* Work supported by the Air Force Office of Scientific Research, A.F.S.C.,
U.S.A.F., Grant No. AFOSR-74-2736.
WEAK CONVERGENCE OF SO~E QUANTILE PROCESSES ARISING
IN PROGRESSIVELY CENSORED TESTS*
by
Pranab Kumar Sen
University of North Garolina at ChapeJ Hill
Institute of Statistics Mimeo Series No. 1092
October 1976
,
"
_
,
(,
WEAK CONVERGENCE OF SOME QUANTILE PROCESSES ARISING
IN PROGRESSIVELY CENSORED TESTS*
by
Pranab Kumar Sen
University of North Carolina, Chapel Hill
ABSTRACT
For progressive censoring schemes pertaining to a general class of
(parametric as well asnonparametric) testing situations, one encounters
a (partial) sequence of linear combinations of functions of order statistics where the coefficients are themselves stochastic variables.
Weak
covergence of such a quantile process to an appropriate Gaussian function
~
is studied here, and the same is incorporated in the formulation of suitable (time -) sequential tests based on these quantile processes.
AMS 1970 Classification NOS: 60BlO, 62G30, 62L99.
Key Words and Phrases: Brownian motions, linear combinations of functions
of order statistics (with stochastic coefficients), martingales, progressive censoring schemes, quantile processes, sampling from a finite population, time-sequential tests, weak convergence.
* Work supported by the Air Force Office of Scientific Research, A.F.S.C.,
U.S.A.F., Grant No. AFOSR-74-2736.
1.
INTRODUCTION
Xl, ... ,X , the survival times of N(> 1) items under lifeN
testing, be independent random variables (rv) with continuous distribution
Let
functions (df) F , ... ,F , respectively, all defined on the real line
l
N
(_00,00). In a life-testing problem, the smallest observation comes first,
the second smallest next, and so on, until the largest one emerges last.
Thus, the observable random variables can be represented as
(1.1)
is the j -th smallest observation among
where
for
(1. 2)
j
ZNi)
(l ~ j ~ N) and
ties among the
F. ,
1
may be neglected, in probabi Ii t Y, so that the
uniquely (in probability) defined by (1.2) and
a permutation of
N
= 1, ... ,N
by virtue of the assumed continuity of the
hence, the
Xl"" ,X
(l, ... ,N).
(QNl, ... ,QNN)
X.1
(and
QNj
are
represents
Since a complete collection of (1.1) demands
the span of the experimentation until
ZNN
is observed, while practical
considerations often set time and cost limitations on the duration of
experimentation, the experiment may be terminated at the r-th failure
ZNr'
(1. 3)
([s]
where
r
=
[Np]
1 for some .. 0 < p < 1
being the largest integer contained in
vable random variables are
(1.4)
+
s).
Thus, here, the obser-
e
~
-3-
[We also know the complementary set
Q(N)
",N
Q(r) '
of the order in which the elements do appear.]
statistical hypotheses concerning the df's
based on
(Z(r)
"'N
nCr))
';:$N
but without any idea
- ",N
For testing suitable
Fl, ... ,F ,
N
a terminal test
is termed a censored test; we denote the corre-
sponding test statistic by
T .
Nr
In a progressive censoring scheme (PCS) , the experiment is monitored from the very beginning with the objective of an early termination
(prior to
T
Nk
ZNr)
whenever statistically feasible, i.e., one observes
at each failure time .ZNk - (1
:i;
k ::; r), and, if for some
k (::; r), T
Nk
provocates a clear statistical decision in favor of one of the hypotheses,
experimentation is terminated at that time-point; if no such
the experimentation is stopped at the r-th failure
priate statistical decision.
ZNr
k«
r)
exists,
along with an appro-
Thus, by constitution, a PCS test is based on
the entire partial sequence
(k) ' Q(.k)
{Z N
N ' 1 ::; k ::; r } ,
(1. 5)
and is time-sequential in nature.
1::; k::; r}
Since the updated sequence
{T
,
Nk
involves dependent random elements and the PCS involves repeated
testing on these dependent statistics, statistical analysis of such a-problem, often, becomes complicated.
principles for
{T
Nk
In this context, suitable invariance
, 1::; k::; r} - provide us with convenient tools for formu-
lating a PCS test and studying its (asymptotic) properties.
In the context of nonparmetric Ufe testing, Chatterjee and Sen (1973)
have studied PCS tests based on a general class of linear rank statistics;
the theory rests on an invariance principle for PCS linear rank statistics.
For the case of
F
I
= ... = FN = F
involving an unknown parameter-e (form
-4-
of
F assumed to be
HO: 6 = eO
vs.
HI: 6
specified)~
~
Sen (1976) has constructed PCS tests for
(or> or <) 6 based on PCSLR (likelihood ratio-)
0
statistics; here also, the theory is based on an invariance principle for
the PCSLR.
The object of the present investigation is to focus on a
general class of location, scale and regression models where the PCPLR
statistics yield suitable quantile processes (QP) and to develop suitable
invariance principles for such PCQP's.
These models are introduced in
Section 2 and the corresponding PCSLR statistics are derived and incorporated in the construction of appropriate PCQP's.
By nature, such a PCQP
involves a partial sequence of linear combinations of functions of order
statistics with stochastic coefficients depending on the various 'censoring
stages.
Invariance principles for the PCQP are studied in Section 3.
The
last section deals with some application of the main theory to some time-
e·
sequential tests.
2.
Let
e
A CLASS OF PCQP'S
be an open interval containing
0
and let
{f(x; 6), 6 E El}
be
a family of absolutely continuous probability density functions (pdf), and
for every
f: -
00
< x <
00,
g(x)
(2.1)
let us denote by
= -(a;a6Jlog
f(x;6)
10
and
G(x)
= [1-F(X;O)]-lfoog(Z)dF(X;O)
x
where
F(x;O)
X
f(Z;6)dZ.
=J
_00
We conceive of the model where the df
F.1
-5-
admits of the pdf
f.
and
1
- oo<x<oo,
(2.2)
where
~
l::;;i::;;N.
cI"",c N are given constants (not all equal),
is an unknown parameter.
H :
O
(2.3)
~
-
cN=N
-1\N
L.i=IC i
and
We intend to test
= 0 vs.
HI:
~ ~
(or> or <) 0 ,
Let us also denote by
Then, the likelihood function for
·e
(2.5)
Simple computations leads us to
(2.6)
TNk =
c~I{(a/a~)lOg LN,k(~~k), g~k)) I~=o}
k
=
I
i=l
and
T =0.
NO
c
" [g (ZNi) - G(ZNk)]' k = l, ... ,N
NQN1
Note that
Thus, the LMP (locally most powerful) test statistic based on
is
TNk~
and in the setup of Progressive censoring, the sequence
{T
; 0::; k ::; r} relates to a sequence of linear combinations of functions
Nk
of order statistics with the coefficients {c
} all stochastic ,in nature .
NQNi
. By reference to Hajek and Sidal (l967~ pp. 70-71), we may remark that the
model (2.2) includes as special
cases~
the classical two ... sample location
and seal models as well as the so called regression model in location and
scale
We are primarily concerned here with weak convergence of suitable stochastic processes constructed from the partial sequence
{T
where
often~
r
satisfies (1.3).
In statistical applications,
Nk
;
0::;
k ~ r}
we face
so that by (2.1)
(2.8)
G(x)
e·
= -{I-F(x;O)r If x g(z)dF(z;O),
_00
Let now
u(t)
be equal to
1 or
0 according as
t
is
~
or
< O~
let
(2.9)
be the empirical df.
We define
(2.10)
Then ~ in (2.6) ~ \ve replace G(Znk)
sequence
by GN(ZNk) ,
and obtain a related
and
-7-
o
~
T~k =I~=l
(2.11)
k
C
=0
NQNi
[g(ZNi) - GN(ZNk)]
1::;; k::;; N.. l
k
TN '
Note that. one can rewrite
T* (1::;; k ::;; N-l)
Nk
=
N•
as
(2.12)
so that it is a linear combination of a function of order statistics with
~
stochastic coefficents depending on the censoring stage.
We conclude this section with the asymptotic stochastic equivalence
of the two sequences {T
Nk
; 0::;; k ::;; r}
and
{T~k; 0::;; k ::;; r}.
We specifically
provide the proof for the null hypothesis situation and we shall see in
Section 3 that the conclusion remains true for contiguous alternatives.
We denote by
Lemma 2.1.
FO(X) = F(x;O)
Let
{d
Ni
and consider first the following.
, 1::;; i ::;; N; N::;; l}
be a tY'iangular arY'ay of Y'eal numbeY's
satisfying
,N
Li=l
dNi = 0 and
(2.13)
Also ~ Zet
q
= {q (t):
0 < t < l}
(2.14)
2
= 1 .
Ni
be a continuous ~ non-n~gative~ U-shaped and
square integY'able function inside
takes on each peY'mutation of
,Ni=l
d
L
I
= [0.1].
(l •..•• N)
FinaZZy~
let
g = (Ql' ... ,QN)
with equal pY'obability
(N!)-l,
Then
~8-
Proof.
N!
Let
PN denote the uniform probability measure over the set of
(l~ ••• ~N).
permutations of
Then~
1 toN
E[dNQ.IPN] = 0 and
1
d2 _ 1
Na - N ~
NLa=l
4It
-1
i = j
(2.15)
i
;t
j
.
thus; if we let
(2.16 )
.
.'
UNk = (N-k)
-1 ,k
Li=ldNQ. ~
1~ k
~
N-l ,
1
we have
(2.17)
Further; under
PN,
o ~ k ~ N-l ,
e·
(2.19)
=U Nk ,
so that under
P , {U
N
(2.20)
Nk
h = (N-k) q (k/N),
Nk
is
k=0,1, ... ,N-2
} is a martingale.
Then, by the U-shapedness of q,
(N-k)q(k/n)
for
~
Let
1::; k ~ N-1
there exists an
a: 0 < a. < 1,
such that
)a in k for 1::; k ~ Na. Hence, by the Chow (1960)
extension of the Hdjek-Renyi inequality,
•
-9 ..
e
(2.21)
P{l~n;:~a. q()</N) Ir~=ldNQil ~ 1}= P{l~~~Na hNk IUNk I ~ I}
~ {h~lE(U~l) + r~~~]h~k[E(U~k) - E(U~k_1)]}
={N- 1q(1/N) + r~~~]q2(k/N) [(N-k)/(N..1) (N-k+1)]}
(as
~J:q2(t)dt,
as
q
I~=ldNQ. = -r~=k+ldNQ.'
Since
1
(N-k)/(N-1)(N-k+1)
is
~ N.. I , k·~ 1) ,
U-shaped.
Is k s N-l,
the case of
Na < k
S
N-l
canbe
"
1
reduced by reflection and an inequality for this complementary part be
obtained in the sarne manner.
In particular, if we let
Q.E.D.
q (t)
= K- l ,
0S t S 1
where
0 < K < 00
and choose
K large, we obtain from (2.14) that
(2.22)
Note that if the
Cantelli Lemma, as
I
Lemma 2.2.
(2.24)
Proof.
are i.i.d. with df
<
F '
O
then by the Glivenko-
00 ,
/ I
max FO(ZNi) - k N -+ 0
l~k~N
(2.23)
Jig IdF0
N -+
Xi
If the Xi are i.i.d.
almost surely (a.s.) .
~ith
df FO'
then under (1.3) and
co ,
max 1.. l~k
I
lskSr G(ZNk) + (N-k) L.i=l g (ZNi) -+- 0
a.s., as N-+-!lO.
First note that under the hypothesis of Lemma 2.2,
-l.O-
the proof is straightforward [see, for example, Basu and Borwankar (1971)],
and hence, is omitted.
Secondly, under (1,3),
r/N -+
p: 0 <p < 1,
by (2.23).
The rest of the proof follows from (2,8) and (2.24)-(2.26),
Q.E~D.
Xl'" "XN (with df Fa)' g~N) ~ (QN1, ... ,QNN)'
takes on each permutation of (1, ... ,N) with equal probability liN! Thus,
Note that for i.i.d.
by (2.6), (2.12), (2.22) and (2.24), we obtain that under _ H and (1.3)
O
(2.27)
max IT - T* I
l::;k::;r
Nk
Nk
-+
0,
in probability .
With this results at our disposal, we are tempted to consider a
IIOred~l
class of PCQP's and then to study invariance principles for thisc!as$_ Jead!
ing to similar results for
3.
{T
Nk
} as special cases.
AN INVARIANCE PRINCIPLES FOR PCQP
Instead of considering PCQP's derivable from some PCSLR statistics,
we study here a broader class of PCQP's.
Let
J = {J (x), - co <
X
< co}
be absolutely continuous (on finite inter-
vals) and be a difference of two non-decreasing and square integrable (with
respect to
Fa)
functions, so that
(3.1)
Further, let
{d
, •.. ,d
N1
NN
; N;:: l}
be a triangular array of real numbers
satisfying the conditions:
N
N
L dN · = 0
(3.2)
i=l
Finally, let
I
,
'1
1=
1
2
dN1·
{~~k), g~k),
1
=1
I
I
max d
l~i~N
Ni -+ 0
and
~ k ~ r}
and
r
N-+oo •
as
be defined as in (1.3) and
(1. 4), and 1et
0,
k
=0
,k
(3.3)
1 ,k
Li=l J (ZNi) [dNQ . + N-k La =l d NQ
N1
L
_
NN 1
Na
1
]
~
k
~
N-1
k = N •
Our primary concern is to develop an invariance principle for
and we consider first the case of the null hypothesis
are i.i.d.r.v with an absolutely continuous df
tion
and variance under
H by
O
EO
and
O~k~N
(3.4)
and for every
(I = [0,1])
Va'
N(> r ;:: 1),
F '
O
{LNk; 0 ~ k S r} ,
where the
X.
1
We denote the expecta-
respectively.
Let
,
we consider a stochastic process
W = {WN(t), t
N
E
by introducing a sequence of non-decreasing, right-Gontinuous and
integer-valued functions
(3.5)
and then letting
{kN(t), t
E
2
k (t) = max{k: 8
Nk
N
I},
S
where
2
to }
Nr
tEl ,
I}
-12-
(3.6)
Note that
W belongs to the D[O,l] space endowed with the Skorokhod
N
J I-topology (for N= 0,1, WN(t) = 0, V t E 1). Our primary concern is to
show that under suitable regularity condtions,
(3.7)
W
where
W= {W(t), t
N
E.
E
W
I}
in the J -topo10gy on
1
D[O,l]
is a standard Brownian motion in
For absolutely continuous
FO'
the a-quantile
~
I.
/
is defined (uni-
a
quely) by
(3.8)
Let then
~
(3.9)
v~ = f
~
a J2 (X)dF o (X) + Cl-a)-l(J aJ(X)dFo(X)
_00
\)2 <
by (3.1),
a.
Lemma 3. 1.
_00
for every
00
0 < a. < 1.
N -+
be the set of all possible
(N!)
H '
O
0< a< 1
First, we consider the following.
as
Under (3. 1); (3 • 2) and
r
00 ,
(3.10)
Proof.
Let
Then, under
(3.11)
and hence
2N
permutations of
(l, .•. ,N),
H '
O
L
(z (N)
N,N . . .N
~~N), g~N) are stochastically independent with g~N) assuming
each permutation of
insures that for each
(l, •.. ,N)
with the same probability
k(=l, ... ,N), g~k)
is independent of
(N!) -1.
ZeN)
.....N
This
(and
e-
-13-
hence, of
Z(k))
when
~N
H0
holds.
Thus,proceeding as in the rnroof of
Lemma 2.1, we obtain that
(3.12)
(3.13) EO
(L~k)
= EO{EO
(L~k I~~k)) } = EO{V0 (L Nk I~~k)) }
=
Eo{V)~I
J(ZNi)d NQ .
li=l
Nl
=
{N~Jl=l Lfd Ni - k a=lI dN~2}EO{kJ
J2 (ZNi) + N(N1_k) ~I ,l(ZNi Y
~
1=1
li=l
J}
=N~l
+.
Z
Eo{f Nk J2 (X)dS (X)
N
I
l=k+l
+N~k(f
Z
NkJ(X)dSN(X)f},
k/N + a: 0 < a < 1 => N/ (N-k) + (I-a)
Now,
J
-1
< 00, and by (2.23) - (2.25) ,
~a
ZNk
(3.14)
by (2.9) and (3.2).
_00
_00
·e
I J(ZNa~}J
d NQ . (-N=l
Nl
a=l
J
Jr(x)dSN(X) +
_00
Jr(x)dFo(X)
a. s., as
N+
00
(r = 1,2) .
_00
Finally, for
r
= 1,2
,
(3.15)
::;
2
-l,n
2
.
J (x)dSN(x) = N Li=lJ (Xi)
oo
f
_00
(being a reverse-martingale) is uni-
where under (3.1),
form1y (in
N) integrable.
Hence, (3.10) follows from (3.13)-(3.15) and the
Dominated Covergence Theorem [cf. Loeve (1963, p. 124)].
Let
and
k
B*
Nk
B*
Nk
= B(z(k)
=B(Z~N'
(N)
= 1,2, ... ,N
.
~N'
n (k))
~N
n(k))
~N
be the a-field generated by
be the a-field generated by
Q.E.D.
(Z(k)
~N
(~'N(N),
.-
'
n(k))
~N
nN(k)),
~
for
lJndei:'
Lemma 3.;2.
ctPe
map1;-ingatee
Proof.
H t
O
fo't' evei>y
N(2
k
{LNk' B ; 0 ~ k ~ N})
Nk
(and hence,
0::;; k ::;; N}
i).
LNN = L~JN"'l'
Note that
LNk +1 - LNk ::
U Nk > BNk ;
k ~ N-2,
while for
'.
Ai
by (3.2),
J
.~ i
k+l
1 ~.
N. -k-1 XdNQ , - N:-k bdNQ. ' .
d=l Na.
(1=1
Nti
..
L J(Zw)
i=1
1.
(3.16)
(N ...k)
1 k
J' t ' ,
= (N-k...
l) d NQ
+ N-L? dNOL
J(ZNk+l)
.. Nk+1
ex=1 'Nex
t"
Since; under
(k+l)
HO; gN
EO [dNQNk+11 §~k]
.
.
1.S
• .' .' .
'.
(k+l)
wdependent of ~N
'
=EO [dNQNk+llg~k)]
== '- (N-k)""lr:=l dNQNa.'
,1
~t J(ZN'd.)~.
ex=1,
+ N-k
•
while as in elL 18) ,
it follows from (3. 16)
and the above that
e·
(3.17)
Lemma 3.3.
(3.19)
Proof.
Under (3.1), (3.2) and
d,
H
kiN +
I ks= oE {( LNs+ 1 - LNs ) 2 1BNs } f
v
ex
<1:
as
0 < ex < 1
N+
insuJ>es that
00 ,
Note that by (3016) and the stochastic. independence of ",N
Q(N)
we haVe for 0:0; S ~ N-2,
J
Z,(N)
N ;
....
-15-
(3.20)
E{CL
2
+ ~ L ) IB }
NS l
Ns
NS
={CN.S) / (N-s-l) 2 }{l:~=S+l tNQNi • N~S l:~=S+ 1'1;QNJ2} •
EO{tCZNS+l) +N~S(l:~=lJ(ZNi) JT'~~S)} ,
n > 0,
Now, by (3.1), for every
. there exists an
£ > 0,
such that
f,;£
J
(3.21)
IJ (x) Ir dF0 (x) < n
for
r = 1,2
_00
For
s:S; N£,
\'N
2
we note that
[.i=l d Ni = 1,
.
\'N
[d 2
£i=s+l NQ
Ni
__1_ \'N
d
N-s £l=s+l NQ
] 2 < \'N
d2
. - £i=s+~ NQ
N
<
Ni
-
so that
(3.22) dN£]E{(L
- L )21B }
£s=O
Ns+l
Ns
Ns
:s;~..r~[N£]E
N~s=O
{G(Z
) +_1_ IS J(Z .)l2IZ(S)ll~ + O(N- 1
0
Ns+l
N-s s=l
Nl~ ~N
~
L
)1
L
and proceeding as in (3.14)-(3.15), it can be shown on using (3.21) that the.
right hand side of (3.22) can be made arbitrarily small, in probability
(when
N
-+
(0) .
Note that the conditional pdf of
given
ZNs+l
Z(s)
is
~N
(3.23)
It is easy to show that
E[J(ZNs+l)
I~~S)]
exists for all
(3.1)] and further by the absolute continuity of
and the a. s. convergence of
IZNs'- f,;.s/N I
to
0
J(x)
O':s;
~ N-l
5
(on finite intervals)
for every
N£:S;
it follows as in Theorem 3.1 of Sen (1961) that
(3.24)
max
IE[J(ZN l)lz ] -J(ZN)I
Ns
. <s <
s+
S
S.£_
N"-a
-+
0
a.s., as
[under
N
-+
00
•
5
:s; Na,a < 1,
-16-
Let us then denote by
d
( 3.25) U
(
) -,h:N
l.s+l NQ
Ns = N-s
~
(3.26)
UNS = (N - s)
= HN-s)
Ni
-l\'N
2
l. 1d
'
NQ
s+
Ni
It follows from (2.19) that
when
H
O
.. l\,s d
l.i=l NQNi'
{U
Ns
'
1 ~ s ~ N::'l ,
s = 1, .•. , N-1 •
B(g~S)); 1 ~ s ~ N-l}
,
is a martingale
holds and as in (2.14) and (2.26),
I
max UNs
s~No.
(3.27)
I --
0p (N- 1 )
f or every
0 <a < 1 •
Also, note that
I
I
~
(s)
-1
~
2
(5)
.
(3.27) EO[UNs + l B(QN )] = (N-s-l) {(N-S)UNs - EO [dNQNS+l B~gN )]}
= (N-s-l) -l{ (N-s)U
Ns
__1_ LN . d 2
}
N-s o.=s+l NQNo.
,e .
~
=UNs
,
1
~
s
~
N-2 .
Using the martingale property in (3.27) and the Kolmogorov inequality., we
obtain that under
H '
O
for every
£
>0
(3.28)
where
max 2 )\,N. 2
I}
~ [Nk/ (N-l) (N-k)] {[l~i~ndNi l.i=l d Ni + N
~o
by (3.2) and the fact that
k/N~o.:
0<0'.< 1,
as
N
~
00
•
J
~17-
Thus, from (3,28) and (3,29), we have under
HO'
as
(3.30)
N+
From (3.20) through (3.30), we obtain that for
()() •
kin
+
a: 0<0.<1,
·e
while
Hence, by (2.23) and (3.14), the right hand side of (3.31) converges (in
probability) to
2
v ,
as
a
from (3.31) and (3.32).
Remark.
N+
()(),
and the proof of the lemma then follows
Q.E.D.
Note that in (3.32),
follows from the fact that
(3.33)
where the last step follows by standard arguments under (3.1).
-18-
Let nd~
LEHiiiri~ :;. 4.
ItA) debote the indicator fUnction of the set A.
For every
1:;;'
> 0, kiN
+
~: 0 < d < i 1 as
N+
00
(3.34)
pfotH; \qe
by
Hhl1klip the;
~fgUilierlt~ ~imi1ar
t~o sUbsets {s ~
sum itH:ti
Nd
antf {Nc
< S
$
k} .
Then,
to tHat in (3;22),
(3.35)
sihce
is the difference
J(xj
titiUbtE; arid sequare ihtegtahie
every
o<£<ct<i,
(3.30)
iila~
...•. <:5<:"
5
dfi
tH~
.c- N-C1
ot
tWb fibrt-aecreaSin&absoiutely con-
functions; it can be shb\tfu easi 1y
there e:Hsts li
C=ctE;a) «
1
Ij(ZN" "1) +N , ISi·'-lJ{Z"'T.jl
S+
-5
1''11'
other hand; by (;3. 2 j; fot every
~t
00),
a.!L;
Hi~t
for
stich thaL
as
I'4-+&;
g~N) E ~;
,
IlUHc· ' ,
. i .' i
.
IDl1X' .'
i. tN
'
.
(3.31) iSBN{lliNQNi +N-i IS=l dNQNs l = i~H;NldNQNi -N-i z'js::i+ 1liNQNS!}
. 2
~
Hence,
for
r max
11$i~N
every
Id'Ni 11J +
C
anli
b as
£'"
> 0,
N -+
00
'.
there exists an integer
stich that
(3.38)
p{ Max Id.NQNs
s~Na
+
~
r~ _d '. I < £ "/C} = 1
N-s 1-1 NQ
Ni
From (3.16), (3.36) and (3.38), it follows that for
N2:N '
O
NO(=No(c'" ,C))
J
-19 ..
(3.39)
and (3.34) follows from (3.35) and (3.39).
Q.E.D,
We are now in a position to formulate and prove our main theorem of
this section.
Theorem 1.
Under (1.3), (3.1) and (3.2), when the
an absoZuteZy continuous df FO'
X.1
are i.i.d.r.v with
(3.7) hoZds.
Nk }, when HO holds, we
are in a position to use Theorem 2 of Scott (1973), and to prove the theorem,
Proof.
By virtue of the martingale property of {L
all we need to show that
2 kN(t)
(3.40)
0N- r
L·1--1
I
VoLLN'1 BN1· 1]
~t
as
N -+ 00 (0 < t < 1) ,
\
where
kN(t)
and
r
are defined by (3.5) and (1.3), respectively.
Now,
(3.41) follows directly from Lemmas 3.1 and 3.4 (where we note that (1.3)
insures that
O<p=a< 1).
Since, by (3.4), (3.5) and Lemma 3.1,
.('.2
(3.42)
u
Nr
-+ \l
2
p'
as
N -+ 00 ,
the proof of (3.40) follows from (3.42) and Lemma 3.3.
Remarks.
The condition that
J
Q.E.D.
is the difference of two non-decreasing
functions, through quite general, can be dispensed at the cost of
ing (3.1) to
(3.43)
JOO!J(X) ImdFo(X) <00
_00
for some
m> 2 •
strngthen~
In that case, in Lemma, 3.4, a Liapounoff-type condition can be obtained
(which implies (3.41), and the rest of the proof remains the same.
~econdly,
by (1. 3), we have limited ourselves to
arbitra~
rily close to
1,
equal to one.
Note that
V
X ->
V
2
p
0 < P < 1.
Though p may be
there are a few technical barriers for allowing
=p/(l~p)).
v 2 may tend to
P
as
00
p
1
+
(viz, ,
p to be
J (x)
=1
,
If, however, we impose the additional condition [as
before (2.8)] that
(3.44)
fooJ(X)dFO(X)
=0
,
_00
we obtain from (3.9) and (3.44) that for every
v~=
(3.45)
~
I
2
PJ (X)dF O(X)
+
2
(l-p)-l[r J (X)dF O(X))2
~p
_00
~
I
0 < P < I,
~pJ 2 (x)dFO(x) + [. J 2 (x)dFO(X) = 02
lim 2 .1' 2
1 v =u
pt
P
and
~p
_00
Even so,!ooJ(x)dSN(X)
is not necessarily equal to
0;
I
in fact, it is
_00
Op(N-~).
Further,
{l~~~NJ2(ZNi)} = EO{1~~~NJ2(Xi)}
while under (3.9) and (3.44),
=
o(N),
-1 2
EO (f OOJ(x)dSN(x) ) 2 = No.
_00
from (3.13) that for every (fixed
under (3.9),
Hence, it follows
.
s (~l),
as
N+
00 ,
(3.46)
This apparent anomaly can be straightened out with the help of (2.8) and
(2.10).
as
N+
Note that if g
00 ,
satisfies (3.1), then for every (fixed
s(~l),
.
(3.47) G(ZNN -s )
.,
=
0p ([N/s]
1<:
2)
while
_00
1
Thus, whereas (2.8) relates to a term (for x = ZNN_s) 0p([N/S]~),
(2.10)
...21-
_k
~(ZNN-S)
k
0p((N/S)N 2) = 0p(N 2/s)
leads us to a term
1
and hence,
variance of
s
_ )
NN S
and
are of different (stochastic) orders of magnitude, and the
stochastically larger order of magnitude fro
But, if
GCZ
L _ ;
NN s
= seN)
GN(ZNN_S)
pushes up the
in fact, here (2.24) and hence, (2,27) may not hold.
be any (slowly varying) function with
=
lim..N+OO s (N)
00 ,
it can be shown that
2
2
EO(LNN_S(N)) = 0 (1 + liseN) + 0(1)) +0
(3.48)
as
Thus, as regards (2.27), we can proceed as follows.
N+oo.
First, doing the
same line (of proof) as in Lemma 3.1 of Sen (1976), it can be shown that
under
H '
O
(3.49)
e
while for
n > 0,
shown that
arbitrarily small, on letting
2
EO (TN)
=VI2
and
EO (2
TN
)~2
r
~ VI
N
inequality for martingales, for every
(3.50)
P{rNm:kx~N TNk - TNrN I > E:} ~
-n '
On the other hand, for
r
~
r
N
E: -2 (TN
{T
Nk
O
= [N(l-n)],
; 0 ~ k ~ r },
N
resul t for the entire sequence
it can be
-T
)2
NrN
0(>0)
= 0,
2
V _ + 0(1)]
l n
by choosing
by (3.4.5),
n > 0,
(2.27), so that the invariance principle for
the same for
= [N(l-n)],
E: > 0,
which can be made smaller tahn any given
JgdF
N
so that by the Kolmogorov-
= E: -2 [VI2 -
dently small [and noting that as
r
n(>O)
siffi-
limn+oV~_n = V~].
we are in a position to use
{T~k; 0 ~ k ~ r }
N
leads us to
and this along with (3.50) yields the desired
{T
Nk
; 0 ~ k ~ N}.
In a similar manner, by the
-,,2martingale property (Lemma 3,2) df {L Nk } and (3,48)~ for any slowly varying {s (N)} ,; we can replace {LNI<; a ~ k ~ N-s (N)} by an appropriate
{L
Nk
; O~k~N(l-n)}
that
EO (L
(n>O)
and apply ouf Theorem I,
2
NN
(not to
- L _ (N))2 + 0 ,
NN S
replace N-s(N)
p quite beloW
t<.J
we are, however, unable to
0),
tn actual practice,
by N in this case.
irivbives it terniinal censoring humber
In view of the fact
PCS
mostly
cottesponding to a iTalue of
(t)
and hence, thiS teeJiiIical1ty is not of much concern
1,
us.
Let us nOW proceed on to the non-riull case.
We shaii confine
our~
seives to Ideal (contiguous) alternatives where pataeiiel resuits tan
dEiriiTed arid these wiii
be incorporated in the next section for the study
dfas,mptdtie power of some
tfiatiguia'r
array UN·'1 j i
.
and assume that
he
Pcs
~ i ~
tests based
Nj N~ i}
of
on
such PCQP's.
(tow~wisej
Consider a
independent riT' s
xNi has an absolutely cotitiriuotisdf FNi with an
i:iHsolutely continuous pdf f~.Ji and
(3.51)
where
i
f,
A
arid
c
Ni
Note that, in (3.51),
=1, ... ,N
ate all defined as in the beginning
~
Seetioh 2.
is regarded as fixed while hy (2.4), the
t
Ni
We denote stich a sequence of alternative hypo-
ali go to
b as
N+
thesestiy
{~},
while HO relates to
00,
~f
b,= O.
Our concern is to study
the weak toirtrel'gence of {W }, defined by (3.5) - (3.6) , when {H } hold.
N
N
We defihe the d
Ni as in (3.2), the .c *1" as in (2.4), and assume
N
that they satisfy the limits
(3.52)
IN. 1dN·1 cN*1" -- P* ( - 1 <- P* <- 1) )
N400 1=
. ..
1,1m
max
l~i~N
IcNi
* I+ 0
d . = C * ., 1
Nl '
N1
in fact, for
and
(3.53)
and
e
i
$;
N, p*= 1
a(t1p) = max{a:
v 2 is defined by (3.9).
a
222
va(O,p) = 0, Va(l,p) = V p '
also, we denote
F(x;O)
g(x)
as in (2.1).
and
G(x)
and
V
for almost all
fe(x;O)
ting
as
e
0,
+
x,
2
a
2
$;
tv }
p
t
co:
[0,1]
€
[0,1] ,
2
f(x;O)
by
F and
O
fO'
a(l,p) =p.
Here
respectively, and
Further, we define
f(x;e)
is absolutely continuous in
(a/ae)f(x;e) = fe(x;e)
and further, defining
FO(x) = F(x;O),
t
I
Note that v a (t,p )
so that a(O ,p) = 0 and
We also assume that the pdf
e(E8)
For every
(by (2,4)),
we define
0< P < 1 ,
where
$;
g(x)
exists and converges to
as in Section 2 and let-
we assume that
(3.56)
_00
00
Finally, let us denote by
(3.57)
nad note that by assumptions made on
J
and
g,
]J E:
C[0,1]
space.
Then,
we have the following.
Theorem 2,
Let
{W } and
N
W be defined as in (3,5)-(3,7),
(1.3), (3.2), (3.52), (3.56) and
{H }
N
in (3.51), as
N+
Then~
00,
under
. . 24-
v
(3.58)
WN -].l -+ W
.
Proof.
Let
P and
N
N be
P
in the J 1-topoZogy on 0[0,1]
respectively the joint df of
(~~N) g~N))
1
when HO (i.e. Fi = Fo' Vi =1) and ~ in (3,51) hold. Then, under
(3.2), (3.52) and the assumed regularity conditions on f, it can be
shown
guousto
{P }.
N
(3.59) wo(x)
Since,
{p*} is conti-
[cf. Hajek and Sidak (1967, pp. 239-240)] that
For
oE(O,I),
let us define
= sup{min[lx(t)-x(s) I, Ix(s)-x(u) I]:
O:su<<S<t:su+o~l}
WN(O) = 0,
xED[O,I]
and
N
with probability 1,
and, by Theorem 1, ~nder H ,'{W }
O N
is tight, it follows that
lim
Mo
(3.60)
lim
N P{W<s (W ) > e: IH } =0,
N
O
V
e: > 0
•
W is a mapping of (z(N), Q(N)) into the space 0[0,1].
....N
....N
N
the contiguity of {P~} to {P } and (3.60), we conclude that
N
Also,
lim
MO lim
N P{W<s (WN) > e: I~ } =0,
(3.61)
Hence, by
Y e: > 0 ,
{WN} remains tight under {~}. Thus, to prove (3.58), we need to
establish only the convergence of the, finite dimensional distributions of
that is
{W -].l} to the corresponding ones of W.
N
For this purpose, for any k: kiN -+ a: 0 < a:S p < 1,
(3 .62)
~k' i tdNiJ (Xi) I (Xi<ZNk) • {N~kit J (ZNi)
where
I(A)
and J*
as in (3,8) and (3.54), we introduce
LN*k
N
as
L
Nk
I (Xi<ZNk)} •
stands for the indicator function of the set A.
N
(3.63)
}tt'1fi
we' rewrite
= L dN1
, J (X.
) I (X. :S ~ ) + J* (~) L dN, I (XI :S ~ )
11
'1
a
a1=' l l ,
a
1=
Defining
~a
-25~
then by (3,62), (3.63) and the defintion of
If we write
._9(.k),
N
h
weave
L~k = LNkN + {J* (~~)
(3.64)
-1
Note that N kN +
Ct,
k
k
N
N
-
-N_1kN iL J(ZNi) }t~ldN~i}
in probability, under
H
O
[viz., (2,23)], so that
by the same technique as in Lemaa 2.2,
!;
(3.65)
0
under HO '
I = 0 (1), under HO' Hence, the second
NQNi
p
term on the right hand side of (3.64) converges in probability to 0 as
d
N+
00
when
H
O
holds.
Further, by the martingale property (Lemma 3.2),
we have by the Kolmogorov-inequality,
(3.66)
(3.67)
L~k
- L
Nk
g0
as
N+00 when
Again, by virtue of the contiguity of
clude that as
(3.68)
N+
O
to
holds .
{PN} and (3.67), we con-
00 ,
L~k -
L
Nk
!;
Thus, for finitely many
under
0
k's,
fying
(3.69)
{P~}
H
N-lk. + Ct·(t .,p )
J
J
,
{~} as well .
say,
kl:::; ..• :::;k , m(;:::l)
m
given,
satis~
to
st~dy th~
dis.t~ib~tion
joint
LZk
consider th~ joint qf of
ind~~end~n.t f~ndom
., "'1
h"tion with
"~m
fol~ows
[L~k , .•. ,L~~]
<
~,. \~~~~
variables, by the
central limit theorem, it
and (3.56),
+ '
me~n v~ctor
In
(2.27)~
and
{TN:I<; OS1<sr}
we can
of
p~()ceed ~
{~N}
to {P
\
ai~c.~ th~
it suffices to
L~k
'
involve ~ i4m. ov~r
c1~ssica+ (m~ltivar~at~
version of the)
that under (3.1), (3.2), (3.51), (3.52)
converges in law to a
~ultivariate nor~~l
distri-
'm
[~(tl)" ~.,~(tm)]Vv
V~((tj Atl ))j,l=l, ... ,m whi~h
Remar~s.
WN(~~)! .• ~~WN(tm)'
of
we haVe
and Qispersio~ matrix
confo:rn\s tp tlw
~roved
d~sir~d
pattern.
Q.E.D.
the stochastic. equivalence Of
{T 1<; Osksr}
N
the sameline as. in
when HO ~nd 0.3.) hold.
(3.62)-(3~~4)
l1e~e ~lso,
and use the contiguity
N} to show that (2.27) remains true when {"N} , [in (3..51),
(3.52) and (3.56)) hqldS alpng with (1.3).
One
of
~ould
have
non-identical~y
cq~~lic~tions
~xtended
the results of Sen (1976) to the
distributed. rv's.
in the proof a 1pngwith
tions
[vi~.,
vi4es
~~ ~ll~~ative ~nq
H.ow.ever,
siwp,le
that would have induced mOre
SO~e extr~
(2.8) and (3.36) of $en (1976)].
(mild.) regularity condi-
T~e c"rren~ ~p~r9ac~ p~o­
sol4ti9n~
4. APPLlCATIONS TO TIME-SEQUENTIAL TESTS BASEP ON
A v~~~etY of
~urrent s~tup
~CQP
r~n~ based PCS ~ests is available in the liter~ture~
Hajek (1993) has d.eveloped the asumptotic theory of Kolmogorov-Smir:Qov
(~S-)
type tests for regression
a1terna.tives~
adapted readily in a PCS provided we let
n~ll distrih~tions
and his
r/N -+ 1.
re~4lts
The simple
can
b~
limiti:Q~
of these KS-type statistics [viz., (3) and (4) 9n
~
v~lid
page 189 of Hajek and Sidak (1967)] are not
if
r/N+p; O<p< L
How-
ever, some recent tabulations of the critical values of the truncated KS ...
type
tests by Koziol and Byar (1975) provides us with these (approximate)
critical values.
In Chapters V and VI of Hajek and Sidtik (1967), some
related tests are also considered; in particular, the Renyi-type and Cramervon Mises type tests for regression alternatives deserve mention and they
can also be adapted in a PCS when we let
0< p < 1,
r/N + L
Again, for
rln
+
p:
the limiting distributions of these statistics are no longer
very simple and extensive simulation studies are being made to provide
approximate critical values in such cases.
Chatterjee and Sen (1976) have
studied the weak convergence of PC linear rank statistics to a Brownian
motion and their procedure can be used for any
r IN + p: 0 < p :;; 1 with sim-
pIe limiting null distribution theory provided by them,
~
procedure is better than the Hajek's ones.
one common feature:
All these procedures spare
namely, they are based solely on the vector
disregarding any information contained in the vector
order statistics.
Usually, their
z(r)
....N
(r)
Q
~N
'
of associated
Thus, it is quite intuitive to extract this information
and in Section 2, we have shown that a PCPLR statistics sequence relates to
PCQP's which again can be approximated by more convenient linear combinations of functions of order statistics with stochastic coefficients.
Thus,
in the same spirit as in Chatterjee and Sen (1973) and Sen (1976), we may
be interested in employing the process
W,
N
defined by (3.5)-(3.6) and
use as a test-statistic
(4.1)
where
M(x) = M(x(t): 0:;; t
~
1)
is a suitable functional,
For example, we
-28-
may take the KS-type statsitics as
(4.2)
•
_ sup I
I _ max I I
~ - 0:::;ts1 WN(t) - Osksr LNk /Vp
(4.3)
. and obtain the limiting distributions of
+
or M with the aid of our
N
Theorem 1 and the well known distributional results on O~~~l~(t) or
sup I
I.
OStSlW(t)
test.
M~
Theorem 2 provides us with the asymptotic power of such a
We may also consider other functional (such as the Renyi-type or
Cramer-von Mises type) of W and purpose the same as test-statistics.
N
This leads us to the study of the asymptotic behavior of differentfunctiona1s of PCQP's with different
{J},
and will be studied in a subse-
quent paper.
REFERENCES
[1]
Basu, A.P., and Borwonkar, J.D. (1970). Some asymptotic results, based
on censored data, for the maximum likelihood estimators of parameters subject to restraints. Sankhya:, Sero. A 33, 35-44.
[2]
Chatterjee, S.K.,and Sen, P.K. (1973). Nonparametric testing under
progressive censoring. CaZautta Statist. Assoa. BuZZ. 22,13-50.
[3]
Chow, Y.S. (1960). A martingale inequality and the law of large numbers.
Prooa. Amero. Math. Soa. !!, 107-111.
[4]
Hajek, J. (1963). Extension of the Kolmogorov-Smirnov test to ~egres­
sion alternatives. Proc. Bernoulli-Bayes-Laplace Seminar,·
Berkeley (ed. L. LeCam).
[5]
Hajek, J., and Sidak, Z. (1967),
New York.
[6]
Koziol, J ,A., and Byar, D. P, (1975). Percentage points of the asymptotic distributions of the one and two-sample K-S tests for
truncated or censored data. Teahnometroias!I, 507-510.
Theoroy of Rank Tests, Academic Press,
e·
-29-
ppobabiUty Theopy..
[7]
Loeve, M. (1963),
(2nd Ed.)
[8]
Scott, D.J, (1973). Central limit theorems for martingales and for
processes with stationary increments using a Skorokhod representation approach. Adv. AppZ. Ppob. ~, 119-137.
[9]
Sen, P,K. (1961). A note on the large sample behavior of extreme
values from distributions with finite end points. CaZautta
Statist. Asso. BuZZ.
[10]
·e
~,
Van Nostrand 1 p'rinceton
106~115.
Sen, P.K. (1976). Weak convergence of progressively censored likelihood ratio statistics and its role in asymptotic theory of
life testing. Ann. Statist. !, No.6, in press.