Work supported by the Air Force Office of Scientific Research, AFSC, USAF, Grant No. AFOSR ...

25 Pages·2008·0.57 MB·English

*
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 popula-
tion 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 pre-
assigned-strength sequential test for a suitable parameter based on jack-
knife estimators.
AMS 1970 Classification No: 60BIO, 62E20, 62G35,62LIO
Key Words &Phrases: Almost sure convergence, Brownian motions, bounded
length 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 asympto
tic 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 approxima
tions for the tail-sequence of jackknife estimators. These results are
then incorporated in providing (i) a bounded-length (sequential) con
fidence interval and (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. Section 4 displays parallel results for
general estimators. The two sequential problems of estimation and test
ing are treated in the last two sections of the paper.
-3-
2. Preliminary Notions
Let {X., i ~ l} be a sequence of independent and identically distri
1
buted random vectors (i.i.d.r.v) with a distribution function F, defined
on the p(~l) - dimensional Euclidean space RP. Let
e
(2.1) = T (Xl' ... 'X ), n ~ 1
n n n
be a sequence of estimators of a parameter 8, such that
(2.2) E8n = e+ n-181 + n-2S2 + (=> E(8An-8) = O(n-1))
where the B. are unknown constants. Let us denote by
]
Ai
(2.3) 8 = T l(X '· .. ,X. l' X·l,···,X), l~i~n ,
n-l n- l 1- 1+ n
A Ai
(2.4) 8 = nS (n-l)8 l' l~i~n ,
n,i n n-
(2.5) 8* = n-lIn. 18A . = ne - (n-l){n-lIni=18Ai- }
n 1= n,1 n n l
Then, 8* is termed the jackknife estimator of 8. Clearly, by (2.2),
n
(2.3), and (2.5),
-2
(2.6) E8~ = 8 - B/n(n-l) + ... (=> E(8*-8) = O(n )).
n
Further let,
n I I
(2.7) v* = __1__ I [8 ._8*]2 = (n-l) (Si - 1 8j 1)2.
n n-l. 1 n,1 n i=l n-l n j=l n-
1=
In support of the Tukey conjecture, various authors have shown that under
suitable regularity conditions,
!-.< !-.< V
(2.8) n 2(8n*-8)/ [vn*] 2 + N(0,1) as n + 00 •
-4-
Our contention is to obtain stronger results concerning (i) the almost
sure (a.s.) convergence of v* and (ii) Wiener process approximations
n
for the tail-sequence {e~-e; k ~ nl.
For simplicity, we assume that p = 1 i.e., the X. are real valued
1
and R = (_00, 00) • For every n(~l), the order statistics corresponding
to Xl"'" Xn are denoted by Xn,1:0; ••• :0; Xn,n Let Cn = C(Xn,1"'" Xn,n,
Xl"") be the cr-field generated by (X l""'X ) andby X .,j~l.
n+ n, n,n n+J
Then, C is non-increasing in n(~l). Note that given C , X ., j ~ 1
n n n+J
are all held fixed while (Xl, ... ,X ) are interchangeable and assume all
n
possible permutations of with equal conditional proba-
(Xn,1"" ,Xn,n)
-1
bility (n!) . Hence,
I
-l~n ~i
(2.9) E(eA 1 C ) = n L'_le 1 a.e. ,
n- n 1- n-
and, therefore, by (2.5) and (2.9),
(2.10) e* = ne - (n-l)E(8 llC) = 8 + (n-l)E{8 -8 llC} a.e.
n n n- n n n n- n
e .
Clearly, if {8 ,C } is a reverse-martingale, e* = otherwise, the
n n n n'
jackknifing consists in adding up the correction factor
e
(2.11) e* - = (n-l) E{(8 -8 1) IC }
n n n n- n
It follows by similar arguments that
(2.12) v* =n(n-l) Var{(8 -8 1) Ic }
n n n- n
= n(r-l){E[(8 -8 1)2IC ] - (E[(8 -8 l)IC ])2} .
n n- n n n- n
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. In
fact, we have also a second interpretation for 8* e** etc. from the
n' n
weighted least squares point of view. In most of the cases to follow,
we shall observe that for some VI > 0 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
'" -1-2
Also, by (2.2), E(en-e) = n ~\ +n 8 + . .. . Thus, neglecting terms
2
-2
of O(n ), the weighted least squares method of estimating
consists in minimizing
2'" 1 2 "" 1 1
(2.16) n (en - e - n f\) - 2n(n-l) (en - e - nSl) (8A _ - 8 - n-l Sl)
n l
'" 1 2
+n(n-l)(en_l-e-nS )
l
with respect to e and Sl; the simultaneous equations yield
(2.17) 8 = n§ - (n-l)8
w n n-l
Ie ).
and our e* = E(e Similarly, by (2.14)-(2.15), on writing
n w n
(2.18) (k-l)8 _ , k~ 2 ,
k l
(2.19) Var(Z ) Cov(Z ,Z 1)
n n n-
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Thus, Z , Z 1 are asymptotically uncorrelated and by (2.2),
n n-
EZ :: e- 8/k(k-l) + O(k-3). Hence, considering
k
(2.20)
and minimizing with respect to (e, ( ), we obtain the weighted least
2
squares solution
(2.21)
In fact, we obtain the same solution by working with (8,8 1,8 2)
n n- n-
and applying the weighted least squares method directly on it. Again,
e** :: E(8 Ie ). In general, if we want to reduce the bias to the O(n-k -l) ,
n w n
for some k 2: 1, then we need to work with (eA ,... ,eA k) and the k-th
n n-
e )
order jackknife estimator is the conditional expectation (given of
n
e,
the weighted least squares estimator of neglecting 8 ., j 2: 1 in
k+J
(2.2) and terms in (2.14)-(2.15). Thus, we have the following.
Theorem 2.1. Under assumptions (2.14)-(2.15), the jackknife estimators
e )
(of different orders) are the conditionaZ expectations (given of the
n
weighted Least squares estimators obtained from the originaZ (biased) esti-
mators 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 ), symmetric in its m arguments, be a
l m
Borel measurable kerneL of degree m(2:l) and consider the regular func-
tionaL (estimable parameter)
(2.22)
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I I
where F = {F: I;(F) < oo}. Then, for n ~ m, the U-statistics correspond-
ing to I; is defined by
[n)
(2.23) U = -lIe <j>(X., .•• ,x. ) e = {I::;; i 1 < . • • < i ::;; n}
n m n,m 11 1m n,m m
Note that EU = I;(F), V n ~ m. Further, let
n
(2.24)
for h =0, ... ,m, where ?::O = 0 and cPO = 1;. We assume that
(2.25) O<l;,?:: <00 (where
1 m
3. 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 R, have a bounded second derivative in
some neighborhood of 1;, and
(3.1)
(ii) For some positive integer q, we have
e Iq oct ~
(3.2) = U(s) , n m ,
n s= n,s n
where U(O) = U is an unbiased estimator of e = I;(F),
n n
-1 -2 -3
(3.3) ctn,O=l+n cO,l+n c ,2+ 0(n ),
O
-8-
U~l) ,u~q)
, ... are appropriate U-statistics with expectations 8 ", .,8
1 q
(unknown but finite) and
-h -h-l
=
(3.4) CI.n,h n ch,0 + 0(n ), h ~ 1
the c . are real constants; possibly, some being equal to o. The
S,]
classical von Mises' (1947) differentiable statistical function (corre-
sponding to E,(F)) is a special case of (3.2) with q =m and cO,1=-(;) .
First, we consider the following.
Theorem 3.1. For {e} defined by (3.1) or (3.2)-(3.4),
n
(3.5) Vn* -+ y2 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),
(3.7) n(n-l)E[(Un-l-Un)2Ien] -+ m2~1 a.s., as n -+ 00 •
On the other hand, as in Section 2,
2Ie ] \,n i 2
(3.8) n(n-l)E[(U l-U) = (n-l)l,'_l[U l-U]
n- n n 1- n- n
where the are defined as in (2.3) with T being replaced by
n-l
U l' Hence, from (3.7) and (3.8), we obtain that
n-
-9-
(3.9) a.s., as n + 00 •
Further, {U , C , n ~ m} is a reverse martingale, so that U + E,(F)
n n n
a.s., as n + 00, and hence, by (3.9),
I
max I i
(3.10) 1-<l'<-n Un_l-E,(F) + 0 a.s., as n + 00 •
First, consider the case of (3.1). Then, we have
e
8 -
(3.11) = g(U )-g(U )
n-1 n n-1 n
]+~g"(hU
= g'(U )[U 1-U +(l-h)U l)[U 1-U ]2, O<h<l.
n n- n n n- n- n
Note that E[U 11C] = U a.e. and further by (3.7), (3.8), (3.10) and
n- n n
the boundedness of g" (in a neighborhood of E,), we have
}!
(3.12) IE{g"(hU +(l-h)U l)[U 1-U ]21C
n n- n- n n
$; m~x I fIChU +(l-h)Ui )I{.!. \'~ [Ui -U ]2}
l$;l$;n g n n-1 n L1=1 n-1 n
-2
= O(n ) a.s., as n
+ 00 •
Hence, we obtain from (3.11) and (3.12) that
I
(3.13) E(8" n-1-8An Cn) = O(n-2) a.s., as n + 00 •
Similarly,
"" "" 2 -4
(3.14) Var{(8 1-8)IC}=E{(8 1-8) C}+O(n) a.s.
n- n n n- n 1 n
-l\,n i 2-4
= n Li=l[g(Un_1) - g(Un)] +O(n ) a.s., as n + 00 •
Again as in (3.12), for some 0 < h < 1