* Work supported by the Air Force Office of Scientific Research, A.F.S.C.,
U.S.A.F., Grant No. AFOSR-7402736.
SOME INVARIANCE PRINCIPLES RELATING TO JACKKNIFING
AND THEIR ROLE IN SEQUENTIAL ANALYSIS*
by
Pranab Kumar Sen
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1055
February 1976
\
SOME INVARIANCE PRINCIPLES RELATING TO JACKKNIFING
AND THEIR ROLE IN SEQUENTIAL ANALYSIS*
Pranab Kumar Sen
University of North Carolina, Chapel Hill
ABSTRACT
For a broad class of jackknife statistics, it is shown that the
Tukey estimator of the variance converges almost surely to its population counterpart.
Moreover, the usual invariance principles (relating
to the Wiener process approximations) usually filter through jackknifing
under no extra regularity conditions.
These results are then incorporated
in providing a bounded-length (sequential) confidence interval and a preassigned-strength sequential test for a suitable parameter based on jackknife estimators.
AMS 1970 Classification No:
60BIO, 62E20, 62G35,62LIO
Key Words &Phrases: Almost sure convergence, Brownian motions, boundedlength confidence intervals; invariance principles, preassigned strength
sequential tests, jackknife, stopping time, tightness, Tukey estimator of
the variance, U-statistics, von Mises' functionals.
*Work supported by the Air Force Office of Scientific Research, A.F.S.C.,
U.S.A.F., Grant No. AFOSR-74-2736.
-2-
1.
Introduction
The jackknife estimator, originally introduced for bias reduction by
Quenouille and extended by Tukey for robust interval estimation, has been
studied thoroughly by a host of workers during the past twenty years; along
with some extensive bibliography, detailed studies are made in the recent
papers of Arvesen (1969), Schucany, Gray and Owen (1971), Gray, Watkins and
Adams (1972) and Miller (1974).
One of the major concerns is the asympto-
tic normality of the studentized form of the jackknife statistics.
The
purpose of the present investigation is to focus on some deeper asymptotic
properties of jackknife estimators and to stress their role in the asymptotic theory of sequential procedures bdsed on jackknifing.
Specifically,
the almost sure convergence of the Tukey estimator of the variance is
established here for a broad class of jackknife statistics and their
asymptotic normality results are strengthened to appropriate (weak as
well as strong) invariance principles yielding Wiener process approximations for the tail-sequence of jackknife estimators.
then incorporated in providing
fidence interval and
These results are
(i) a bounded-length (sequential) con-
(ii) a prescribed-strength sequential test for a
suitable parameter based on jackknife estimators.
Section 2 deals with the preliminary notions along with some new
interpretations of the jackknife estimator and the Tukey estimator of
the variance.
For convenience of presentation,
in Section 3, we adopt
the framework of Arvesen (1969) and present the invariance principles
for jackknifing U-statistics.
general estimators.
Section 4 displays parallel results for
The two sequential problems of estimation and test-
ing are treated in the last two sections of the paper.
-3-
2.
Let
~ l}
{X., i
1
Preliminary Notions
be a sequence of independent and identically distri-
buted random vectors (i.i.d.r.v) with a distribution function
on the
p(~l) - dimensional Euclidean space
en
(2.1)
E8n
where the
B.]
=
e + n -1 81 + n -2 S2 +
8,
such that
A
(=> E(8 -8) = O(n
n
are unknown constants.
A
8
= nS
n,i
n
(2.4)
8*
n
-
-1
))
Let us denote by
Ai
(n-l)8 n- l '
8* = n -lIn. 18A . = ne
n
1= n,1
n
(2.5)
Then,
Let
Ai
8
= Tn- l(X l '· .. ,X.1- l' X·l,···,X),
1+
n
n-l
(2.3)
defined
= Tn (Xl' ... ' Xn ), n ~ 1
be a sequence of estimators of a parameter
(2.2)
RP.
F,
l~i~n
l~i~n
,
,
(n-l) {-lIn
n
i=18Ain - l }
is termed the jackknife estimator of
Clearly, by (2.2),
8.
(2.3), and (2.5),
E8~ = 8 - B/n(n-l) + ...
(2.6)
(=> E(8*-8) = O(n
n
-2 )).
Further let,
(2.7)
I
I
n
j
v* = __1__ I [8 ._8*]2 = (n-l)
(Si - 1
8
1)2.
n
n-l.1= 1 n,1 n
i=l n-l n j=l n-
In support of the Tukey conjecture, various authors have shown that under
suitable regularity conditions,
(2.8)
!-.<
n 2 ( 8*- 8) / [v*]
n
n
!-.<
2
V
+
N(0 , 1) as n +
00
•
-4-
Our contention is to obtain stronger results concerning
v*n and
sure (a.s.) convergence of
(ii) Wiener process approximations
{e~-e; k ~ nl.
for the tail-sequence
For simplicity, we assume that
and
to
R = (_00, 00) •
Xl"'" Xn
For every
p=1
n (~l),
are denoted by
Xn, 1:0; ••• :0; Xn,n
be the cr-field generated by
Then,
is non-increasing in
n (~l).
possible permutations of
(X n, 1"" , Xn,n )
-1
.
are real valued
1
Let
Cn
=
C(Xn, 1"'" Xn,n ,
andby
Note that given
(Xl, ... ,X )
n
(n!)
X.
(X n, l""'Xn,n )
are all held fixed while
bility
i. e., the
the order statistics corresponding
Xl"")
n+
C
n
(i) the almost
Xn+J.,j~l.
C , X ., j
n
n+J
~ 1
are interchangeable and assume all
with equal conditional proba-
Hence,
A
(2.9)
I
E(en- 1 Cn )
= n
-l~n
e~in- 1 a.e. ,
L'_l
1-
and, therefore, by (2.5) and (2.9),
(2.10)
e*n
Clearly, if
=
ne - (n-l)E(8 llC) = 8
n
nn
n
{8n ,Cn }
+
(n-l)E{8 -8 llC}
n nn
is a reverse-martingale,
e*n
=
en'.
a.e.
otherwise, the
jackknifing consists in adding up the correction factor
e*n - en
(2.11)
= (n-l) E{(8 -8
n
n-
1)
IC }
n
It follows by similar arguments that
(2.12)
v*n = n(n-l) Var{(8n -8n- 1) Icn }
=
n(r-l){E[(8
n
-8n- 1)2 IC n ]
- (E[(8 -8 l)IC ])2} .
n nn
These interpretations and representations for jackknifing are quite useful
for our subsequent results.
-5-
For further reduction of bias, higher order jackknife estimators
have been proposed by various workers (see [6, 12]). The second order
jackknife estimator (see (4.20) of [12]) can be written in our notations
as
(2.13)
and a similar expression holds for the higher order jackknifing.
fact, we have also a second interpretation for
weighted least squares point
of view.
VI > 0
we shall observe that for some
8*
n'
In
e** etc. from the
n
In most of the cases to follow,
and real
v '
2
(2.14 )
(2.15) Cov{n!z(8 -e), (n-l)!z(8
-e)} =jn-l [v + n-lv + O(n -2)]
n
n-l
n
1
2
Also, by (2.2),
of
O(n -2 ),
'"
E (en -e)
= n
-1-2
~\ + n
8 + . .. .
2
Thus, neglecting terms
the weighted least squares method of estimating
consists in minimizing
(2.16)
2'"
1
2
""
1
A
1
n (en - e - n f\) - 2n(n-l) (en - e - nSl) (8 n _l - 8 - n-l Sl)
'"
1
+n(n-l)(en_l-e-nS l )
with respect to
e
and
8
(2.17)
and our
w
e*
n
=
E(e
Ie ).
w n
the simultaneous equations yield
= n§
n
- (n-l)8
n-l
Similarly, by (2.14)-(2.15), on writing
(k-l)8 _l , k~ 2 ,
k
(2.18)
(2.19)
Sl;
2
Var(Z )
n
Cov(Z n ,Z n- 1)
-6-
Thus,
EZ
k
::
are asymptotically uncorrelated and by (2.2),
Z
n , Zn- 1
e - 8/k(k-l)
+
3
O(k- ).
Hence, considering
(2.20)
and minimizing with respect to
(e, ( ),
we obtain the weighted least
2
squares solution
(2.21)
(8,8
1,8n- 2)
n n-
In fact, we obtain the same solution by working with
and applying the weighted least squares method directly on it.
:: E(8wIen ).
e**
n
for some
k 2: 1,
O(n- k - l ) ,
In general, if we want to reduce the bias to the
then we need to work with
A
A
(e n , ... ,e n- k)
and the k-th
order jackknife estimator is the conditional expectation (given
the weighted least squares estimator of
(2.2) and terms
Theorem 2.1.
Again,
e,
in (2.14)-(2.15).
neglecting
en )
8 ., j 2: 1
k +J
of
in
Thus, we have the following.
Under assumptions (2.14)-(2.15), the jackknife estimators
(of different orders) are the conditionaZ expectations (given
en )
of the
weighted Least squares estimators obtained from the originaZ (biased) estimators for the successive sampLe sizes.
Since in Section 3, we shall be concerned with jackknifing functions
of V-statistics, we find it convenient to introduce the following notations
at this stage.
Let
¢(X , ... ,X ),
m
l
Borel measurable kerneL of degree
tionaL (estimable parameter)
(2.22)
symmetric in its
m(2:l)
m arguments, be a
and consider the regular func-
-7-
where
II; (F) I
F = {F:
ing to
I;
n ~ m,
Then, for
< oo}.
the U-statistics correspond-
is defined by
[n) -lIe
(2.23) U =
n m
Note that
EU
<j>(X., .••
11
n,m
= I;(F), V n
n
1
~
en,m
,x. )
m.
m
= {I::;; i 1 < . • • < i
m
::;; n}
Further, let
(2.24)
for
h = 0, ... ,m,
(2.25)
where
?::O = 0
and
O<l;,?:: <00
1
m
3.
cPO = 1;.
We assume that
(where
Invariance Principles Relating to Jackknifing U-Statistics
We shall be concerned here mainly with the following two types of
estimators:
(i)
Let
g,
defined on
some neighborhood of
1;,
R,
have a bounded second derivative in
and
(3.1)
(ii)
For some positive integer
en = I qs= octn,sUn(s)
(3.2)
where
(3.3)
q,
U(O) = U
n
n
we have
,
n
~m
is an unbiased estimator of
ctn,O=l+n
-1
cO,l+n
-2
,
e = I; (F),
-3
c O,2+ 0 (n ) ,
-8-
U~l) , ... ,u~q) are appropriate U-statistics with expectations 8 1 ", .,8 q
(unknown but finite) and
(3.4 )
the
CI.
n,h
c
.
S,]
= n -h c h , 0
+ 0 (n
-h-l
),
h ~ 1
are real constants; possibly, some being equal to
o.
The
classical von Mises' (1947) differentiable statistical function (corresponding to
E, (F))
is a special case of (3.2) with
q
=m
and
cO, 1 =- (;)
First, we consider the following.
For
Theorem 3.1.
{e}
defined by (3.1) or (3.2)-(3.4),
n
Vn* -+ y 2
(3.5)
a. s., as
n-+
oo
,
where
(3.6)
Proof.
In the context of weak convergence of Rao-Blackwell estimator
of distribution functions, Bhattacharyya and Sen (1974) have shown that
under (2.25),
n(n-l)E[(U
(3.7)
n-
l-U )2 Ie ]
n
n
-+
m2~1
a.s., as
n
-+
00
•
On the other hand, as in Section 2,
(3.8)
n(n-l)E[(U n- l-U)
n
where the
U l'
n-
2
Ien ]
=
i
\,n
(n-l)l,'_l[U
1n- l-U]
n
are defined as in (2.3) with
T
n-l
Hence, from (3.7) and (3.8), we obtain that
2
being replaced by
.
-9-
(3.9)
a.s., as
Further,
{U , C , n
a.s., as
n
n
n
+
00,
~
m}
00
•
U
n
+
E, (F)
and hence, by (3.9),
I
0
+
a.s., as
First, consider the case of (3.1).
(3.11)
+
is a reverse martingale, so that
max
IUi _l-E,(F)
<'<
1-l-n
n
(3.10)
n
n +
00 •
Then, we have
8n-1 -en = g (Un-1 ) - g (U n )
= g'(U )[U 1-U ]+~g"(hU +(l-h)U l)[U 1-U ]2, O<h<l.
n
nn
n
nnn
Note that
E[U
n-
11C] = U
n
the boundedness of
a.e. and further by (3.7), (3.8), (3.10) and
n
g"
(in a neighborhood of
E,),
we have
IE{g"(hU +(l-h)U l)[U 1-U ]2 1C }!
n
nnn
n
(3.12)
$;
m~x
l$;l$;n
= O(n -2 )
i )I{.!. \'~ [U i -U ]2}
I gfIChU n +(l-h)Un-1
n L =1 n-1 n
1
a.s., as
n
+
00
•
Hence, we obtain from (3.11) and (3.12) that
-2
E(8" n- 1-8A n IC)
n = O(n )
(3.13)
a.s., as
n
+
00
•
Similarly,
(3.14)
"
" 1-8)
" 2 C}+O(n)
-4
Var{(8" 1-8)IC}=E{(8
nn
n
nn 1 n
= n
-l\,n
i
2-4
Li=l [g(U n _1) - g(U n )] + O(n )
Again as in (3.12), for some
0 < h < 1
a.s., as
a.s.
n
+
00
•
-10-
[i
2
Li=l g(U n _1 )-g(Un )] -[g
In -l\n
(3.15)
~
{
m~x
l~l~n g
I
I
I
2 -l\n
i
2I
(Un)] n Li=l (Un_I-Un)
}{.!.
(hU +(l-h)U i ) _ I (U ) 1 2
\~ [U i -U] 2}
n
n-1 g n
n L1 =1 n-1 n
= {o(l) a.s.}{O(n- 2 ) a.s.} = o(n -2 ) a.s., as
where by (3.9), (3.8) and the a.s. convergence of
(
3.16 )
[
\,n [Un_I-Un
i
] 2 -+ m2sl [g'
g I (Un ) ] 2 (n-1)Li=1
(~)]
U
to
n
2
n-+
oo
,
~(F),
a.s., as
n-+ oo
•
Hence, from (3.14)-(3.16), we obtain that
(3.17)
= n(n-1)Var{(8
2
I
-+ m sl[g
(~)]
n-
2
1-8 )IC }
n
n
= y
2
a.s., as
n -+
00
•
For the case of (3.2), we note that
a
(3.18)
u(O)_a
u(O)=(U
-U )+c
{(n-1)-l U _n- 1U}
n-1,0 n-1 n,O n
n-1 n
0,1
n-1
n
+O(n
(3.19)
a
-2
)
a. s. ,
1 h u (h)l- a hu(h)=a 1 h(u(h)1- u (h))+o(n- h - 1 )u(h),h ~ 1 ,
n-, nn, n
n-,
nn
n
and hence, the proof of (3.5) follows on parallel lines.
Remark 1.
From (2.11) and (3.13), we obtain that for every
(3.20)
1
n - E Ie* -e"
n
I
-+ 0
a.s., as
n -+
00
Q.E.D.
E
> 0,
•
The last equation is of fundamental importance to the main results of
this section.
-11-
Remark 2.
v*n
By virtue of (3.14)-(3.16),
is asymptotically equivalent to
(3.21)
2
\n
i
2
s n = (n-l)L"_l[U
1n- l-U]
n ;
in case (3.2), (3.21) holds with
g'(U ) - 1.
n
V. = rn-ll)-lI .</l(X.,X. , ... ,X. ),
nl
lmn,l
1
1
1
m
2
(3.22)
where the summation
i.
J
~
i
Let us also denote by
for
\
Ln,i
2:5 j :5 m.
(3.23)
extends over all
Then,
Vn
Sen (1960) has shown that
=
-lIn. IV ..
n = n
1=
nl
-l\n
L'
1=
l[Vnl.-U]
n
2
< •.• <
i
m
::; n
with
Further, let
U
(n-l)
V
n
1:5 i
l:5i:5n ,
2
is a distribution-free estimator of
It is interesting to note that by definition
(3.24)
and, as a result, it follows by routine steps that
s
(3.25)
2
2
2
-2
= m (n-l) (n-m) V , 'if n > m .
n
n
Hence, the a.s. convergence of
(to
Sl)'
V
n
2
n
(to
insures the same for
V
n
However, from the computational point of view, the labor involved
in the computation of
Hence,
s
V
n
whereas for
is
should be preferable to
2
s .
n
2
s ,
n
it is
(3.25) will be of use in Section 5.
By virtue of (3.5) and (3.20) and the invariance principles for Ustatistics, studied by Loynes (1970), Miller and Sen (1972) and Sen (1974b),
we are in a position to present the following results (without derivation):
(i)
(3.26)
Consider a sequence
{w*}
n
of stochastic processes, where
-12-
and
k (t) = min{k: n/k ~ t}, 0 ~ t ~ 1.
Note that
n
with probability
n(~m), W*
n
and for every
1
space with which we associate the Jl-topology.
be a standard Brownian motion on
W*n ~ W* '
(3.27)
. t h e Jl-topology on
ln
2
Finally, if in (2.36), we replace
"* ,
W
process by
(ii)
S = {Set), t
(3.28)
Set)
and let
Further, let
n-+
oo
0
D[O,l]
W*={W*(t),O~t~l}
,
D[ 0,1 ] .
and denote the corresponding
then (3.27) also holds for
n
Let
v*n
by
Y
is equal to
n
belongs to the
Then, as
[0,1].
W* (0)
W = {W(t), t
_ {o,
E
[O,oo)}
0
~t
be a random process defined by
< m+ 1
k [ 8~ - 8 ] / y, k ~ t
E
<
k + 1, k ~ m+ 1 ,
be a standard Wiener process on
[O,oo)}
Further, we assume that for some
[0,00).
r > 2
(3.29)
Then, we have the following
Set) = Wet)
(3.30)
(iii)
A
8 =
n
g(V
k
+
o(t 2 )
a.s., as
8" =
n
In (3.1), we have considered
(1)
n
, . .. ,V
(k)
n
),
for some
k
~
1,
t -+
g(V ).
n
where
g
00
•
It is possible to take
has bounded second order
partial derivatives in a neighborhood of the point
The proof follows as a straight-forward extension of what has been done
before, and hence, for intended brevity, the details are omitted.
(iv)
Jackknifing functions of generalized V-statistics has been con-
sidered by Arvesen (1969).
Here also, as in Sen (1974c), we may consider
-13-
the product sigma-field formed by the individual sample sequence
{C }
n
and express the usual jackknife estimator as the conditional expectation
of a linear combination of original estimators for adjacent sample sizes.
Further, a result parallel to (3.20) holds in this case.
Hence, by virtue
of Theorem 2.2 of Sen (1974a), we are in a position to derive a similar
invariance principle for the jackknife estimators.
Further, by virtue of
(3.19)-(3.23) of Sen (1974a), it can be shown that (3.30) extends to a
multi-parameter Gaussian process.
For intended brevity, the details are
omitted again.
4.
Invariance Principles for General {8*}
n
Structural properties of U-statistics have enabled us to study the
invariance principles in Section 3 without having any extra regularity
conditions.
If
en
is not a function of U-statistics, we need, however,
a few extra regularity conditions to derive similar results.
These will
be studied here.
Concerning the original sequence of estimators
A
8 -+ 8
n
(4.1)
(4.2)
n-+
a. s. , as
{§},
n
oo
lim
2
2
2
n0 = 0
0 = Var(S ) i- 0 as n -+ 00 ,.
n-+oo
n
n
n
0<0<00
Let use also define
(4.3)
A
A
Y = n(n-l) (8 1-8)
n
nn
2
, n
~
2 ,
and assume that
(4.4)
E[Y n Icn ] -+ 02 a.s., as
n
-+ 00
,
we assume that
-14-
(4.5)
Y
n
is uniformly (in
n) integrable
\Ln~N IE(en -8n+l Ien+l )1
(4.6)
as
1
N-+co
Consider now a sequence of stochastic processes
,
{w },
n
where
(4.7)
(4.8)
Then, we have the following.
Theorem 4.1.
w Jl. w*
(4.9)
where
Under the assumptions made above, as
n
W*
'
~n
the Jl-topoZogy on
Qk
Let us write
=
-+ co
D[O,l] ,
[0,1].
is a standard Brownian motion on
Outline of the proof.
n
6
8k
8k + l , k
~
1.
(4.10)
Also, by (4.2), (4.4) and (4.8), for every
(4.11)
-+
Further, by (4.2) and (4.6),
(4.12)
Finally, by (4.5), for every
£
> 0,
t
t
E
[0,1]
a. s., as
n-+ co •
Then, by (4.1),
-15-
(4.13)
-2
\'
= (on )(O(l))(Lk~n
1
k(k+1)) = 0(1) ,
and hence
(4.14 )
Reversing now the order of the index set
{k: k
~
n}
(to
{k:
_00
<
k
~
-n}),
the rest of the proof follows by an appeal to the general functional central
limit theorem of McLeish (1974) where (4.11), (4.12) and (4.14) insure the
satisfaction of his underlying regularity conditions.
Q.E.D.
Since (4.4) corresponds to (3.7), virtually repeating the proof of
Theorem 3.1, it follows that under the same conditions on
g,
as in Section
3,
le*-8
1 = (n-l)\E{(8n- 1-8n )!en }I
n n
(4.15)
a. s. ,
by (4.6), and
(4.16 )
a.s., as
n
Hence, if in (4.7), we replace
{8k }
{e*}
ing process by
w~,
by
then (4.9) holds for
may also be replaced by
!.:
+
00
k
{w*}
n
•
and denote the correspondas well.
Further,
!.:
n 2(V*)-2.
n
The conditions (4.1), (4.2), (4.4), (4.5) and (4.6) are most conveniently verifiable if
where
n
can be expressed as
+ r
(4.17)
as
8n
{m
n
+
00
,en }
•
n
is a reverse martingale and
Irn- l-r n I = o(n
-3/2
) a.s.,
.,..16.,..
5.
Asymptotic Sequential Confidence Intervals Based on Jackknifing
Tukey proposed the use of (2.8) for a robust confidence intervals for
e.
By virtue of our invariance principles, we are in a position to consider
the following robust sequential interval estimation problem.
2
e*
e, y , e
n' n
A
Suppose
ing df
F,
and hence,
and
e
v*
n
and
y
2
are defined as before.
being unknown, it is desired to deter-
mine (sequentially) a confidence interval for
2d, d > 0
For every
n
o
T
be the upper
a
(~n
variabZe
(5.2)
n
~
1
and
I (d) = {e: e* - d ~
n
n
(5.1)
let
e
having a maximum-width
being predetermined, and a preassigned confidence coefficient
1 - a, 0 < a < 1.
and let
The underly-
0
(d))
d > 0,
e
~
e*
n
n
Finally,
Then, we consider a stopping
defined by
N = Smallest integer n(~nO) such that v~
if no such
d} ,
+
100a% point of the standard normal df.
be the initial sample size.
N(=N(d)),
let
exist, we let
fidence interval for
e
is
N=
00.
IN(d),
Whenever
~
N<
2 . 2
nd /T a / 2
00,
the proposed con-
defined by (5.1) for
n = N= N(d).
The above procedure is a direct adaptation of the Chow-Robbins (1965) procedure under our jackknifing setup.
Theorem 5.1.
Under the hypothesis of Theorem
3.J
(or 4.1),
(5.3)
limd+Op{e ( IN (d) (d)} = 1 - a ,
(5.4)
limd+o{(N(d)d2)/(T~/2y2)} = 1 a.s.
If, in addition
<
00
, then
-17-
(5.5)
Outline of the proof.
(1965).
We follow the line of attack of Chow and Robbins
We need to show that
holds and
(c) for every
n* ,
and an
£
such that for
(5.6)
P{ n , ..
>0
(a) V*
n
Y2
-+
a. s "
n > 0,
and
as
n
-+
00,
there exist a
(b) (2 . 8)
0 (0 < 0 < 1)
n ~ n*
max' I<~
In-n
_un
n
!:z 18 * ,-8 * I > £ } < n .
n
n
Now (a) has already been proved, (b) is a direct consequence of (3.27) and
finally (c) follows from the tightness property of
which, in turn,
Q.E.D.
is insured by (3.27).
The condition that
E (SUP
n
v*)
n
<
00
,
needed for (5.5), however, does
not follow from the hypothesis of Theorem 3.1 (or 4.1); nor it is a very
readily verifiable one.
It is possible to obtain (5.5) under a somewhat
different condition which is more easily verifiable.
If {8}
n
Theorem 5.2.
is defined by (3.1) or (3.2)-(3.4) and the hypo-
thesis of Theorem 3.1 hoZds~ then
E{ I<P (Xl' ... , X ) Ir} <
m
00
for some
r> 4 ,
insupes (5.5).
Proof.
Consider the estimator
Sproule (1969) that
V
n
Vn '
defined by (3.23).
can be expressed as a linear combination of
several U-statistics whose moments of the order
EI<p1
2q
<
As such,
00.
for every
£
> 0,
It follows from
q(>O)
exists whenever
using Theorem 1 of Sen (1974c), it follows that
there exist a positive
K
£
«
(0)
and an
n O(£)'
such
-18-
Further,
g' (t)
has a bounded derivative in a neighborhood of
t =C
and
hence, by Theorem 1 of Sen (1974c), again,
(5.8)
From (3.15), (3.21), (3.25), (5.7) and (5.8), it follows by some standard
steps that for every
such that for
n
~
n > 0,
there exist a constant
K « 00)
n
nO (n) ,
(5.9)
K n
-s
n
. s = r/2 > 1 •
'
Having established this, we may proceed as in the proof of Theorem 3.1 of
Sen and Ghosh (1971) [namely, as in their (5.16)-(5.19)], and complete
the proof of (5.5) by using (5.3) and (5.9).
For brevity, the details are
Q.E.D.
omitted.
6.
Sequential Tests based on Jackknife Estimators
The embedding of Wiener processes in (3.30) and the strong convergence
v*n
0f
2)
Y
(to
in (3.5) enable us to construct the following type
of asymptotic sequential tests; for further motivation of this type of
procedures, we may refer to Sen (1973) and Sen and Ghosh (1974).
Consider a suitable parameter
{e*}
n
e
(for which the sequences
{§ }
n
and
of estimators are available sequentially), and suppose that we desire
to test
(6.1)
where
eO
and
scribed strength
f:..
are specified and we like the test to have the pre-
(a,S).
Since the df
F
is not known, no fixed sample
-19-
size test sounds feasible and we take recourse to the following sequential
procedure:
Suppose that
0 < a , 8 <
(A,B): 0 < B < 1 < A <
~
and consider two positive numbers
where
00
with an initial sample of size
8/(1-a)
~
nO(~))'
n (=
O
Band
~
(1-8)/a
A.
Starting
continue drawing observations
one by one as long as
bV* <
m
(6.2)
where
a = log A , b = log B (=> _
knife estimator of
Since
~ bV*
is
m
m
and
~ aV~),
(or
a.s., as
totical1y normal with mean
8
< b < 0 < a < (0), 8*
1S
the jackV* is
m
Xl' ... , Xm [viz. , (2.5)] and
m -+
(by (3.5)) and
00
and variance
0
accept
H
(or
O
HI);
m=N
the
N(~).
N is denoted by
m-~V * -+ 0
for every fixed
00
If, for the first time, (6.2) is vitiated for
~[8~-(80+8l)/2]
stopping variabZe
based on
8
defined as in (2.7).
and
+8 )/2] < aV*
,
mOl
m ' m ~ nO(M
~[8*-(8
y2,
!,;
m2 (8*-8)
is asymp-
m
it is easy to see that
~,
(6.3)
-+ 0
as
n
-+
00
,
For the OC
and hence, the proposed test terminates with probability one.
and ASN function, as in Sen (1973) and Sen and Ghosh (1974), we consider
the asymptotic situation where we let
~ -+
0
(comparable to
Section 5) and set
(6.4)
(6.5)
8 = 8 + <I>~
0
where
<I>
e
¢
=
{<I>:
I<I> I < K <
00 }
d
-+
0
in
-20-
(6.6)
e
a
= A = (1-8)/0.,
Finally, let us denote by
e
LF(~'~)
8 = 8 + ~M
0
H
when actually
O
b
= B = 8/(1-0.),
0<0.
8<~
.
the OC (i.e. probability of accepting
of the test based on (6.2).
Then, we
have the following:
Under (6.4)-(6.6) and the hypothesis of Theorem 3.1 (or 4.1)
Theorem 6.1.
lim
(6.7)
~-+O
and hence, asymptotically the OC does not depend on
(6.8)
P(O) = I-a
and
pel) = 8 ,
(0.,8).
so that the test has asymptotic strength
Proof.
Let us choose a sequence
n*(~)~K~-2 as
(6.9)
~-+O,
{n* = n * (~) }
where
K«oo)
Then, by (3.27), (6.4), (6.5) and (6.9), for
u~
=
F. Further
such that
is arbitrarily large.
8 = 8 + ~~,
0
defining
{U~(t)=~n[e~-~(eO+el)]/y,~2n:'5:t<~2(n+l), no(~)sn<n*(L'l)},
~
lows that as
-+ 0,
U~ Q {Wet)
(6.10)
where
(3.5),
{W(t),
t> o}
+
(~-~)t/y,
0 < t:'5: K}
is a standard Wiener process on
sup{lv~// -II: nO(~) :'5:n:'5:n*(~)} g
for every
(6.9) with
it fol-
n > 0,
K=K ,
n
there exists
we have
0
a K= K e:: (0),
n
as
~
[0,(0) .
-+ O.
Also, by
Finally, by (6.3),
such that defining
P{N(~) >n*(L'l)}<n.
n* (M
by
Hence, using (6.10), (6.2)
and the classical result of Dvoretzky, Kiefer and Wolfowitz (1953) on the
boundary crossing probability for Wiener processes, (6.7) follows readily.
-21-
Finally,
(6.8)
and
Q.E.D.
1.
follows
from
(6.7)
¢
by substituting
As in Theorem 5.2, for the study of the ASN (i. e. ,
for
II
~
0)
function,
= 0
Ie = 80 + ¢lI}
E{N (lI)
the weak (or a.s.) convergence results of Section
3 (or 4) are not enough and we need some analogous moment convergence
results which, in turn, may demand more restrictive conditions on the df
F.
Suppose that as in Theorem 5.2, we assume
~
= g(U )
n
n
and that
(6.11)
Then, not only, we have (5.9), but also, it can be shown by steps similar to those in Section 3 that
p{
(6.12)
for
n
Ie*n -8 I > d
sufficiently large.
:s; C n -s,
£
Further, for
s >1 ,
nO(lI)
:s;
n
:s;
n*(lI),
we may write
(6.13)
where for every
£
> 0,
(6.14)
and where
{Z , B ; n
n
n
erated by
Xl' ... ' X , n
n
C:
I}
C:
is a martingale;
1 (=> B
n
is /l
in
B
n
n).
being the
a-field gen-
As such the method of
attack of Sen (1973) and Ghosh and Sen (1976) can directly be adapted to
arrive at the following.
Theorem 6. 2.
(6.15)
Under the asswrrptions made earlier~ for every
¢
E
cI>,
-22-
where
(6.16)
and y2
_{{bP(<p)+a[l-p(<p)]}{y2/(<p_!z)} ,
ljJ (<p ,y) 2
-y ab ,
is defined by (3.6).
<p=!z
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Ann. Math. Statist.
[1]
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[2]
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[3]
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[4]
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[6]
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n
Ann. Math. Statist. 43, 1-30.
[7]
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[8]
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[9]
Miller, R.G., Jr. (1974).
~, 880-891.
An unbalanced jackknife.
Ann. Statist.
[10]
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[11]
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[14]
Sen, P.K. (1973). Asymptotic sequential tests for regular functionals
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[15]
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[17]
Sen, P.K. (1974c). On LP-convergence of U-statistics.
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[18]
Sen, P.K. and Ghosh, M. (1971). On bounded length sequential confidence intervals based on one-sample rank order statistics.
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Sen, P.K. and Ghosh, M. (1974). Sequential rank tests for location. Ann. Statist. ~, 540-552.
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