RESAMPLING METHODS FOR THE EXTREMA OF CERTAIN SAMPLE FUNCTIONS
by
Pranab Kumar Sen
Department of Biostatistics, University of
North Carolina at Chapel Hill, NC.
Institute of Statistics Mirneo Series No. 1896
March 1992
RESAMPLING METHODS FOR THE EXTREMA OF CERTAIN SAMPLE FUNCTIONS
PRANAB KUMAR SEN
UnivVL6ily 06 NoJLth CaJtOlina. at Chapel fU.ll.
For a general class of extrema of sample functions, including bundle strength
of filaments, for variance estimation, the classical delta method along with
jackknifing and bootstrapping are critically examined and their relative merits and demerits are discussed.
1.
INTRODUCTION
We motivate our study with the following simple model. Consider a bundle of
n parallel filaments whose individual breaking strengths are denoted by Xl , ... ,
Xn . These Xi are assumed to be independent and identically distributed (i.i.d.)
nonnegative random variables (r.v.) having a continuous distribution function
(d.f.) F, defined on R+ = [O,~}. Let Xn: 1 < •••< Xn:n be the associated order
statistics, and as in Daniels (1945), define
(1.1 )
On = max{ (n- i+l) X .' ; 1 < i < n }
n.'
--
as the bundle ~~ngth 06 fLlame~. Daniels prescribed a very elaborate analysis leading to the asymptotic normality of a normalized version of On' Based on
the behavior of the empirical d.f. (e.d.f.) Fn (x) = n-1E~ 1 I(X. < x}, x E R ,a
greatly simplified approach to this study was initiated by Sen, Bhattacharyya
and Suh (1973). We may note in this context that
Zn = n- 1Dn = max{ [1 - (i-l)/n]X .' : 1 < i < n }
n.' +- = sup{ [l-F n (x} ]x: x E R } , n ~ 1,
(1.2)
so that the asymptotic behavior of Fn may provide us with all the necessary
tools for the study of the same for Zn' Keeping this in mind and proceeding as
in Bhattacharyya and Sen (1976) and Sen (1976, 1981), we may conceive of a
sequence {X.;i -> 1} of i.i .d.r.v.'s with a continuous d.f. F, defined on RP,
for some p ~ 1, along with a smooth function $ : RP x [0,1] + R, and consider
a functional
e = eCF) = sup{ $(x,F(x}) : x E AC RP },
(1.3)
,=, -
,
AMS Subjec:t
Key
Wo~
c.fJu.6i.frtc.a:ti.On6. NOli :
62E20, 62E99
& Ph.!ta.5u : A6ymptoUc. noJrmaLi.t.y; a.6ymptoUc. vaJli.anc.e; b-i.a.6; boo:tAtJta.-
pp.i.ng; bundle
~:tJtength
06 6ilame.n.-U; deUa. me.:tho~; j a.c.kk.ni6J-ng .
Let Fn be the e.d.f. based on Xl' ... ,X n ' and define
"
(1 .4)
an = sup{ ~(x,F n (x» : x E A}, n -> no .
We assume that there exists a unique xo ' an interior point of A, such that
~ < a = ~(x ,F(x » < +00
and a < IT 0 = F(x 0 ) < 1.
(1.5 )
o
0
Also, we assume that for for every E > 0, there exists an n
~(x,F(x» < a - E
,for every x :11 x - xoll > n.
>
0, such that
(1.6)
Moreover, we assume that
xS~PA
I ~(x,Fn(X» - ~(x,F(x»1 ~
0, on the set
I IFn-FI I ~
O.
(1.7»
In many cases of practical interest, (1.6) may be verified by incorporating
Hadamard or compact continuity of functionals [viz., Sen (1988)]. Finally,we
assume that ~al(x,y) = (a/ay)~(x,y) exists for every x E A and y E (0,1),
and further, there exists a neighborhood of (x o ,IT 0 ) in which ~nl(.)
is uni...
formly continuous; we denote by
~
= ~01(Xo,ITo)
and
y
2
= ~ 2IT o(
(1.8)
1 - ITo)·
First, we may note that by (1.3) and (1.4), for every n ~ no '
I en - a I ~
/~PA I ~(x,Fn(x» - ~(x,F(x»1 ,
(1. 9)
so that by <.1.7) and the fact (that by the Glivenko-Cantelli leoma)
~ a almost surely (a.s.) as n~
we obtain that
II Fn-F II
00,
"
la n - al ~ a a.s.,asn~oo.
Also, if we set
an = ~(xon' IT on ) where IT on = Fn(x on ) ,
""
A
A
(1.10)
A
(J .11)
then, by (1.10), (1.11) and the a.s. ~onvergence of IIFn - FII to a , we have
Ilx on - Xo II ~ 0 a.s., and IT on - ITo ~ O. a.s., as n ~
(1.12)
"
"
"
Furt~er, b~ (1.3) and (1 .4), ~(xo,Fn(xo» - ~(Xo,F(Xo» ~ an - a ~ ~(xon,ITon)
- ~(xon,F(xon»' with prObabi1ity l. Moreover, if we incorporate the compactness
part of the weak convergence of nl /2(F n - F), we may claim that for every E > 0,
there exist an n. > 0 and a sample size no (= no(E,n», such that
P{II x-x
SUP'I <E n1/ 2 1Fn (x)-F(x)-F n(x 0 )+F(x 0 )1 > E} < n
,V n -> no. (1.13)
o
At this stage, we make use of (1.12) and the uniform continuity of ~01 (.) in a
neighborhood of (xo,IT o), and obtain from the above analysis that as n increases,
"
2
n1/2 Can-a)
~
N(O,y).
(1.14)
00.
This result extends Theorem 8.1.6 of Sen(l98l) under slightly less stringent
•
regularity conditions. In order to draw statistical conclusions on e based on
en ' the use of (1.14) rests on the estimation of y2. Moreover, e is a highly
n
non-linear estimator, and hence, its bias may not be of the order n- 1 . As such,
there may be a need to study the nature of bias, so as to make plausible corrections for moderate sample sizes. We shall look into both these Yariance estimation and bias estimation problems.
A
A
2.0RDER OF ASYMPTOTIC BIAS
We have already noticed that with probability one,
~(xo,Fn(xo)) - ~(xo,F(xo)) ~ en - e ~ ~(~on,Fn(~on))- ~(~on,F(~on))' (2.1)
In the particular case of the bundle strength of filaments, ~(x,y) = x(l-y) ,x
E R+ and y E [0,1], so that the left hand side of (2.1) is equal to x [F(x ) o
0
Fn(x o )] which has mean zero. Hence, en has a nonnegative bias. In fact, in this
special case, by the use of the reverse martingale property of{Fn }, it has been
shown in Sen et a1. (1973) that {e lis a reverse submartingale, and further, the
bias of en is 0(n- l / 2 ), as n ~ 00. ~ven in this special case, directAmanipulation
of the right hand side of (2.1) seemed to be quite complicated (as x is stochastic), and a bias of the order n- 1 did not appear to be evident
on
To study the bias term 1n a general setup, we rewrite (2.1) as
A
A
A
o
~{en
- e - [
"
~(xo,Fn(xo))
A
A
-
~(xo,F(xo))
] }
(2.2)
A
~ { [ ~(xon,Fn(xon))- ~(xon,F(xon)) ] - [ ~(xo,Fn(xo))- ~(xo,F(xo))]} .
Given enough smoothness conditions on ~( .. ), we may be tempted to make use of a
Bahadur (1966) type representation for the right hand side of (2.2) wherein we
may need to use suitable rates of convergence in (1.12) and (1.13). Further, as
we need to take expectations , we may have to use the Duttweiler (1973) approach
to the Bahadur representation along with Theorem 7.3.1 of Sen (1981) to cope with
a slower rate than n- l / 2 ( for ~ ~ x ). The end product is that typically the
A
on
0
bias of ( en - e ) is O(n- ), for some A : 1/2 < A <3/4. In this context, we may
note that the Hadamard continuity in (1.7) does not necessarily insure the Hadamard
differentiability, and hence, the usual (first or second order) differentiability
properties of the functional under review may not hold. Expecting a second order
differentiability property to hold would restrict the class of ~( .. ) and hamper
the generality of the result in (1.14). Also, in the negation of such differentiability, the classical jackknifing may not work out well in reducing the bias to a
lower order. For this reason, we need to explore other possibilities. Among these
the sub-sampling method considered by Sen (1990) may work out well.
A
In this context, we may note that if for a statistic Tn based on n observations,
(2.3)
E( Tn ) = e + a(F ) n"A + o(n -A ), for some A : a < A <1 ,
where a(F) is an unknown functional of the df F, then
E{ nT n - (n-l)T _ } = e + (1- A)n-A.a(F) + o(n- A ),
(2.4)
n l
so that for A < 1, the order of the bias term remains the same, although its
contribution is discounted by the factor (1- A). This very simple observation
implies that for an order of bias O(n- A ), A < 1, there may not be enough incentive in using the classical jackknife for bias reduction, although its utility
in estimating the asymptotic variance may still remain in tact. The situation is
no better for the classical bootstrap. Since the bootstrap sampling allows a
possible duplication of some of the units in the original sample ( with a positive probability) , the effect of ties arising in this manner may actually make
the bias term even worse! For this reason, we may find it convenoent to use a
delete-d jackknifing method (where d is not too small) as a better compromise in
controlling the bias term without making any real difference in the asymptotic
variance term. We may refer to some of the details in Sen (1989).
3.
RESAMPLING PLANS FOR VARIANCE ESTIMATION
First, we consider a naive estimator of y 2 . Using (1.8), (1.10), (1.11)
and (1.12), we propose
"2
Yn =
"2 "
~n
on ( 1 - 7T on
as an estimator of y2 , where we let
"
~n
7T
""
(3.1)
= tP Ol ( xon ' 7T on ).
(3.2)
This is actually based on the classical delta..method, and the estimator is con..
sistent under the assumed regularity conditions. However, as we have noticed
"
"
that the original estimator en in (1.4) is highly non ..linear, the estimator Yn
in (3.1) may inherit the sensitivity of the delta..method to basic nonlinearity
of the functional ,and as such may have significant bias component. For this
reason, we may like to explore alternative methods based on suitable resampling
plans with due emphasis on the classical jackknifing and bootstrapping methods.
Let X(i ) be the sample of size n-l obtained by deleting the ith observation
-n .... l
(X.) from the base sample of size n, for i=l , ... ,no We denote the empirical d.f.
as~ociated with ~~.~~ by F~~i ,and the estimator of e based on this subsample
is denoted by e(~)l ' for i= 1, ... ,no In the classical jackknife method, we
n"
then introduce the pseudo-values en ,1., i= 1, ... , n, as
f·
1
6'"n,l. = n 6'" n _ (n-.l) 6"'(i)
n- 1 ' or 1 = , •.• ,no
Then the classical jackknifed version of the estimator 6n is given by
'"
-1 n
'"
6nJ
= n r.1= 1 6n,l.
'"
1 n
"'(")
(3.3)
"
(3.4 )
= 6n + (n - l){ n- L 1--1 ( 6n-11 - 6n )}.
It follows from (2.3) and (2.4) that the bias of the jackknifed version is O(n-A)
where A is greater than 1/2, but not more than 3/4. In this sense, jackknifing
has reduced the bias to some extent, but not tOI any lower order of magnitude. Let
us also define
-1
n
'"
"2
VnJ = (n-l) r i =l{ 6n ,i - 6nJ }
= (n -1) E~ -1 ( e (i 1) - n-1 r~ 6(j) ) 2
(3. 5)
1nJ=l n-l
as the jackknifed variance estimator.Since the sUbsample (nonlinear) estimators
e~~~ , i=l, ... ,n and en are directly involved in (3.5), VnJ is expected to be
less sensitive to the inherent nonlinearity of the functional than the classical
delta method. Let Cn = C(Xn : l"'.'X n:n ;X n+.,j
J -> 1 ), n > 1 be the sequence of
tail sigma field. Then, we may rewrite (3.5) as
'"
'"
2
VnJ = n(n-l)E[{ 6n~1 - E( 6n_1 1 Cn )} I Cn ]
(3.6)
so that a sufficient condition for the consistency of V J is the a.s. convergence
'"
n
of the standardized conditional variance of 6n ' given Cn+1 ' as n ~ 00.
In the particular case of the bundle strength of filaments, we have already
noticed that {6'" ;n > 1} is a reversed submartingale, and hence, we may even link
n (3.6) as a vital component towards the asymptotic normality result in (1.14).
'"
Motivated by this, our goal may be to approximate the sequence {6 n ; n ~ 1} by
a reversed martingale, and use this cha:acterization in",t~e co~v~rgence of VnJ
in C3.5} or (3.6). For the subsample ~~:~ , we define 6~~~ , x~:~_l and IT~~J_l
as in (1.11), for every i( =l, •.. ,n). Then, note that
max{ sup[1 F(i )(x) - F (x) I : x E: R] : 1 < i < n} = n- 1
(3.7)
n- 1
n
with probability one, and hence, we can easily extend (1.12) as follows: As. n -+.00 ,
max I IT'" (i)
I 0
()
- Xo II ~ 0 a.s., and l.::,i<n
o ,n-l - ITo ~ a.s. 3.8
o
This enables us to make use of the uniform continuity result on W01(·) for every
subsample ~~~.~ , i=l, ... ,n, when n is large. Note that for every i"( =1, ... ,n),
" (i) =
~ ei)
(i) '" (i) )
(3. 9)
6n_1
wCx o ,n_l' Fn - 1 (X o ,n-l) ,
so that
max
l.::,i<n
II x (i)
,n-l
(3.110)
On the other hand, for every x
RP,
€
F(i ) (x) - F (x)
n- 1
n
= (n_l)-l{ nF (x) - I(X. < x) } - F (x)
n
1n
-1
()
.
(3.1lJ)
= (n-1) {F x - !(X~ < x)} ,1 =l, ... ,n.
n
.Thus, using the uniform continuity of ~01(.) in a neighborhood of (x o ' ~o)
along with (3.8), we conclude that the left hand side of (3.10) behaves as
(3.12)
~Ol(~on,Fn(~on)){ (n_l)-l[ Fn(~on) - I(X i ~ ~on)]} + op(n- 1 ),
uniformly in i = 1, ... ,no In a similar manner, the right hand side of (3.10)
can be expressed as
~
(~(i)
F (~(i) )){ (n_l)-l[ F (~(i) ) - I(X. < ~(i) 1)]} +
o,n-l' n o,n-l
n o,n-l
1 - O,n0p (n -1) ,
(3. 13 )
for every i (=l, ... ,n). Incorporating (3.8) along with the uniform continuity
of ~01(.) in a neighborhood of (xo'~o), the first factor in (3.12) as well as
(3.13) can be replaced by ·~Ol(xo'~o) , and the difference can be absorbed in
the second term i.e., with 0p(n- 1 ), uniformly in i (=1 , ..• ,n). Secondly, if we
sum over i (=1, ... ,n), the second factor of the first term in (3.12) becomes
identically equal to 0, so that, by (3.10) and (3.12), we obtain that
n
01
-1
n
(
r·_
1- 1
"(i)"
8n- 1 - 8)
n
>
0
-
p (n
-1
(3.14)
).
Similarly, if we let
"*
xO,n
F (~(i) ) and x"**
n o,n-l
o,n
"*
max
Fn(xo,n) = l<i<n
then
"**
"*
Fn(Xo,n) - Fn (x o , n) -<
n-lr~1=1 [F n (~{i)
)
o,n-l
_ I(X. <
1-
"
"**
~(i)
)]
0,n-1
Fn(X~,n) - Fn(xo,n ),
<
(3.16)
where, by (3.8), both the left and right hand sides of (3.16) are oClla.s:. ~sothat
n -1 E~
(e (i)
_
e)
<
0
(n - 1) .
1=1
n-l
n
p
From (3.14) and (3.17), we obtain that as n
(3.17)
+
00
,
(3.18 )
n -1, Eni =l ("(i)
8n_1 - 8"n) I = 0p (-1)
n .
1
In fact, in (3.12) and (3.13), the remainder terms are o~) a.s., as n + 00, and
1
1
hence, in (3.18) too, we may replace op(n)
by o(iii)
a.s., as n ~. As such, by
(3.5), (3.6) and (3.18), we conclude that
VnJ = (n_l)-l~~J =1 (e(j)
n-1 _
L.
SA
n )2 + 0(·1) a. s., as n
-+-
00.
(3.19)
This enables us to use again (3.10) through (3.13) with the 0 (n- 1 ) terms being
replaced by 0(n- 1 ) a.s., as n -+- 0 0 . Towadrs this, using (3.8), we first write
(3.13) as
~Ol(;on' Fn(;on)){ (n-l)-l[Fno,
(;(i) 1) - I(X, < ;(i) )]} + 0(n- 1) a.s. (3.20)
n1 - 0 ,n -1
Secondly, we write
Fn (;(i)
o,n-l ) - I(X,1 -< ;(i)
o,n-l ) = Fn(;on) - I(X 1, ._< ;on) +
for every i(=l, ... ,n), where
Rni = Fn(;~~~_l) - Fn(;on) - I(X i ~;~:~-1)
+
I(X i
~nl'
~ ;on
(3.21)
), i=l, ... ,n. (3.24)
Note that by (3.15) and (3.22),
-1 n
-1 n
A2
A(i )
A
n E.1= 1 Rnl. -< 2 n E.1= 1[I (X.1 -< x
l ) - I (X,1 -< xon )]
o,n+ 2 n-1E~ [F (;(1)
) F (A )]2
1=1 n o,n-l
- n xon
-1 n
A**
A*
< 2 n E. 1 I( x
< X. < x
1=
o,n 1 - o,n )
A*
A*
+ 2[ Fn (x-O,n ) - Fn(x o,n ) ]
A*
A**
2
2
A*
A**
= 2[F n(x o ,n) - Fn(Xo,n ) ][ 1 + Fn(xo,n) - Fn(xo,n) ]
= 0(1) a.s., as n -+- 00 ,
(3.23)
where in the last step we have made use of (3.8), (3.15) and the Glivenko-Cantelli
lemma. Consequently, by (3.12), (3.13), (3.20) - (3.23), we may write V in
nJ
(3.19) as
...1 n
2
A
A "
"2
VnJ = (n-1)~ E·i=l~Ol(xon,Fn(xon))[Fn(xon) - I(X i ~ XOn )] + 0(1) a.s.
= n(n-1)-l ~~l(;on,Fn(;on)) Fn(;on)[l-Fn(;on) ] + 0(1) a.s.
= n(n-1) -1 y"2 + 0(1) a.s., as n -+- 00 ,
n
(3.24)
where ~~ is defined by (3.1), and by (1.12) and (3. 1), ~~ is a strongly consistent
estimator of y2, defined by (1.8). Thus, we obtain that
VnJ -+- y2 a.s., as n -+- 00 •
(3.25)
Although the (strong) consistency of VnJ has been established here through the
asymptotic a.s. equivalence result in (3.24) and the a.s. convergence of the naive
estimator ~2 to y2, as has been noted earlier, V J may be much less sensitive to
n
n
nonlinearity of ~(.), and hence, should be more robust.
Let us look into the bootstrap variance estimation problem. Corresponding to
the given sample Xl , ... ,X , we define the empirical d.f. F as in earlier, and
*
*
n
n
let Xl"",X n be n (conditionally) i.i.d.r.v. 's drawn from the d.f. F ; we
n
denote the empirical d.f. for the X.*1 by Fn* . Then, we define
~*
*
en
= sup{ 1jJ(x, Fn(X)) : x
E:
A CoRP
}.
(3.26)
For latter use,
"*
en = 1jJ(
we write
"'*
"*
"*
* "*
xon ' TT on ) where TT on = Fn(x on )
(3.27)
For some positive integer M (usually large), we draw Msuch (conditionally independent) bootstrap samples from Fn , and denote the corresponding estimators by
A*
A*
en,l , . .. , en, M
(3.28)
Let then
*
-1 M
"*
2
(3.29)
VnB = nM
E.1= 1(e n,l. - en )
be the bootstrap estimator of the asymptotic variance of nl / 2 (
e ) . We
*
n
like to study the convergence properties of VnB and also compare it with the
jackknifed variance estimator VnJ .
Recall that by definition,
"'"
*
A
A* * A*
A*
A*.
1jJ(xon,Fn(xon)) - 1jJ(xon,Fn(xon)) ~ en - en ~ 1jJ(xon,Fn(xon)) - 1jJ(x on ,Fn (X on )),
(3.30)
and further,
•
A
e-
A
A
A
l 2
II
A*
II
(3.31)
a.e. (F).
n
By virtue of (3.31) and the assumed regularity conditions on 1jJ , it can be shown
that as n -+
"*
"
"*"
Ilx on - xon II -+ 0 and ITT on - TT on I -+ 0, in probability,
(3.32)
and further, the classical Bahadur (1966) representation of sample quantiles holds
for the bootstrap empirical d.f.[ see for example, Gangopadhyay and Sen (1990)J.
As such, using the uniform continuity of 1jJ01(') in a neighborhood of (x o ' TT O)'
it foll ows from (3.30) through (3.32) that as n -+
n /
00
F*
n
-
F
n
= 0 (1)
p
,
00,
1/2
n
"*
A
" "
(en - en ) - 1jJ01(xon,Fn(xon)) n
1/2
*
A
"
[Fn(x on ) - Fn(xon)J
0, in probability ( a.e. Fn ) ,
(3.33)
so that using the conditional i.i .d. nature of the set in (3.28), and the representation in (3.33), we obtain from (3.29) that as n -+
*
2'"
"
VnB - 1jJ01(x ,F (x ))Fn(x )[l-F (x n)J -+ 0, in probability. (3.34)
on n on
on
n 0
-:"2
Therefore, from D.l), (3.34) and the consistency of the naive estimator Yn ' we
-+
00
A
A
,
'.
conclude that under the assumed regularity conditions, as n +
00
,
* ' V and Y"'2 are equivalent in probability, and they
VnB
nJ
n
2
estimate y consistently.
(3.35)
It may be remarked her.e that~~ ~n~ VnJ are both strongly consistent estimators
of the asymptotic variance of n~(6n - 8 ), whereas, we have been able to establish only the weak consistency of the bootstrap variance estimator. This is
mainly due to the fact that in bootstrapping we consider the conditional law
given the base sample, and hence, an a.s. result will not only require the a.s.
part on the conditional setup but also on the part of the base sample. Thus, we
would have to deal with a double sequence, and hence, we may need extra regularity conditions. On the other hand, for the conditional law, there are some
nice recent developments [ viz., Gine and Zinn (1990)] which may be incorporated
to obtain some uniformity results. However, we shall not probe into these here.
4.
PERFO~1ANCE
CHARACTERISTICS UNDER ADDITIONAL SMOOTHNESS CONDITIONS
For simplicity of presentation, we assume that in (1.3)-(1.4), x or the Xi
are real valued; the results to follow continue to hold even if x £ RP, for p
> 2. We assume that in a neighborhood of (x ,TI ), ~(x,y) has continuous second
· 0 0
order partial derivatives (with respect to x,y), and these are bounded. We denote
the first order derivatives by ~10(.) and ~Ol(.)' while the second order ones are
denoted by ~20(.)' ~ll(·) and ~02(·) respectively. Also, we denote the density
function for the d.f. F by f(x), and assume that f(x) is continuous in a neighborhood of Xo . Then, note that by definition of 8 in (1.3) and (1.5),
~lO(xo' TI o)
+ f(x o) ~01 (xo,TI o)
= o.
(4.1)
We have already assumed that ~Ol(xo,TIo) is nonzero ( as otherwise y2 will be
equal to 0 and we would have a degenerate normal law in (1.14)). Then, by a
local expansion of ~(x,F(x)) around ~(xo,F(xo))' we obtain that as x + Xo '
Cx",xo)~l 0 (x o ,F(xoD = -[F(x) -FCx o )]~O,l (x o ,F(xo )) + O((F(x) -F (x o) )2) .
On the other hand, oy (4.1), and a local expansion of
we obtain that as x + x ,
~(x,F(x))
(4.2)
around (xo,F(x ))'
o
o
~(x,F(x)) = ~(xo,F(xo)) - O( [x - xo]2 + [F(x) -F(Xo )]2 ).
(4.3)
Combining (4.2), (4.3) and the Bahadur (1966) representation for { Fn(x) - F(x)
- Fn(x o) + F(x o) }, Ix - Xo I :If O(n- 13 (logn) 1/2), for some 13 13 ~ 1/2), [see,
Theorem 7.3.1 of Sen (1981)] , we obtain that
-1/3 (log n) 1/2)_ a .s ., as n +
(4.4 )
I xon ,.. x0 I = 0 (n.
A
00
•
As such, by an iterated use of Theorem 7.3.1 of Sen (1981), we obtain from (3.1),
(4.4) and the assumed boundedness of W02(.) in a neighborhood of (xo,F(x o» that
;~ = y2 + 0(n- 1/ 3 (10g n)1/2) a.s., as n + 00 •
(4.5)
"
This exhibits the good performance of the naive estimator y in (3.1) even when
the functional is not so smooth to induce a bias term of th~ order n- 1 .
Let us next look into the picture of the jackknifed variance estimator VnJ .
As in before (3.7), we introduce the notations ~(i) 1 ' F(ii etc.- for i=l, ... ,n.
Using (3.10) along with the fact that w(x on,Fn(io'nn) ~ ~(x(i)
.o,n _l,F
. n(;(i
0, n)-1»' for
every i (=l, .... ,n), with probability 1, we obtain that for every ;(=l, ... ,n),
o ~ (e~:~
- en ) - [ W(Xon,F~:~(~on»
"(i)
(i) "(i)
W(~on,Fn(~on»
"(i) "
]
"(i)
"(i)
- w(x on ' Fn_1(x on » - w(x o,n_l,F n(X o ,n_1»
"
"
+ Wexon' Fn(xon » }.
(4 .6)
{w(x o,n_l,F n_1(x o ,n_l»
<
-
Making use of the Taylor expansion on the right hand side of (4.6) and the fact that
F(i 1)(x) - F (x) =-{I(X. < x) - F (x)}/(n-1), for every x and i, we obtain that the
nn
1 n
right hand side of (4.6) can be written as
W01(~on,Fn(~on»(n-1)-1{
- I(X i ~ ~~~~-1) "(i)
-1
Fn (X o ,n_l) } + o(n ) ,a.s., as n + 00,
I(X i
~ ~on)
Fn(~on)
+
(4.7)
so that
(n-1)[
(e~~~
- en) 2"
{W(~on' F~~1(~on»
"
= 1P 01 (Xon,Fn(x on ) )
where
-
W(;on,Fn(~on»
} ]2
2
An i '
(4.8)
L~ =1 A~ i = a(n -1/31og n) a. s . , ,a s n
+
00
(4.9)
•
In the derivation of (4.9), we have made use of (4.1) through (4.4) along with
"*
"**
-1/3
(3.15) and (3.16), whereby we are able to show that Fn(xo,n) - Fn(xo,n) = O(n
.
log n) a.s., as n + 00 • By virtue of (4.8) and (4.9), we obtain that as n + 00 ,
n
" ( i) "
VnJ = (n-l) Li=l{ ~(xon,Fn_l(xon»
_ *
-1/3
log n) a. s . ,
- VnJ + a(n
"
"
- ~(xon,Fn(xon»}
2
+ O(n-
1/3
log n) a.s.
(4 .10)
where by a simple Taylor expansion, it follows that as n + 00 ,
V*nJ -- Y
r.2 + 0(n-1/2 log n) a.s..
(4.11)
n
Thus, (4.5) extends to VnJ as well. On the other hand, in (4.10), the nonlinearity
of ~(.) is retained to a greater extent (than in (3.1», so that V~J and hence
,
VnJ are likely to be more robust than ~~ when n is not so large.
Let us consider next the bootstrap variance estimator VnB defined by (3.29).
Note that given Fn , the estimators in (3.28) are conditionally i.i.d., so that if
we define
(4.12)
en ) I Fn },
and
V~ = V~(Fn) = E{ n (e~
Fn },
(4.13)
n )2
e
then, by (3.29) and the conditional i.i.d. nature of the estimators in (3.28), we
conclude that given Fn ,
* - V0
VnB
n
a
a.s., as M +
(4.14)
However, a suitable rate of convergence in (4.14) may demand extra regularity
f'o*
conditions ( particularly higher order monents of the en)' On the top of that,
we may write
+
"'*
00
'"
V~ = n.Vad ( en'" en)
I
•
2
Fn } + an '
(4.15)
so that the behavior of V~ is not only dependent on the conditional variance in
(4.15) but also on the as:ymptotic bias tenn in (4.12). In this context, we may
note further that although llJ.cx,F(x)) is twice differentiable with respect to x
C as has been assumed earlier in this secti.on) with (d/dx)llJ(x,F(x)) = a at ~ = xo '
the functi on llJ(x ,Fn(x)) may not have a derivative (even fi rst order ) at xon
This may entail some non-smooth nature for which the rate of convergence for
* (x)) may be somewhat slower. For example, in (4.1), if we replace F(x )
llJ(x,F n",
o
by F (x )) and want to have a similar identity, that won't work because Fn may
n on
'"
not be differentiable at x . As a result, in (4.3), with F(x) replaced by Fn(x),
on
'" 2
2
we may have a slower rate than (x - xon ) + [Fn(x) - Fn(x on )] • Moreover, even
if F(x) - F(y) is O(n- l ), Fn(x) - Fn(Y) may not be of the same order [Bahadur
(1966) representation generally yields an order n- 3/ 4 0og n)1/2 a.s. ]oSimilar
*
A*
slower rates of convergence remain applicable to the Fn(x) and xon . Thus, we
may not be in a position to claim the same rate of convergence as in (4.11).
We may remark in passing that the empirical distribution of the nl/2(~~ - en)
in (3.28) may be used to provide a confidence interval for a . But, any claim that
such an interval behaves better than the conventional one based on (1.14) and (3.1)
may demand the validity of the bootstrap Edgeworth expansion, which in this nonsmooth case remains largely open for verification. The. standard tools developed for
functions· of the mean type statistics may not be usable here.
A
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[lJ
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[2J
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[3J
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[4J
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[5J
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[6J
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[l]
SEN, P.K. (981). Sequential Nonparametrics: Invariance Principles and
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[8J
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[9J
SEN, P.K. (J988b). Functional approaches in res.amp1i.ng plans: A review of
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\~
[10J SEN, P.K. (989). Whither delete-I< jacKKni'fing for smooth statistical functiona1s? In Statistical Data Anal~sis and Inference (ed: Y. Dodge),
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•