1]
space (having only jump discon-
tinuities) endowed with the Skorokhos Jl-topology.
a standard Wiener process on
o
w(l) -.. W
n
L '
ul
00
and (3.1)-(3.2) hoZd, then under
in the Jl-topoZogy on
0[0,1] .
< ••• <
urn
{W(l)}
n
~
1,
let
to those of W.
For every (fixed)
k j = k Cu.), 1 ~ j
n
J
~
m(~l) and
m and for an arbitrary
k ..
(3.7)
be
First, we establish the convergence of the finite-dimensional distribu-
tions (f.d.d.) of
o~
O~u~ l}
= y = 0,
(3.6)
Proof.
W= {W(u),
Then, we have the following.
If Ee(t) = 0, Ee 2 (t) = 0"2 <
Theorem 3.1.
H* : a = S
[0,1].
Let
r
~ I ~m
~ J
Z = ~m d wCl)Cu ) = CynO"~CI))
L.j=ldjL.i=ICtni-tnk.)Yi
n
L.j=l j n
j
J
let
-8-
e
where
(3.8)
f
n
I'
= L~-ld j [(t .,.t" k ) I (i s k,)], 1 sis n .
Jnl n j
J
It follows by some routine steps that
t~ Ifnl. = 0
ll=
and
(3.9)
1
n ~ [max{ If,
nl I: 1 sis n}]
(3.10)
'+
0 as
Note that the rhs of (3.9) equals the variance of
*'
HO
the
Yi
n
-+
00 •
L~=ldjW(uj)' Further, under
are independent and identically distributed (i.i.d.) rv's with
2
mean and variance
0 •
Hence, using (3.10) and a special version of the central
limit theorem in Hajek and ~idak (1967, p. 153), it follows that
Zn
is asyrnp-
This establishes the convergence of the f.d.d. 's
totically normally distributed.
.,
0
of {WeI) } to those of W. It remains to show that {well}
is tight. ~ince by
n
n
definition, W(l) (0) = 0, with probability 1, it suffices to show that for
n
every e: > 0 and n > 0, there exist a 0: o < tS < 1 and an nO' such that for
and every k: 0 s k s n- [ntS], q = k + [no],
p{k<msq
max IUm- Uk I > e:0~(1);n} < no
(3.11)
see Theorem 8.3 of Billingsley (1968, p. 56).
os
;
For this note that for every k,q:
k < q s n,
tm
max n ~IU
-U I < It"nq -t"nk In ~{max
k<msq
~ m k k<msq Ili=k+l Yi
+n
(3.12)
= C(l)
kq
I}
~{k~:~qIt"nk-t"nmll r~=l Yi I} + n ~{k~:~q IL~=k+l (tnk -t"nq)Yi I}
+
C(2)
C(3)
• kq + kq
By the nondecreasing nature of
say .
t
nl.
(in
i),
-9-
max ,I=
t -t
nm nk
(3.13)
k<m~q
limitn[ns] -tn[ns+no] I
..
n-.oo
Y.
Further, the
0
nk
o '"
as
are LLd.rv's with
l.
nq
'
1l(8), 8E [0,1],
where by (3.1) and the continuity of
(3.14)
1-t -t- I
0
VSE[O,l)
0,
mean and variance
Donsker Theorem [cf. Billingsley (1968)], for every
0: 0 < 0 < 1
exist a
q
e
=k +
and an
nO'
such that for
P{ n
~ kmax
<m~q Ilo~mi =k + 1Yi I>·
E
(3.16)
P{ n
~ I:s;k~n
max I~k
loi=l Y
Kn,o
i
is a positive number
«
> 0, n ' > 0
and
p{e(l) + e(2)
kq
kq
(3.17)
where
0: 0<0 < 1,
0(> 0)
nO
and every
> 0,
there
k: O:s; k
~
n- [no] ,
(0),
depending on
E
<
!.2
no·
'
EI
> 0
and every
such that for
I
< no
,
> 0
n'o.
n > 0,
and
n' =
and
Thus, it follows
there exist
q - k = [on], n
!.
n E
4' '
limit -t klK I~ < E/4.
n-.oo nq n
nu
it suffices to show that for every
and an
I
< no,
such that for
>!.2 E}
is so small that
}
I > Kn,o }
from (3.12) through (3.16) that for every
E'
n'
> 0 and
E'
~
so that by the
[no],
(3.15)
where
n
(J 2 ,
n I > 0,
k: 0
~
k
~
~
nO'
= E/4
Hence, to prove (3.11),
there exist a
0: 0 < 0 < 1
n-[no], q = k+[no],
(3.18)
V. = (t .-tu )Y., k~i~q, then the V. are independent, EV. =0,
nl.
nl. nq l.
nl.
nl.
~q
2 _ 2~q
2 _ 2{ 2
~k
2} _ 2{ 2
2
2}
lo· k ·lEV . - (J lo· k 1 (t . -t ) - (J T
- (J Tnq -Tn k-k(t n k- t nq )
l.= +
nl.
l.= +
nl. nq
nq - lo·l.= 1 (t nl.. -tnq )
If we let
~ (J2(T~ -T~k) ~ n(J2~2(*)
(1n(J)-lr~q
llo s =k + IVns )
-
s2(~U' {L~=k+lVns' i ~k+l}
is a martingale and, finally,
is asymptotically normally distributed with
0
mean and variance
-10-
_
~
[~2(S-)
n
_
~2(k1
n·
_
(k/n)~(~)
~ n
_
1.l(~)12).
(s;[~2(q/n)
n~
_
~2(k/n)]).
Hence, (3.18)
follows by using Lemma 4 of Brown (1961) and proceeding as in the proof of
Q.E.D.
Theorem 3.
Let us now consider the process
where
W(2)(u) = S k ()/o, O~u~l ,
n
n,
u
(3.19)
n
k (u) is defined by (3.5) and the S
Note that in (2.7) and, elsen
n,k by (2.7).
where, we replace the t. by t . and T2 by T2 , 1 ~ k ~ n. Also, let wO =
1
nl
nk
k
O
{w (t) = Wet) - tW(l), O~t~l} be a standard Brownian bridge on [ql] . Note
~
Bk are invariant under shift and regression i.e.,
that by definition in (2.5), the
if we work with
Y.1 - a - bt.,
1~ i
1
~
the ones in (2.5) for every real
n,
(a,b).
n
*'
HO
H in (1.4) will be the same as under
O
~
B will be the same as
k
Hence, the distribution of w(2) under
then the resulting
Further, by definition,
(3.20)
Also, note that
T~O
=
222
(3.21) (Tnk+ 1- Tnk)/Tnn
while by (3.2),
=
0,
while for
k ~ 1,
= [(k+l)/k][t nk+1- -t nk+1] 2/Tn2
I:n~(l)/T
n
+
1 as
n
+
00.
+0,
sup{ IR (u) I: 0 ~ u ~ l}
n
the following.
+
O.
by (3.3) ,
Hence,
W(2) (u) = (I:n~(l)/T )[w(l) (u) - uw(l) (1)]
n
nn n
n
(3.22)
where
T~l
+
R (u)
n
,
From Theorem 3.1 and (3.22), we arrive at
-11-
Theorem 3.2.
2
= 02
Ee (t) = 0, Ee (t)
If
< 00
and (3.1)-(3.2) hoZd~
then under
"0 in (1.4),
w(2 )
(3.23)
wO
L·'
-+
n
in the Jl-topoZogy on D[O,l] .
For the standard Brownian brigde
(1968, p.85)] that for every
t
~
it is well known [viz., Billingsley
0
(3.24 )
I
P{ 0 SUP
u 1 W0 (u)
(3.25)
I~ }
t =
k
2 2
l-2l.~oo
k =1 (-1) exp(-2k t ) ,
where the rhs of (3.25) is bounded from below by
e
2
1-2 exp(-2t )
and is practi-
On equating the rhs
cally equal to this lower bound when
t
is not very small.
of (3.24) and (3.25) to
£
is the desired level of significance
(0 < £ < 1),
1-£,
where
we denote the solutions by
l1;
and
l1£
respectively.
Then, by
Theorem 3.2, we have on noting that
M+/a
n
(3.26)
=
sup
O~u~l
w(2)(u)
n
and
M /0
n
sup Iw(2)(u)
n
=
O~u~l
I '
(3.27)
and hence, the asymptotic critical values of
of significance
£: 0 < £ < 1)
are
011
+
£
and
M+
n
and
0l1,
M
n
(at the desired level
respectively.
£
Thus, if
is specified, we have the following test procedure:
Compute the
(3.28)
Sn, k' k:sn,
l:Sk :sn-1, S
"0
n,k
in (1.4).
(or
defined by (2.7).
ISki)
n,
If, no such
exceeds
k
If, for at least one
A+
Ot.l£
exists, accept
"0'
k:
0
-12-
In
practice~
mostly,
a
is not specified,
(3.29)
(where
Under
H
O
A
an;
V
n
=!n L~1= lv.l
1)
~n ~ a. As such, we may proceed as in (3.28) where we replace a
the asymptotic level of significance remains equal to
Let us now consider the behavior of these tests when
hold.
H
O
E.
in (1.4) does not
Suppose that (1.1) holds with
(3.30)
Let
.
in (1.4), we have a simple regression model, and hence [viz., Sen and
Puri (1970)],
by
We consider the estimator
= t m for some m:
T
V?
1
= V.1
+
o (t. -t ) I (t
1
m
I
<m~n-1
and
B~y
=0
0) .
(;t
~
t ), i = 1, ... ,n and in (2.2), (2.4), (2.5) and
m
o
by V. and denote the resulting quantities by
(2.7), we replace the Y.
1
1
"0 AO .... 0
o
Sk' Yi , Bk and Sn, k' respectively.
Then, we have by some direct computations
that
(3.31)
where
2
22·
-2
S k~o(Tk/T ){l-T /T ~m(t ~t ) (t· -t )/T }, k
n,
n
m n
m m n m n
(3.32)
2
2
m T2]
m
T
S
-o(T /T). -2- - -met -t)
n, k
k n
T T2
m m
{!
k
n
~m
,
~ nm
-to - t km
-t~} ,
T2
T2
n
m<k~n
k
is bounded away from 0 and 1, then under (3.1), (3.2) and
0
(3.30) , max{S k- S k: l~k~n}-+oo as n-+ oo when 0 > 0 and max~ S k-S k I:
n,
n,
n,
n,
0
1 ~ k ~ n} -+00 as n-+ oo when o ;t O. On the other hand, for the V. , the simple
As such, if m/n
"
regression model holds, so that Theorem 3.2 applies to the
o
Sn, k'
1
and hence,
-13-
max{S
0
n1
or max{ Is
k; 1::; k ::; n}
1 ::; k ::; n} -+
00
O
n,
k I: 1::; k ::; n}
1 in probabi lity, if
probability, if
<5
~
O.
and
> 0
<S
is
0p(l)~
Thus,
max{Sn,k:
max{IS k l : l::;k ::;n} -+
n1
00
1 in
Hence, the tests based on (3.28) are consistent.
In view of the consistency of the tests 1 for the study of the asymptotic
power properties, we confine ourselves to a suitable sequence
{H}
n
hypotheses for which the asymptotic power is different from 1.
of alter-
Keeping (3.1)-
(3.2) in mind, we assume that
T=t
(3.33)
where
m
min
2 2
0< V < 1 1 T /T
-+ V:
m n
-+
p: 0 < p < 1 •
ActuallY1 if we let
2
p = l;2(V)/l;2(1)
then
2
h (u) = inf {t: l; ( t ) / l; ( 1)
(3.34)
and
H : S = y +T- I <5
n
n
(3.35)
Let us then define
~(v) = ~(h(p)).
for some real
a<5 = {a<5 (t), 0::; t ::; I}
~
u L 0::; u ::; 1 1
We consider then
<5,
where (3,3) holds
by letting
<5t[l-P-V(T-~(V))(~(I)-~(V))/l;2(1)]icr,O::;t::;p ,
(3.36) a<5(t) =
1
cr
<5[p(l-t)-V(T-~(V)){(l-t)~(v)-~(h(t))+t~(I)}/l;
In (3.32), replacing
<5
by
ofTn
2
(1)], p<t::;l .
and then using Theorem 3.2 (for the
along with (3.5) and (3.33)-(3.36), we arrive at the following.
Theorem 3.3.
then under
If Ee(t)
{H}
n
= 0,
2
Ee (t) =cr
2
<
00
and (3.1), (3.2) and (3.33) holds,
in (3.35),
in the J1-topology on
(3.37)
D[O,l] .
By virtue of Theorem 3.3, the asymptotic power of the test based on M+ 1
n
under
{H}
n
in (3.35)1 is given by
-14-
(3.38)
+
~
and for the test based on
M,
n
for some
Ae: I
t
E
(0,1) 1 ,
the corresponding expression is
(3.39)
Since
a
<5
is not a linear (in
t) boundary (in general), closed expression for
(3.38) and (3.39) are not generally available.
4.
Asymptotic properties of the tests based on
D+
and
n
D.
n
We need to develop some invariance principles for aligned LRS for the study
of the distribution theory of
(3)
(4.1)
Wn
and
W~3) =
defined by (2.8), and define
e
D:
(u)
D . Consider first the case of
n
{W~3) (u), 0 SuS I} by letting
{Lkl,
-1 -1
= Tn
An Lk* (u )' 0 SuS 1
n
(4.2)
2
2
r n and A are defined by (2.1) and (2.18), respectively.
where
n
belongs to the
W(3)
n
Theorem 4.1.
grabZe
~,
space.
Then, we have the following.
For scores defined by (2.9) with nondecreasing and square inteunder (3.3) and
W(3) -+- W
(4.3)
n
L
'
O: 6
H
= Y = 0
(refer to (1.1)),
in the JI-topoZogy on
D[O,l] .
* ' the Y. are i.i.d.rv with a continuous df
HO
1
(4.3) directly follows from Theorem 2.2 of Sen (1975).
Proof:
Ow
D[O,l]
Here also,
Since under
We proceed on to the case where
(i) the df
H~
may not hold.
F(x-a),
Here, we assume, that
F admits of an absolutely continuous probability density function
~15~
~
(pdf)
f
with a finite Fisher information
I(f) = f[f.(x)7f(X)]2 dF (X)
(4.4)
«00).
_co
Also, let
ljJ(u) = -f'(F~l(u)/f(F-l(u)), O<u<l ,
(4.5)
2
A •
(4.6)
f>2(U)dU - U>(U)dUr
and A(¢,¢) • U:¢(U)¢(UJdU)/AI"(f) .
We assume that
(4.7)
Further, we assume that (3.1), (3.2) and (3.33) hold and consider a sequence
{K*
n,.!2 }
of alternative hypotheses where
(4.8)
and
b ,b
l 2
are real numbers.
Theorem 4.2.
Under
n
w
= {w(u),
(4.10) w(u)
where
and the assumptions made above,
{K* b}
n,....
W(3) -+ W + w
(4.9)
where
Then, we have the following.
L
0 Sus l}
' in the Jl-topoZogy on
0[0,1] ,
is given by
=
v, p, s(l)
and h(u)
are defined as in (3.1), (3.2), (3.33) and (3.34).
'.
Proof.
Let us denote the joint distribution of
so that
P
n,Q
(Y1, ••• ,Y )
stands for the null hypothesis
n
(H*)
o
under
case.
K*
n,£
by
Then, by an
'Il
appeal to the basic results in Hajek and Sidak (1967: Ch, VI), we conclude that
{Pn ,£}
under the hypothesis of this theorel1\.
(4,3) insures the tightness of
(3
W )
under
n
is aont~guous to
{p
{K*
n,£ } as well. Hence, to prove
vergence of f.d.d,'s of
Also,
a}. Hence, proceeding as in
n, ...
the proof of Theorem 2 of Sen (1976), we conclude that
under
{Pn,Q}'
(4.9). it
W(3)
suffi~es
{W(3) } to those of W + W,
n
remains tight
n
to prove the conholds.
when
Towards this, we define
-l\k
S*
T i.' l(t .-t k)<P(F(Y.-ex)), 1
n1 n
1
n.k = nn 1=
(4.11)
Then, for every (fixed)
by (4.2),
1 ~j ~m,
mC~
1)
and
0
~ul
< .,. <urn
~k ~n
~
1,
.
defining
k. = k*(u.)
J
i t follows by an appeal to Hajek (1961) that under
f3 = y = 0 T-lL = W(3) (u.) is equivalent in quadratic mean to
• nn k.
n
J
By contiguity ~f {p b} to {p O}' Iw(3) (u.) - s* k I ~ 0, V
n,...
n,
n
J
n, j
{K* } as well. Finally. under {K* b}' the asymptotic joint
n,E
n,,..,
S*
n , k. '
1
~ /~ m
J
n
H~:
V 1 ~ j ~ ID.
under
normality of
(s* k •...• S*·k ) can be derived by an appeal to a theorem on page 216 of
n, 1
v n, m
Hajek and Sidak (1967). In view of the similarity of this proof with that of
Theorem 3.1 of Sen (1977), the details are omitted.
For every real
d;
let us define
(4.12)
where the
Lk(b)
are defined by (2.11).
Also, in (4,1), replacing the
W(3)
by ~*(u)(d), we define the corresponding stochastic process by
n,d
fi(3 ) (~). 0 ~ u ~ l}. Thus. w(3) = w(3 ).
n. d
n
n, 0
Theorem 4.3.
Under
every (fixed)
d: -
n.£
00
< d <
00,
n. d L
wet)
=
n
} in (4.8) and the hypothesis of Theorem 4,2, for
W(3 ) -+-
(4.13)
~here
{K*
Lk*(u)
{W(t)-tw(t)-dA*t, 0 ~t ~ l}
is defined by (4.10).
subsets
among
,
d
) under {K* b}' de;fined for Borel
ITn,...,b be the distribution of W<3
n,d
n , ...,
V of 0[0,1], Note that by denoting by R~k(d) ~ rank of Yi-dti/T
n
Let
Yi-dt/Tn' .• qTn . . dtn/Tn'
for
i
c
1,. .. ,n,
~(d) c
(R~1(d),,,.,R~n(d))~,
it follows that
(4.14 )
[R* (d),
under
"'fi
and hence, for every
K*
n,12
] L [R* (0)
K~ b-d1] ,
under
1
"'fi
'...,
...,
0 € V,
(4.15 )
TId b(O)
n,...,
=
p{w(3)
n,s!
-(3)
= P{W
n 0
,
E
OIK*
E
0 K~ b-dl
n,12
}
I
',..""
}
t"'W
o
= TIn '...,b_d1(0)
...,
which insures that
n~{3)
n,d'
under
(4.13) follows from Theorem 4,2.
Under
Theorem 4.4.
£
fixed
and
under
K~ b-d1}'
'..., ...,
and hence,
Q.E.O.
n,Q } in (4.8) and the hypothesis of Theorem 4.2, for every
{K*
00),
sup{\w(3)(t) _W(3)(t) +A*dtl: O~t~l, Idl ~K}
n,d
n
(4.16 )
Proof.
K«
K* } ~ {W(3)
n '...,
b
n'
&0
•
We may virtually repeat the proof of Theorem 3.3 of Sen (1977).
We note
that (3.24) and (3.25) of Sen (1977) hold here, by virtue of the basic result of
v
Jureckova (1969) and our Theorem 4.3 here.
Hence, the details are omitted.
We are now in a position to study the invariance principles for our aligned
LRS{L* . 0
n,k'
'.
(4.17)
£ = (b l ,b 2)
o~
~
k ~ nL
Consider a sequence
K b: (1.1) holds with
n,...,
(~Q).
{K
b}
n,...,
of alternative hypotheses, where
a
Lk~(u)
Further, in (4,1), we replace the
u ~ 1 and denote the resulting process by
W (4)
n
=
by
{W~4)(u),
L*
n,k~(u)'
o~u~l}.
~18~
Note that
1 ~k ~n}
under
under
K~ b
'......
as. the same di.stribution
{K* b}
n, . . .
as
Hence, for the study of the distribution of w(4)
Kn ,b'
under
n
we may, without any loss of generality, set [in (4.17)],
{Kn ,£}'
under
{~(J)+a), 1 ~k ~n}
in (4.8).
By
a
=0,
Le. ,
virtue of (2.13), (2.14), Theorem 4.4 and the above
discussion, it follows by some standard steps that under
{K* b}'
n,.. . .
A*Tnnn
B* = W(3)
0
n (1)
u +p
.(1) ,
(4.18)
so that by Theorem 4.2,
(4.19)
IT
B*I
nn n
= 0 p (I),
under
{K; b} •
' ...
By Theorem 4.4, (4.18) and (4.19), we obtain that under
{Kn,Q}
under
1
~
i
~n,
as
W~4) remains in variant if the Yi
for any real
{K* } (or equivalently,
n,b
are replaced by Y. - at. ,
1
1
a),
(4.20)
so that, if we define
{K
n,Q
}
W(S)
n
(u) = W~3) (u) - uw~3) (1) ,
= {W(S)
n
and
w(S)
n
,
w(4)
n
(4.21)
are convergent equivalent .
Finally, by an appeal to Theorem 4.2, we conclude that under
W(S) -+ wo + w0 ,
n
L
(4.22)
where
o~u~l},
Wo
in the Jl-topol,ogy on
is standard Brownian bridge on
[0,1]
and
{K*n, .b}'
..
D[O,l] ,
then under
-19",
where
s2 ,
lJ,
T, P
and
h(t)
are de;fined by (",1)
J
(3,2), (3,33) and (3,34).
This leads us to the main theorem of this section.
Theorem 4.5.
Under
{K
b}
n, ....
and the hypothesis of Theorem 4.2,
in the J -topology on
(4.24)
1
O
w
where
= {wO (t),
0
D[O,l]
~ t ~ I} is given by (4.23).
We notice that by definition in (2.17),
(4.25)
and
Hence, by Theorem 4.5 and (4.26), under
D
n
=
sup Iw(4)(t)
O~t~l
H : S=y
n
I
+
i,e.,
Kn,....0' Dn and Dn
have the asymptotic distributions given by (3.24) and (3.25), respectively.
"
O
This leads us to the following test procedure:
Compute the
(4.26)
L*n, k'
(or
n,k
in (1.4).
defined by (2.16).
exceeds
L*
If, no such
k
A 6.+
n e:
If, for at least one
(or
A 6. ),
n £
exists, accept
reject
k: 1
~k ~n-1,
H '
o
H '
O
As in Section 3, we confine ourselves to local alternatives for the study
of the asymptotic power properties of the tests based on (4.26).
the same sequence
{H}
n
We assume that
of alternative hypotheses in (3.35) holds.
Then, by an
appeal to Theorem 4.5, we conclude that under the regularity conditions of
Theorem 4.5,
(4.27)
(4.28)
where
O
lim P{D+n >6.+IH}
n-+<lO
£
n:Z: p{w ()
t
=Wo()
ot
>6.+£
lim P{D n >6.£ IHn } = p{lwO(t)
I >6. £
n-+<lO
. +w~(t)
v .
for some
for some
~
t
t
[0, I]} ,
E
[O,l]} ,
-20-
W~(t)
(4.29)
A*O[p(l-t)-V(T-~(V)){(I-t)~(V)-~(h(t))
=
l
Note that
°
wo(t)
5.
is linear in
is not linear in
t
t
for
~(h(t))
[as
but 1 in general 1 for
os t s P1
need not be linear in
t].
Asymptotic comparison of the LSE and LRS procedure.
It may be remarked that both
+
M /a
n
and
+
D
n
b (t) = {(t+1)
M /a
n
(or
same limiting null distribution (3.24) [or (3.25)].
(5.1)
pst S 1 1
and
D)
n
have the
If 1 we let
[l-p-V(T-~(V)) (~O.)-~(V))fI:;2(I) 1,
0 <t 2< pi (I-p) ,
p-V(T-j.l(V){j.l(v)+tj.l(I)-(t+I)j.l(h(t/(t+I)))}/~ (1)1 t ~p/(l-p)
{(t+I)WO(t~d
and note that
°St
1
+
a standard Wiener process on
R,
that the asymptotic power of
M+
(5.2)
p{WO(t)
~6; - ao(t)
n
<co} = {X(tL
°st
<OO}1
is equal to
for some
~ (t+I)6;
-
t
E
[O,l]}
~ bet)
1
for some
for some
t
E
R+}
•
Similarly 1 for
(5.3)
Mn ,
{X(tL t
~O}
we obtain then from (3.35), (3.36) and (3.38)
= P{(t+l)WO(t~lJ ~ (t+I)6; - (t-l)ao[t~d
= p{X(t)
where
we have the asymptotic power is equal to
t
E
R+}
is
.,21.,
Likewise, from (4.27)-(4.29) and (5.1), we have the asymptotic power of the
0+
n
test given by
p{X(t)
(5.4)
while for the
0
n
+
~
(t+l)ti - A.*ob(t)
e:
t
E
R }
for some
t
E
R } •
test, it is equal to
I
P{ X(t) + A* ob (t)
(5.5)
+
for some
I ~ (t + 1) ~ e:
+
Note that (5.2) though (5.5) are computed for the common sequence
{H}
n
of
alternatives [in (3.35)] when all the statistics are based on the same sample
size
0+
n
n.
or
0
n
are based on
M
and
n
Mare ba.sed on sample size
n
{N = N(n) } ,
+
ON
and
ON
statistics with
f or some
N = N(n)
as in Section 4 where we replace
W(3 )
W~5),
n =N,
respectively, defined for
K b' K* band
n,~
n
n,~
H.
n
{n}
while
where
lim N(n)/n _- e- l
n-+oo
(5.6)
For
+
Suppose now that
W(4)
'
n
0 < e <00
satisfying (5.6), we may proceed
W(5)
and
n
b
Y
W(3) W(4)
N ' N
and
while we stick to the same alternatives
Since,by (3.2) and (5.6),
(5.7)
lim
1
= n-+oo [N(n)/n] = ein (4.10), (4.23) and (4.29),
o
W(u) , w (u)
and
0
w (u)
are to be replaced by
e-~w(u), e-~wO(u) and e-~w~(u), respectively. Thus, the asymptotic power of
•
{H }
n
the test based on
in (3.33), when (5.6) holds, is given by
(5.7)
for some
Consequently, if we choose
(5.8)
e-
~
e
A*
by letting
= cr -1
i.e.
+
t ER } •
-22-
then (5,2) and (5.7) are equal (for any
Q),
Hence~
e
in the usual
Pitman~
sense, the asymptotic relative efficiency (A,R,E,) of the LRS procedure
with
respect to the LSE procedure is
(5.9)
The same efficiency result holds for
D
n
relative to
M,
n
Noew (5.9) agrees with the classical Pitman-efficiency of the two-sample
rank order test (for location) relative to the Student's t-test.
conclude that if
</> (u)
==
u: 0 < u < 1,
then the corresponding rank procedure has
an A.R.E. with respect to the LSE procedure equal to
..
Thus, we may
3/TI when
F is normal,
is bounded from below by 0.864 for all continuous
F and is usually
~
F has havier (than normal) tails.
= q, -1
(i. e, ,
Also, if
</> (u)
normal scores), then (5.9) is bounded from below by
is attained only when
F
is normal.
(u), 0 < u < 1
1
when
1 where the lower bound
Thus, from the A.R.E. point of view, the
LRS procedures are attractive, they do not require the estimation of
cr
2
(as
is need for the LSE procedure) and they are expected to be robust.
We conclude this section with the remark that
in
t
Since
(5.10)
..
,
for
t
ll(S)
~
p/(l-p),
is /1
in
bet)
in (5.1)
is linear
while for
t > p/(l-p),
in general, it is not so.
s
we have for
t > p/(l-p)
E:
[0,1],
P-P(T-ll(V))t(ll(l)-ll(V))
~b(t) ~P+V(T-ll(V))(ll(l)-ll(V)) •
Hence, for (5.2) and (5.4), bounds for the asymptotic power can be obtained
by using the segmented linear boundaries in (5.1) and (5.10) and then using
~
the results in Anderson (1960, Section 6).
plicated to be expressible in closed form,
In general, these are quire com-
-23-
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[1]
Anderson, T.W. (1960). A modification of the sequential probability ratio
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[2]
Bhattacharyya, G.K. and Johnson, R.A, (1968). Nonparametric tests for shift
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[3]
Billingsley, P. (1968).
[4]
Brown, B.M. (1971).
42, 59-66.
[5]
Chernoff, H. and Zacks, S. (1964). Estimating the current mean of a normal
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~, 999-1018.
[6 ]
Hajek, H. and Sidak, Z. (1967).
York.
[7]
Jureckova, J. (1969).
v
~
v
~
Convergence of Proability Measures.
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Theory of Rank Tests,
Wiley, New York.
Ann. Math. Statist.
Academic Press, New
Asymptotic linearity of rank statistics in regression
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..
[8]
Kandel', Z. and Zacks, S. (1966). Test procedures for possible changes in
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[9]
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[10]
Page, E.S. (1955). A test for a change in a parameter occurring at an
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Sen, A. and Srivastava, M.S. (1975).
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[12]
Sen, P.K. (1969). On a class of rank order tests for the parallelism
several regression lines. Ann. Math. Statist. 40, 1668-1683.
[13]
Sen, P.K. (1975). Rank statistics, martingales and limit theorems. In
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[14]
Sen, P.K. (1976). A two-dimensional functional permutational central limit
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Sen, P.K. (1977). Tied-down Wiener process approximations for aligned rank
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Sen, P.K. and Puri, M.L. (1970). Asymptotic theory of likelihood ratio and
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•
On tests for detecting change in mean.
of