e'
-
"
,
'"
ASYMPTOTIC THEORY OF SOME TIME-SEQUENTIAL TESTS BASED ON
PROGRESSIVELY CENSORED QUANTILE PROCESSES*t
by
Pranab Kumar Sen
Department of Biostatistics
University. of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1267
February 1980
•
ASYMPTOTIC THEORY OF SOME TIME-SEQUENTIAL TESTS BASED ON
PROGRESSIVELY CENSORED QUANTILE PROCESSES*t
by
PRANAB KUMAR SEN
Department of Biostatistics
University of North Carolina, Chapel Hill
ABSTRACT
In the context of time-sequential tests, a general class of (parametric
as well as nonparametric) testing procedures rests on progressively censored
linear combinations of functions of order statistics with stochastic coefficientvectors.
Invariance principles for such quantile processes, developed in Sen
(1979), are extended here to more general models, and these are incorporated in
•
the study of the asymptotic properties of the allied time-sequential tests.
AMS Subject Classification:
Key Words
62L99, 62G99, 62E20,60F17.
&Phrases: Asymptotic tests, Bessel processes, linear combinations
of functions of order statistics with stochastic coefficients, progressive
censoring schemes, quantile processes, sampling from a finite population, sUbmartingales, time-sequential tests, weak convergence.
•
*Work supported by the National Heart, Lung and Blood Institute, Contract
NIH-NHLBI-7l-2243 from the National Institutes of Health.
tThis research is dedicated to Professor C.R.Rao on the occasion of his
60th birthday.
1.
INTRODUCTION
In aZiniaaZ tpiaZs and Zife testing ppobZems, usually, one draws statistical
inference from aensoped or tpunaated data.
•
I
In this context, a ppogpessive
oensoping saheme (PCS) allows a monitoring of the experimentation continuously
from the beginning with the objective of an eapZy ter.mination whenever the
accumulating outcome
ppoaedupe.
provocates so; hence, a PCS involves a time-sequentiaZ
For a simpZe pegpession modeZ (containing the classical two-sample
problem as a special case), Chatterjee and Sen (1973) have studied the general
theory of PCS testing procedures based on Zineap pank statistios.
For some
parametric models, Sen (1976b) and Gardiner and Sen (1978) have employed the
ZikeZihood patio ppinaipZe in a PCS and the resulting test statistics are based on
certain ZineaP aombinations of funations of opdep statistias.
Sen (1979) has also
shown that for a general class of location, scale and regression
mode1s~
ppogpessiveZy aenBoped Zikelihood patio statistios (PCLRS) are based on some
quantiZe ppoaesses which involve partial sequences of linear combinations of
functions of order statistics with stochastic coefficients depending progressively
on the censoring stages; along with some martingale characterizations of these
PC quantile processes, some invariance principles for them have been derived
. there and these have been incorporated in the study of the asymptotic properties
of some proposed tests.
Majumdar and Sen (1978) have extended the results of Chatterjee and Sen
(1973) for the multiple pegpeBsion modeZ (containing the several sample problem
as a particular case); see also Sinha and Sen (1979b) in this context.
A
mU1tiparameter extension of the results of Sen (1976b) has been considered by
Sen and Tsong (1980).
•
The object of the present investigation is to formulate
suitable multiparameter generalizations of the model considered in Sen (1979)
and to develop the asymptotic theory of PCS tests relating to these models.
-3-
Thus, the results in the current paper are the mUltiparameter extensions of
those in Sen (1979).
In view of this, in the sequel, whenever possible, we shall
attempt to minimize the technical manipulations by suitable cross reference to
the earlier paper.
Along with the preliminary notions, the proposed progressively censored
quantile processes (PCQP) are introduced in Section 2.
these PCQP's are considered in Section 3.
Invariance principles for
Section 4 is devoted to the proposal
and study of some PCS tests based on these PCQP's with special emphasis on
their asymptotic properties.
In particular, the multiple linear regression model
and the location-scale model are treated elaborately in this context.
2.
Let
THE PROPOSED PCQP'S
X1"",Xn be the sUPVivaZ times (with distribution functions (d.f.)
Fl, ... ,Fn respectively) of
n(~
1)
items under life testing and let
Z =(Zn l""'Znn ) be the vector of order statistics corresponding to ~
=(X l ' ......):
·-u
~
The Fi are all assumed to be continuous (and defined on the real line (_00, (0)), so
"'11
that ties among them are neglected, in probability.
Let
2n = (~l'" "~n),
the
vector of anti-ranks be defined by
Znj = X~j
for
j =1 , ... , n .
(2.1)
From time and cost considerations, often, the experimentation is terminated at
the
r~th
faiZure
Znr'
where
r = [np] +1,
for some p
£
([s] being the largest integer contained in
(0, 1);
(2.2)
s), so that for such a censored
case~
the observable random vectors (r.v.) are
~ (r) = (ZnP .•. ,Znr) and Q(r) =(Qnl" .• '~n)'
and a test statistic
L
nr
= (~r), ~r))
depends on
~(r)
.-u
and oCr).
~
(2.3)
-4-
In a PCS, the experiment is monitored from the beginning'. so that at each
failure
Z
L (z(r) o(k)) is constructed: if L
favors a clear cut
nk
nk' n n ' '11
terminal decision, experimentation is curtailed at that time-point and if no
,.
such k(s r)
exists then experimentation is stopped at the preplanned r-th
Z ; here. r may even be taken as n. Thus, in a pes, one confronts
nr
a time-sequentiaZ testing procedure based on the partial sequence {~k), ~kJ; k sr},
failure
where neither the
L
nk
Zni nor the
are mutually independent. and hence. the
~i
may not have independent increments.
To introduce the PCQP·s. we consider a multiparameter estension of the model
considered in Sen (1979) and assume (for the time being) that the d.f.
-e
Fi has a
continuous probability density function (p.d.f.) f.J. (almost everywhere), where
f.(x) =f(x; 8(C. -c )1), i =l ..... n. x ER =(_00,00),
(2.4)
J.
'" "'J. "'n
~ is an m~q matrix of unknown parameters, the
~
are specified q-vectors
and -c =n -IL . lC"
The two (as well as several ) samples location"1'l
J.= "'J.
scale models and the mUltiple regression models are special cases of (2.4).
(m
~l.
q
n
~l)
Suppose now that we intend to test for
H :
O
The likelihood function of
~
=Q, against
(2.5)
n·n
e
(2.6)
be an mq-dimensional open rectangle containing Q, as an inner point.
.hl •..•• q
'"6= «6'J-t.h))'1
J = •...I ,m.-t.=
absolutely continuous p.d.f.·s.
and let {f(x; 6), 6
'"
G. (x)
J
s.(x)
= [1
'"
For every x ER and
gj (x) = -(o/a6 jl )10gf(X; ~)
e
;&Q,.
(~k). ~k)) is given by
k
l
~
o(k)) =n
... f(ZnJ.'; ~(£n. -e)I)
[1 -F(Z k;~(~
-e)I)]
~
J.=l'<nJ. ""'11 i=k+l
n
''ni "'n
L (Z(k)
nk ""'11
Let
HI:
E
e}
j
be a family of
(= 1, •• '1m).
IQ •
we let
(2.7)'
(2.8)
-F(x; Q)]-IJoogj(Z)dF(Z; Q),
x
= (gl (x) •..•• gm (x)),
and [(x)
= (G
1 (x), .... Gm(x))
I •
(2.9)
-5-
Then~ by (2.4) through (2.9L we obtain that
T k =-Cololl)logL. C.z(k)
"'Il
nk "'Il
:::I ,..,....,
= I~=l [~(Zni)
o(k))
~
'
III =0
"'"
'"
-~(Znk)] (~i -Sn)' ~
1 sk
Sn~
(2.10)
Proceeding as in (2.8) through (2.12) of Sen (1979)~ we may
consider the related sequence
and TnO =0.
o~
k =0
l:~1= 19(Z
.}[(c
'" n1""Q.
T* =
nk
-c)
n1
""n
+_1_ \'k Cc
-c""'n )]' ~ l
n -k [.s=l ""'0
. ins
S
k s n-1
T
"'Iln- l' k =n.
(2.11)
We shall find it more convenient to study the asymptotic properties of
Let
{d.,
1 si sn}
"'J.
D
",n
A1so~
let
be q-vectors for which
\'~
[.1=;
d.1 =0
{Irik}'
and
=l:~1='"
1d 1""'1
. d ! is positive definite (p.d.)
g=(Q1'"
common probability
.~~)
assume each permutation of
(nl)-l.
_
""
"11""
\'k
1- '"
~i
, -1 \'k
-[([.i=l~.) Q,
1
(I, ... ,n)
with the
e-
Note that for every k: 1 sk Sn,
sup(l'D .e)-~II~_ll'~
~~Q
(2.12)
I
([.i=l~.)]
1
~
_
*
-Unk~
say,
(2.13)
where
*2
-1
k
\'k
EUnk =Trace{Q, [(Ii=l~Q.)([.i=l~Q.)']}
1
(2.14)
1
1 [D ken -k)/n(n -I)]} =qk(n -k)/n(n -1)~ 1 sk Sn.
=Trace{D"11 "11
Further~
proceeding as in the proof of Lemma 2.1 of Sen
{ U k =(n -k) U*} I
~
1}
sk <n
n
nc
(1979)~
is a non-negative submartinga1e.
direct vector-extension of the proof of Lemma 2.1 of Sen
(1979)~
we claim that
Hence, b ya
we arrive at
the following:
Let
w ={w(t)~ 0 <t <I}
be a continuous, non-negative, U-shaped and square
integrable function [inside (0, 1)] and
{£i}~
g
be defined as in above.
Then,
e
-6-
k·
p{sup max wCiiH~'D~)
YQ
,.
l~k<n
"~
.k .
2IL'_l;f'~
"11
'<i
1-
I ;::l}
1 2
~qfow .(t)dt.
(2.15)
Note that Lemma 2.2 of Sen (1979) also directly extends to the case where both
g and G are q-vectors (as is the case here)~ and hence, by using (2.15) and
some standard inequalities, we obtain that under (2.2) and H in (2.5), as
O
n
-+00,
max
k~r
sup sup la I (T k - T*k)ll (l'C l) ~ - + 0 ,
o.J.O· "'" "11
"11 "'" "" "11""
P
a:a ' a=l -Lr
(2.16)
where
C =~~ l(c.
"11
l.1=
""'1
-c"11 )(c.
""1
_C)I
"11
is assumed to be p.d.
(2.17)
{~k}bY {~k}'
Also, to be
(2.16) provides the justification for replacing
more general, we consider a class of hex) =(hI (x}, ... ,hm(x)) I ,
arbitrary and need not be g,
""
x e:R,
(quite
defined by (2.7)), where (i) each h.(x)
J
is
assumed to be expressible as a difference of two nondecreasing functions and (ii)
~= JOO[h(X)] [h(x)] 'dF(x)
(2.18)
is assumed to be p.d. and finite;
_00
here
F(x) =F(x; 0).
~ =J
t,;
Further, let
t,;:
a
F(~
...
a
) =a,
and let
0< a< 1
t,;
~
ani(x)][h(x)]'dF(X) +(1 -a)-l(f ah(X)dF(X))(J ah(X)dF(X))
_00
1
matrix by C-·~.
"11
Q'(Z.)
lie
n1
by h(Z.)
""'. n1
and post-multiply the
The resulting matrix is denoted by T*k* i.e.,
I
T** -- (T*
NJlk
-nk g=h )C-~
. . .n '
"11
f or k = 0., 1 , ••• , n .
(2 . 20).
"'" "'"
I;~
into an mq-vector and denote it by ""**
Ink'
3.
,
(2.19)
_00_00
Now, in (2.ll), we replace the
We roll out
I.
INVARIANCE PRINCIPLES FOR {T**'
0
"11k'
o ~k
~k ~r}
~n.
-7-
the expectation under
HOI we have
EO(~;) = QI
V 0 ~k ~ll
(3.1)
and proceeding as in the proof of Lenuna 3.1 of Sen (1979L we obtain that
[kin -+a] "'=> E { cT*k*) (T*k*) ,} -+v
O
for every 0 <a <11
n(~
For every
stochastic process
v
where
1)
""().
and
rl
~
belongs to the
Skorokhod J 1 -topology.
Gaussian function on
£
[0 1 I])
Dmq[O, 1]
Also I let
[0, 1]1
(3.2)
x I I
... q
(2.2)1
we introduce an mq-variate
by letting
kin ~Pt}1 t £ [0 1 1] .
net) =max{k:
"'11.
Thus I
satisfying
"'11.
W (t) =~~(t)1
....a
'""'n
is defined by (2.19).
W ={W (tL t
"'11.
NJ1
(3.3)
space I endowed with the (extended)
~
= {~(tL t
such that
E [ (~( t) )(1Y, (5 )) '] = ~ (s At ) x
E~(t)
!q
I
£
[0 I I]}
be an mq-variate
=QI V t. [0, 1] and
V 5 , t E [0 I 1 ]
(3.4 )
•
Then, the main theorem of this section is the following, where we impose the
following condition:
as
n -+00,
(3.5)
[(3.5) may be regarded as the vector-version of the classical Noether condition.]
Theorem 3.1.
Proof:
Under HO' (2.2) and (3.5L as n -+00,
~ -v+~, in the extended Jl-topoZogyon
Dmq[OI 1].
(3.6)
Since this theorem is a multiparameter generalization of Theorem 1 of
Sen (1979), we shall only provide a sketch of the proof by repeated cross
reference to the proof in Sen (1979).
We need to show that (i) the finite-
dimensional distributions (f.d.d.) of
{W}
"'1l
(li)
{~) is tight.
Let
·u
B =B(z(k) oCk))
nk
"'11.
I
~
respectively the cr-fields generated by
both are nondecreasing in
converge to those of Wand
k(O
~k ~n).
and
B* =B(Z(n) o(k))
nk
....n I ~
be
(~k). ~k)) and (~n). ~)). so that
Then, repeating the proof of Lemma 3.2
\
4It •
-8-
T~k'
of Sen (1979) for each coordinate of
we obtain that
....** I·· *} ....**
EO{!nk+l
Bnk =!nk a.e., V 0 sk sn -1 .
(3.7)
tf:J.*. *
T
"1lk
(3.8)
Thus, if we let
and if
~k'
""** -T
""
=T"1lk
"1lk -I'
0 sk Sn
1 sk sn;
are arbitrary (non-stochastic) mq-vectors, then
(3.9)
d(~
Now, to prove (i), for any (fixed)
A. ,
""1
I), (0 s)tl«···<)t sl
d
(all mq-vectors), consider the linear compound
1 si sd
W~
=
l:~=l~i~(ti)
=
~=l~{r~~(t.)
and arbitrary
(by (3.3))
1
~ ..... { l:d• 1 I (k Sn (t . )) AnTA**k
= L.k
~u
1=
1 ....1 "1l
(by (3.8))
(3.10)
where the
~k
depend on
AI' ... ,A d and n(t l ), ... ,n(t d),
and by (3.9),
(3.11)
* ' we may readily extend the proof of the Lemma 3.3 of Sen (1979)
Wnk
and show that under (2.18) and (3.5),
For the
~
*2
*
L.ksnEO{Wnk 1Bnk _1 }
where
2
00
(is the variance of
-p+
2
°0'
~~ lA!W(t.),
1
1
[.1=
o~ =l:~=ll:~=l~i (~(t. At.)
1
J
as
n
(3.12)
-loCO,
see (3.4), and)
x !q)
is given by
~j'
(3.13)
Further, by a direct extension (using the Cr -inequality,
of the proof of
.
Lemma 3.4 of Sen (1979), we obtain that for every e: >0,
~L.ksnEO{WnkI(
*2 IWrik I >e:
Hence, the asymptotic normality of
IBnk
* _ }p
l
W~
0,
as
n
-loCO
•
(3.14)
follows from (3.10) through (3.14) and
Dvoretzky's (1972) dependent central limit theorem.
Finally, to prove (ii), we
note that the tightness of each (of the mq) component (s) of
{Wn } follows
-9-
from Theorem 1 of Sen (1979) and since mq
is insured by the tightness of its mq
is fixed, the tightness of
""n
components. Q.E.D.
In the above theorem, by (2.2), we have limited
P <1.
{W}
r =[np] +1
for some
The remarks following Theorem 1 of Sen (1979) also apply to this more
general case (coordinatewise) when we like to take
p =1.
Let us now consider the non-null case and conceive of a model similar to
(2.4); but, we confine ourselves to local alternatives for which the limiting
results are nondegenerate.
D =L~ ld .d ' .
1= n1 n1
ofq-vectors, define
"11
L~ 1d . =0, Vn,
1= ""'111 ""
Rn
is
{d ., I si sn;
We conceive of a triangular array
"'IU
and assume that
supTr(D)
n
""'11
n(~
p.d. for every
(3.lS)
<00
(3.16)
nO)'
(3.17)
lim{ max d'kD-ld k} = 0 .
n-+oo I sk::::n""'11 "11 ""'11
Let then
{K
n
} be a sequence of alternative hypotheses, where
K:
n
~
f.1 (x) =f(x; NNn1
Rd .),
1 si sn,
(3.18)
x e:R,
is an m Xq matrix of (fixed) unknown parameters (all finite) and
satisfies all the regularity conditions, stated after (2.6).
~
Let us also define
etc., as in (2.7) -(2.9).
C
"'a
f
We also define
(0 <a <1)
~,
as in after
(2.18) and define
t;;a
£(a)
=f (~(x))(I(x))
'dF(X) + (1 -a)
(f
t;;a
~(x)dF(x)Y.
(3.19)
_00
_00
Further, we assume that the
d. and
""'111
lim('i'~ ld . (c.
n-+oo
t;;a
n(X)dF(X))g
L. 1
= "'Ill
""1
c.
""1
1
-c)
I)C-~ =P
NJl
n
""
satisfy the condition that
exists
(3.20)
..
,
Let then
M(a)
= [1;;
(a)] (3[P],
,...,
1"'tJ,...,
,...,
0 <a <1.
(3.21)
e
-10~
,
~(a)
We roll out
we define
r
into an mq-vector and denote it by M(a) ,
as in (2.2) with
0 <p <1
Finally,
and denote by
0 ~t ~l},
M = {Mt =l:!.(pt),
i~
0 <a <1.
(3.22)
Then, we have the following
Theorem 3.2.
then~
Let
{~},
and ...,~ be defined as in (3.3), (3.4) and (3.22) •
W
...,
under (2.2), (3.5), (3.15), (3.16), (3.17) and (3.20),
!
finite Fisher information (matrix)
~ -~ ~~,
Proof:
for
having a
f
=E[~(x)][~(x)]',
in the extended Jl-topoZogyon nmq[O, 1].
..
(3.23)
v
It follows from Hajek and Sidak (1967, Chapter VI, pp. 239-240) that
p*
n' the probability measure (for
~n), ~n)) under K, is contiguous to pO the same under H ' Hence, as
n
n'
O
in the proof of Theorem 2 of Sen (1979), the tightness of {~} under H '
O
proved in Theorem 3.1, and the contiguity of {P~} to {P~} insure that
under the assumed regularity conditions,
tightness of
.-\
{~}
under
as well. Hence, to prove the theorem, it
n
suffices to show that the f.d.d.'s of {W -~} converge to those of W. Suppose
{K }
~
that in (2.8), we replace the
ii.,
1
J
~J' smand let
.
g.
J
by h.,
J
~.
~
denote the resulting quantities by
H(x)
= (HI (x), ... ,iim(x)),,
...,
x e:R.
T*k = L.\'~1 =lI(x.1 sf,;kl n...,
){h(x.)
+H(x.
)}ec.
-n
1
...,
1
....1
where the
~k/n
are defined by (2.19) for
Let then
-c)
-n 'C-\
-n
a =k/n.
0 sk sn,
Then, by repeating the
proof of Theorem 2 of Sen (1979) for each coordinate of f~k
for any k: kin +a:
0 <a <1,
as
T** -t*k
-nk
.
On the other hand,
v
~k
-n
(3.22)
-Irik'
we obtain that
n +00,
--+
p
0,
....
under
{K } •
n
(3.25)
involves independent summands (random matrices) and
the classical multivariate central limit theorem along with a theorem of Behnen
{K }
and Neuhaus (1975) yields that under
n
letting k. = [npt.] +1,
J
J
for any (fixed) a (~ l)
1 Sj sa,
y*
v*
k , •.• , T k
-n 1
n a
T
and
have (jointly)
-11-
asymptotically a mqa-variate normal distribution with means
1
~(ptj)~
~j ~a,
defined by (3.21) and covariance functions conforming to that of ,...,W.
Hence,
(3.25) and (3,3) along with the above lead us to the desired result.
Q.E.D.
We may remark that both Theorems 3.1 and 3.2 remain true if the Tri;
replaced by lnk
are
=Eo(~lg(k)) which amounts to replacing the h(Zni) by
EOh(Z
,..., n1.) =h~ (i),
say,
1
~i
sn.
In this case,
v
in (2.19) may also be
~
replaced by
..ex =n-
)2....
1
I·<
1_nex [h~ (i)] [h~ (i)]'
~O
=Q.
+n- 2 (1 _ex)-l(\.<
h (i))
L1 _ n ~
Note that, by definition, the
(I.1=1
..-~
h~ (i))' (3.26)
for
0 <ex <1;
Lnk depends only on
the
~k) and therefore are PCS rank statistics. The necessary modifications
in the proofs are quite stright-forward, and hence, the details are omitted.
4.
ASYMPTOTIC TIME-SEQUENTIAL TESTS
The quantile processes, studies in the previous sections, depend, in
general, on both
m =q =1),
where
~
and
2n.
For the simple regression model (i.e., in (2.4),
Hajek (1963) considered an invariance principle related to the
h(t) =1,
Tnk ,
wherein a tied-down Wienep ppoaess approximation was developed
and the same was incorporated in the study of the asymptotic properties of some
Kolmogorov-Smirnov, Cramer-von Mises' and Renyi type statistics (based on
alone) for testing the hypothesis of no regression.
2n
Sinha and Sen (l979a) have
developed a reverse-martingale approach to this problem, extended the Hajek
results in a PCS setup and also considered the use of some weighted test
statistics.
Sinha and Sen (1979b) have also extended their theory to the multiple
regression model, which is again a special case of (2.4) with m =1,
q
~1.
Their
procedure is based on an invariance principle related to the tied-down Winer
process in the vector case.
We shall see later on that an alternative formula-
tion of this problem involving the BesseZ ppoaesses can be made by using our
results in Sections 2 and 3.
4It.
L
-12We shall find it convenient to classify the PCQP's into two-types:
(1)
Type A PCQP 8:
only on
(k, n)
statistics.
Here, as in the end of Section 3, we let !l(Znk)
(but not on
Znk)'
depend
so that the !;k becomes censored rank
These statistics have been studied in detail by Chatterjee and Sen
(1973), Majumdar and Sen (1978) and others.
At the same time, the needed
asymptotic theory also follows from our Theorems 3.1 and 3.2.
These are
generally distribution-free procedures (under H )'
O
Z(k) and ~k), so that
(2) Type B PCQ~8: Here the ~k depend on both the ""ll
one needs to estimate the ~ in (2.18) -(2.19), and the test procedures are
only asymptotically distribution-free.
These procedures will be mainly 'discussed
here.
Note that by (3.2) and the martingale property in (3.7), for every
o <a.
<a. I <1,
v
""'().
of m =1 first.
is positive semi-definite (p.s.d.).
I-V
"""CY.
For a Type A PCQP,
for every a. £(0, 1),
We consider the case
\)
a.
is·· known and
we may choose
L
nk
Let then
= (T**) I (v x I ) -1 (T**)
"'11k
p "'Cl
""llk
L* = max L •
nr lskSr nk
L~r
where
'"
[T** IT**] .
"'11k "'11k
(4.1)
It follows from Theorem 3.1 that under HO'
sup{[~(t)]I[~(t)]:
W* ={W*(t) =(W(t))I(W(t), t £[0, I]}
Thus, if W**
a.
..
11
=V -1
0 st sO =W**, say,
(4.2)
is a q-variate Bessel process.
be the upper 100a.% point of the d.f. ofW**, then we have the
-13It follows from (4.2) that the Type I error for this pes procedure is
a
asymptotically equal to
(0 <a <1).
4It
Also, from Theorem 3.2, we conclude that
under the hypothesis of Theorem 3.2, the asymptotoc power of the test is given by
p{ [~(t) +·l:!,t] , [~(t) +lI,t] >w~*, for some
Further, it follows from Theorems 3.1 and 3.2 that for
p{[~(t)
P{N >kIK } +
n
Thus, noting that
Znk
~~a
+llt]
I[~(t)
+llt)]
kin +y,
::;;w~*,
-1
for
kin +y, 0 <y ::;;p,
0 <y ::;;p,
V 0 ::;;t ::;;p-ly}.
(a.s. as well as in the t-th mean,
n(~k/n -~(k -l)/n) +[f(~k/n)]
(4.3)
t e: [0, I]}
r
~l)
(4.4)
and that
we obtain from (4.4) that
the average stopping time is asymptotically equal to
lim E(znNIKn ) =ffCl )P{[W(t) +llt]I[W(t) +llt] ::;;w:*,
.0
pe
O::;;t ::;;e}de
(4.5)
n~
A second type of test statistics (more common in RST procedures) may also
be considered where we take
-1 [T** IT~k'
].
nk --vk/n~k
"'L
Note that by (4 ~l) and (4.6),
where
r
l
= [re:] ,
for some
every 0 <e: <1 and under
Ink = Lnk (vpIVk/n)'
e: >0.
e: I = V Iv
~
d.f. of W**,
e:
L
and W**
nk
a
Z
nr l
(or
t-lW*(t)
(> 0).
by
= max L k
nl::;;k::;;r n
Hence, if
~*
W
~a
"'**
e:'::;;tsl } =We:'
(4.7)
say,
be the upper 100a% point of the.
L*nr but replacing the
and We:, a , respectively and starting the RST only when
The reason for choosing an e: >0 is quite clear: as t 1- 0,
Ink
t-l[W (t)]I[W (t)])
~
a . s ., as
this problem.
I*
HO'
then, we may proceed as in the case of
is observed.
t -lW* (t)
e:(> 0)
p
Let than
nr
Parallel to (4.2), we have for
"'*
{ -1 W*(t):
LnrTsupt
where
(4.6)
k ::;;r.
"'Jl
t 1- 0 .
does not behave regularly; in fact,
However, a small exclusion
Sinha and Sen (1979a) have discussed (for
[0, e:) eliminates
q =1) the choice of
in this context and a very similar picture holds for general
The asymptotic power of the test based on
I*
nr
.
q(~
(for the local alternative
1).
-14-
treated in Theorem 3.2) is given by
1
p{t- [l!(t) +k!t]'[l!(t) +k!t] >w~:cy/ for some t E[E'1 1]}1 (4.8)
:J
where get)
and
Elare defined as in before.
Fina11Y1 the asymptotic value
of the average stopping time is
1
1
-1
-**
}
t;PE +p E f(';P8) p{t [~(t) +~]' [!'let) +~] swE1 ex 1 V E' s ts8d8. (4.9)
J
Let us now consider RST based on Type B PCQP 1s.
Since
v
ex
is not known
in advance, we shall find it convenient to use the second type of tests,
described above 1 with the vk / n being estimated by
1 sk <n 1
Z. are defined by (2.1). Thus 1 in (4.6), we need to replace
nl
by Vnk1 while the rest of the discussions in (4.7) through (4.9) remains
~here
V
/
k n
(4.10)
the
true for this case too.
As has been noted earlier, the classical multiple regression model is a special
case of (2.4) where m=l and q > 1. In this case, we have
fi(x) = f(x- ~'(~i - ~n»' x E R , i=l, ••• ,n,
where
~
is a q-vector of regression coefficients and the
(4.11)
~i
are the known regression
constants (vectors) • Both the Type A and Type B procedures described before are
applicable here to test for HO:
~
=
~
against HI:
~ ~ ~
(under progressive censoring).
We may further add that this multiple regression model includes as a special case
the several sample location model: Suppose that there are k(=q+l) samples of sizes
k
nl, ••• ,n k respectively (where n= Li=ln ) drawn from distributions Fl, ••• ,Fk where
i
Fi(x) = F(x - 8 ), i=l, ••• ,k, x E R,
(4.12)
i
II·
and let
~i = 8i - 81 , for i=2, ••• ,k. Then, if we let ~l = ••• = ~nl = ~'Snl+l = •••
= -n
c +n =
2
,
;
(1,0, ••• ,0), ••• , c
+1 = ••• = c = (0, ••• ,0,1), (4.12) corresponds to
-n-nk-n
l
(2.4). Thus, the proposed Type A and Type B tests are applicable for testing the
identity (under PCS) of Fl, ••• ,F
k
(against shift alternatives); PCS rank tests for
this problem are due to Majumdar and Sen(1978). It may be remarked that instead of
-15the location/regression model in (4.11)-(4.12), one could have considered the
regression model in the scale parameter which woul have the several sample scale
model as a special cases. The proposed procedures remain applicable for this scale
model too.
We now proceed to the case of m > 1. The most notable case pertaining to this
model [in (2.4)] is the so-called locatio-scale model where we allow both the
location and scale parameters to vary (under the alternative hypothesis), retaining
the identity of the d.L's under the null hypothesis. Though the theory developed
earlier extends readily to the case of m .::. 1,
q.:::.
in Theorem 3.1 [see (3.4)J, {[W(t)]'[ V~I ]-l[W(t)] ,O<t<l}
-
-p
-q
~
1, for the Gaussian process
-
--
is, in general,
not a Bessel process, and for this process, the distribution of the supremum
may depend on the sequence
o ~a.
{v-a. :
~p}
in a rather involved way, so that
the computation of the critical values may pose a serious problem.
The situation,
however, becomes quite manageable, if we assume that
v
-ex
where y (ex)
p
-p
V
0 ~ex sp,
is a nondecreasing function of
p
(0 <p <1).
=Y (a.)v,
ex (0 Sa. sp),
In such a case, in (4.1), we may replace
(4.6), we need to take
~L
(4.13)
**'
I ]-IT**
~k'.
nk =Ink [~k/n x ~
V
for every fixed
p by
~,
p
while, in
for Type B PCQP's,
vk/ n
need to be replaced by
k
V
~k
for
1 sk sn,
=!..
L [h(Z
.)][h(Z .)]'
n.
~
n~
- n~
+
(4.14)
~=l
[The Bessel process appearing in (4.2) [or (4.7)] will involve
mq processes, instead of the
discussion will be the same.
q ones in the earlier cases.]
The rest of the
We conclude this section with the remark that
Majumdar (1977), Majumdar and Sen (1978) and DeLong (1980) have considered the
W** and -**
W
, for the k-parameter Bessel
e:, ex
a.
and these may be used for the actual RST based on
evaluation of the critical points
process for some typical
the proposed PCQP's.
k
~
e
•
REFERENCES
1.
CHATTERJEE, S.K. and SEN, P.K. (1973). Nonparametric testing under
progressive censoring. CaZcutta Statist. Assoc. BuZZ. 2.2, 13-50.
""""
2.
DELONG,D.M. (198Q). Some asymptotic properties of a progressively
censored nonparametric test for multiple regression. Jour. MuZtivar.
AnaZ • .!£.(1980) ~ in prsp.·
3.
DVORETZKY, A. (1972). Central limit theorems for dependent random variables.
~oc. 6th BerkeZey Symp. Math. Statist. ~ob. ~, 513-535.
4.
GARDINER, J.C. and SEN, P.K. (1978). Asymptotic normality of a class of
time-sequential statistics and applications. Comm. Statist. A?." 373-388.
4.
HAJEK, J. (1963) . Extension of the Kolmogorov-Smirnov test to regression
alternatives. In PY'oc. BernouZZi-Bayes-LapZace Seminar (Ed. L. LeCam).
Berkeley, pp.
~
v ~
HAJEK, J. and SIDAK, Z. (1967) . Theory of Rank Tests. ,Academic Press,
New York.
t
6.
7.
~
MAJUMDAR, H. (1977). Rank order tests for mUltiple regression for grouped
data under progressive censoring. CaZcutta Statist. Assoc. BuZZ.
26, 1-16.
"""'"
8.
MAJUMDAR, H. and SEN, P.K. (1978). Nonparametric tests for mUltiple
regression under progressive censoring. Jour. MUZtivar. AnaZ.
~, 93-95.
9.
SEN, P.K. (1976a). Asymptotically optimal rank order tests for progressive
censoring. CaZcutta Statist. Assoc. BuZZ. ~, 65-78.
10.
SEN, P.K. (1976b). Weak convergence of progressively censored likelihood
ratio statistics and its role in asymptotic theory of life testing.
Ann, Statist, !, 124'7-1257,
11.
SEN, P.K. (1979). Weak convergence of some quantile processes arising in
progressively censored tests. Ann. Statist. ?." 414-431.
12.
SEN, P.K. and TSONG, Y. (1980). An invariance principle for p~ogressively
truncated likelihood ratio statistics, Metrika (to appear).
13.
SINHA, A.N. and SEN, P.K. (1979a). Progressively censored tests for
clini~al experiments and life testing problems based on weighted
empirical distributions. Corron. Statist. A 8, 871-898.
""
14.
•
SINHA, A.N. and SEN, P.K. (1979b). Progressively censored tests for
mUltiple regression based on weighted empirical distributions •
CaZcutta Statist. Assoc, BuZZ. ~ (in press).