HITOTSUBASHI

UNIVERSITY

Center for Financial Engineering Education (CFEE)CFEE Discussion Paper Series No. 2008-1

Smoothed Versions of Statistical Functionals from a Finite Population

Hitoshi Motoyama and Hajime Takahashi

Center for Financial Engineering Education (CFEE)
Department of Economics, Hitotsubashi University
2-1 Naka, Kunitachi Tokyo, 186-8601 Japan
http://www.econ.hit-u.ac.jp/~finmodelSMOOTHED VERSIONS OF STATISTICAL FUNCTIONALSFROM A FINITE POPULATION *Hitoshi Motoyama** and Hajime Takahashi***
We will consider the central limit theorem for the smoothed version of statistical
functionals in a nite population. For the innite population, Reeds (1976) and Fernholz (1983) discuss the problem under the conditions of Hadamard dierentiability ofthe statistical functionals and derive Taylor type expansions. Lindeberg-Feller's central limit theorem is applied to the leading term, and controlling the remainder terms,
the central limit theorem for the statistical functionals are proved. We will modify
Fernholz's method and apply it to the nite population with smoothed empirical distribution functions, and we will also obtain Taylor type expansions. We then apply theErdos-Renyi central limit theorem to the leading linear term to obtain the central limit
theorem. We will also obtain sucient conditions for the central limit theorem, bothfor the smoothed inuence function, and the original non-smoothed versions. Some
Monte Carlo simulation results are also included.
Key words and phrases : asymptotic normality, central limit theorem, dierentiable
functional, empirical distribution function, nite population, functional Taylor series expansions, Hadamard dierentiable, inuence function, kernel smoothing, ocial
statistics, opinion poll, simple random sampling, statistical functional, survey sampling, uniform topology.

1. Introduction

We will consider the central limit theorem for the statistical functionals in a nite
population. A reason why we assume an innite population is to simplify both the theory and the computation. However, in the area of sampling surveys (ocial statistics,
opinion poll, etc.), the sample size is fairly large compared to the population size, andthus it may be inappropriate to apply classical statistical theory, especially large sample
theory, directly to these problems. On the other hand, the progress of modern computer
technology makes it possible to work directly with nite population problems in many
This version 10/10/2008. This paper has been subsequently published in the Journal of Japan
Statistical Society, December 2008, Vol. 38, No.3, pp.475-504.
*This article is a part of the rst author's doctoral thesis submitted to Hitotsubashi University.
**The Graduate School of Economics, Hitotsubashi University, 2-1 Naka Kunitachi-shi Tokyo Japanand Statistical Information Institute for Consulting and Analysis, 6-3-9 Minami-Aoyama Minato-ku
Tokyo Japan. This research is supported in part by the Grant-in-Aid for Scientic Research from the
Ministry of Education, Culture, Sports, Science and Technology 15653014, 17330046 and in part a grantfrom the 21st Century COE Program \Research Unit for Statistical Analysis in Social Sciences" atHitotsubashi University.
***The Graduate School of Economics, Hitotsubashi University, 2-1 Naka Kunitachi-shi Tokyo Japan.
This research is supported in part by the Grant-in-Aid for Scientic Research from the Ministry of
Education, Culture, Sports, Science and Technology 18203013.

1

areas of statistical applications, and methods based on resampling, such as the bootstrap
method, help us to measure the accuracy of the estimators. Here, the justication of
resampling methods may depend on the central limit theorem for the estimators andwe will prove the asymptotic normality of the estimators by applying the theory of statistical functionals developed by von Mises (1947), Reeds (1976), Fernholz (1983), andTakahashi (1988), among others.
We will also consider smoothed versions of the empirical distribution functions. Although it may sound strange in the nite population problem, in some practical situations it is reasonable to assume that the underlying distribution function converges tothe continuous distribution function as the population size goes to innity. Here we have
implicitly assumed the probability space where continuous distribution functions are alsodened. The nite probability spaces are embedded in the space which makes it possible
to consider the limits operation. In this case, the smoothed bootstrap may be applied,
and it is worth considering the smoothed version of the empirical distribution function
and the statistical functionals dened on it. For these reasons, we will derive asymptotic
normality of smoothed statistical functionals for a simple random sample from a nite
population. The non-smoothed version will be obtained as a simple corollary to our
results.
To x the idea, we let  N = fx  : : : xN g be a mutually distinct nite populationof size N , and a simple random sample (X  X  : : : Xn ) is taken from  N without
replacement. More precisely, let (  : : : N ) take all possible permutations of (1 : : :  N )with common probability (N !); , and Xi = x  1  i  n.
Dene a population distribution function (d.f.)
(

)

1

1

(

2

)

1

1

i

N
X
FN (x) = N1 I ;1x (xi)
(

i=1

]

and an empirical distribution function (X  X  : : : Xn ) bynX
Fn(x) = n1 I ;1x (Xi):
i
1

2

(

]

=1

Let T be a statistical functional dened on the set of distribution functions, including
both the population distribution function and all empirical distribution functions (see vonMises (1947), Fernholz (1983), Reeds (1976), Takahashi (1988)), then the parameter ofinterest is expressed by T (FN ) and its naive estimate may be given by T (Fn). However,as a nite population distribution function tends to become a smooth function as N getslarger, it may be more appealing to replace Fn by its smoothed version Fen, the kernel
distribution function estimator, which is to be dened below. This type of statistic
T (Fen) is used in the context of the smoothed bootstrap (Silverman and Young (1987),
Young (1990)) and smoothed quantiles (Falk (1985)). Fernholz (1993) derives asymptotic
2

normality of smoothed statistical functionals in I.I.D. settings. We consider the nite
population counterpart and obtain its asymptotic distribution. In order to ensure that the
population distribution converges to the suciently continuous distribution function, we
need the following Assumption A. This is the standing assumption to be used throughout
this paper
Assumption A. There exist some sequences BN > 0 and constant M > 0 satisfying the condition(1.1)
0 < inf
B  sup BN  M < 1
N N
Nsuch that for any bounded set B  R

jFN (x) ; FN (y)j  BN jx ; yj + O(1=N ) as N ! 1
holds uniformly in x y 2 B.
(1.2)

Assumption A represents the situation where the population distribution goes to the
Lipschitz continuous function uniformly in any bounded set of R as N ! 1. Also wenote that Assumption A assures that the amount of jumps of FN are at most O(1=N ).
Now, we have used and will often use Landau's notation and its probabilistic version
for fpn  n  1g and fqn n  1g, pn = o(qn ), as n ! 1, if pq ! 0, as n ! 1,
(pn = op(qn), as n ! 1, if pq ! 0 in probability, as n ! 1) and pn = O(qn ) as
n ! 1, if j pq j < M for all n  1 and some 0 < M < 1. (pn = Op(qn), as n ! 1, if
P(j pq j < M ) > 1 ;  for all n  1, 0 <  < 1, and some 0 < M < 1).
We are ready to dene a smoothed empirical distribution function. For each n  1,
a kernel d.f. estimator Fen is dened by taking the convolution of Fn with some density
kn  Fen = Fn kn . In our case,
n
n

n
n

n
n

n
n

Z

Fen(x) = Fn kn (x) = Fn(x ; t)kn(t)dt
ZZ= Fn(x ; t)dKn (t) = Kn (x ; t)dFn(t)nX
1= n Kn (x ; Xi )
i
Rx
where Kn (x) = ;1 kn (t)dt.
We will next dene a regular kernel sequence fkn n  1g, which will be used to dene
our smoothed empirical distribution function. Let k be a symmetric kernel function (notnecessarily nonnegative) satisfying R k(x)dx = 1 and let fan n  1g be a sequence of
positive real numbers. The sequence of kernels fkn  n  1g dened by
 
(1.4)
kn(x) = a1 k ax  n  1nnwill be called a kernel sequence if an = o(1) as n ! 1. Note that if fkn  n  1g is aRxkernel sequence, then the sequence of d.f. Kn (x) = ;1
kn(t)dt converges weakly to the
(1.3)

=1

3

d.f. , where8>< 0 x < 0
(x) = >:: 1 x 0(1.5)



We will consider a restricted class of the kernel sequences, which will be called a regular sequence. A regular sequence will be needed when we prove the op(n; 12 ) convergenceof the remainder terms of the Taylor series expansion of the statistical functionals.
Definition 1. Fernholz (1991, 1993)] A kernel sequence fkn g is regular if
there exists a sequence fbn g of positive real numbers such that bn = o(n; = ) and
1 2

(1.6)

Z

jtj>bnjkn (t)jdt = o(n; = ):1 2

Csorgo and Horvath (1995) considers the similar but more restrictive regularity conditions in investigating the asymptotic properties of smoothed empirical and quantile
processes in I.I.D. settings. Imposing these types of regularity conditions on kernel might
be unavoidable to some extent without imposing more smoothness conditions on population distributions (Yukich (1992) and van der Vaart (1994)).
We close this section with some comments. Campbell (1980) proposes the use ofstatistical functionals in the nite population and gives a sketch of the proof for theasymptotic normality in various sampling schemes. She, however, uses essentially the
I.I.D. result in proving the asymptotic normalities. We will ll in the incompleteness ofher arguments and give legitimate proofs for these results. Some of the related results onthe nite population problem are obtained by Motoyama and Takahashi (2003), where
the rate of convergence to a normal distribution of statistical functionals in simple random
sampling is obtained. For L-statistics in survey sampling problems, we refer readers to
Shao (1994).

2. Statistical Functionals

We will briey review the theory of statistical functionals. We start with the denition of statistical functionals for the distribution functions on a nite population. We
then dene the three typical dierentiations of the functionals, and the theory of Taylor
series type expansions. The conditions under which the linear part of the expansions
obeys the central limit theorem, and the conditions and the choice of the topology whichguarantee the convergence of the remainder terms to zero as fast as o(n; 12 ) in probability,
are the main issue of this section. We will modify the arguments of Reeds (1976) andFernholz (1983) for the nite population problems with the usual empirical distributionfunctions, and then for the smoothed distribution functions.
Let be the parameter of interest. We suppose that is a functional of the underlying
distribution FN and we write = N (FN ). We also let Tn = Tn(X  : : : Xn ) be an
14

estimator of . Then it is tempting to write Tn = Nn(Fn). We will formalize this in amanner so that that we can conduct rigorous mathematical arguments in the following
manner(cf. Fernholz (1983, 1993)).
Definition 2. When Tn = Tn (X  : : : Xn ) can be written as a functional T of
the empirical distribution function Fn , Tn = T (Fn), where T does not depend on n, thenT is called a statistical functional. The domain of T is assumed to contain the empiricald.f.s for all n  1, as well as the underlying true d.f. FN . Unless otherwise specied, therange of T will be the set of real numbers. Moreover, we call T (Fen) a smoothed statistical
functional where Fen is the smoothed empirical distribution function considered above.
1

The simplest statistic may be the sample meannX
T (X  : : : X ) = 1 X :Example 1. Sample mean]
n

n

1

ni

=1

i

Then for a general distribution function G, the functional dened byZ

T (G) = xdG(x)satises Tn(X1  : : :  Xn ) = T (Fn).
Example 2. Sample quantile] We dene a statistical functional for a distri-bution function G by

T (G) = G; (q) 0 < q < 1 G; (q) = inf fx : q  G(x)g:1

1

It follows that the q-th sample quantile is given by the statistical functional dened by

T (Fn) = X dnqe ()

where dxe is the smallest integer not less than x and X(1)  : : :  X(n) are the order
statistics of X1  : : :  Xn .
Example 3. L-estimator]

T (Fn) =sX
i=1

Statistics of the form

;1
i Fn (qi)

Fn; (q) = inf fx : q  Fn(x)g1where q1 : : : qs are numbers in (0 1).
Example 4. M -estimators]

and let Tn be dened implicitly by

n
X
i=1

Let

be a real valued function of two variables(Xi Tn) = 0:
5

The corresponding functional is dened as a solution T (G) = ofZ(x )dG(x) = 0:

Estimators of this form are called M -estimators.
Let X1 X2 : : : Xn be a simple random sample without replacement from a nite
population x1 : : :  xN with distribution function FN . Then FN (X1 ) : : : FN (Xn ) is a
simple random sample without replacement from the nite population f1=N : : :  1g. Let
Un?(x) = n1 Pni=1 I0x](FN (Xi)) be the empirical distribution function of fFN (X1) : : :  FN (Xn)g.
ThennX
F (x) = 1 I(X )n i ;1x inX= n1 I F x (FN (Xi )) = Un? FN (x):iNow, the monotone increasing and continuous version of FN with F ; (FN (x)) = x
and FN (FN; (u)) = u will be given by
n

(

]

0

N ( )]

=1

=1

1

1

8
>
>
FN (x(1)) + 2FN (x(1)) arctan(x
>
>
>
>
>
>
>
FN (x(i))
>
>
>
>
>
>
>
>
>
<
+1) );F( x(i) )FNS (x) = > FN (x(i)) + FN (xx((ii+1);x(i) (x
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
: 1 + 1 arctan(x x )(N )N;x ) x  x
(1)

(1)

x=xi
i = 1 : : :  N(2.1); x i ) x 2 x i  x i ] (x i 6= x i )i = 1 : : :  N ; 1;xxN
where x  x  : : :  x N are the ordered characteristics of the population.
FNS dened above is the strictly increasing continuous function (but not a distributionfunction!) satisfying FNS (xi) = FN (xi) for all i = 1 : : :  N and kFNS ; FN k = O(1=N ).
Hence, dening Un(x) = n Pni I x (FNS (Xi)), we also havenXFn(x) = n1 I F x (FNS (Xi))
(2.2)
i
n
X
1
= n I F x (FNS (Xi)) = Un FNS (x):
(1)

(2)

(

1

( )

( )

( )

( )

( +1)

( +1)

(

)

)

=1

0

]

0

N ( )]0N S ( )]

=1

i=1

A statistical functional T induces a functional of d.f.s Un by(2.3)(Un) = T (Fn) = T (Un FNS ):Generally, for each FN , we dene a statistical functional of d.f. G on 0 1] by(2.4)(G) = T (G FNS )6whenever the right hand side is dened. Hence, we can restrict our attention to d.f.s
concentrated on 0 1] and view them as elements of D0 1], the space of right continuous
real valued functions on 0 1] which have left limits.
If T is a statistical functional and is the functional induced in D0 1] by (2.3), theasymptotic properties of T (Fn) and T (Fen) may be determined by the dierentiabilityof . The asymptotic properties depend on the type of the dierentiations. We willconsider three dierent type of dierentiations, and they are dened in the following(cf.
Fernholz (1993))
Definition 3. Let be a functional dened on an open subset A of a normed
vector space V and let g 2 A.
1. The functional is Gateaux dierentiable at g if there exists a continuous linear
functional g0 dened on V such that(g + th) ; (g) ; g0 (th)=0tfor each h 2 V. In this case g0 will be called the Gateaux derivative of at g .
2. The functional is Hadamard dierentiable at g if for any compact subset K  V,
(2.5) holds uniformly for h 2 K . The linear functional g0 will be called the Hadamard
derivative of at g.
3. The functional is Frechet dierentiable at g if for any bounded subset B  V, (2.5)
holds uniformly for h 2 B . The linear functional g0 will be called the Frechet derivativeof at g .
Since singleton is compact, and the compact set is bounded, Frechet dierentiability
implies Hadamard dierentiability which in turn implies Gateaux dierentiability.
For a statistical functional T and a d.f. FN , the inuence function of T at FN is areal valued function IFTF dened by(2.6)
IFTF (x) = dtd T (FN + t(x ; FN ))jt
where x is the d.f. of the point mass one at x, i.e. :
lim
t!

(2.5)

0

N

=0

N

x(s) = (s ;8>< 0 s < xx) = >:: 1 s x

If T and are dened as above, then the Gateaux derivative U0 of at uniform
distribution function U and the inuence function of T at FNS are related by
(2.7)

; )
IFTF (x) = U0 ((x ; FNS ) FNS
1

NS

7

sinceT (FNS + t(x ; FNS )) ; T (FNS )IFTF (x) = lim
t!
t
;
(
U
+
t
(
x ; FNS ) FNS ) ; (U )
= lim
t!
t
; )
= U0 ((x ; FNS ) FNS
NS

0

1

0

1

; (F (x)) = x and F (F ; (u)) = u hold from thewhere we have used the fact that FNS
NS
NS NS
monotone increasing property and the continuity of FNS .
Note: Since FNS is a monotone increasing function, using the (right-)continuity of
FNS we have1

1

; (FNS (x)) = inf fx : FNS (x)  FNS (x)gFNS
= inf fx : FNS (x) = FNS (x)g = x:1

And, from the continuity of FNS , we have
; (u)) = F (inf fx : F (x)  ug) = u:
FNS (FNSNS
NS
1Here and in what follows, we assume
(2.8)

Z

N
X
IFTF (x)dFN (x) = N1 IFTF (xi) = 0
NS

i=1

NS

by appropriate choice of additive constant(Reeds (1976), p.38 and Sering (1980), pp.222223 Lemma A).
Under Assumption A, we will prove the following proposition which corresponds to
Theorem 2.3 of Fernholz (1991) for the I.I.D. case.
Proposition 1. Let X  : : : Xn be a simple random sample without replacementfrom a nite population with distribution function FN . Let Fn be the empirical distributionfunction and let Fen be the smoothed empirical distribution function dened by (1.3) witha regular kernel sequence fkn g. Then, we have
1

(2.9)

pn sup jFe (x) ; F (x)j ! 0 a.s. n N ! 1:nn
;1<x<1

Proof. The proof will be given in the Appendix of this article.

In order to evaluate the linear part of the smoothed statistical functional, we willshow that the inuence function of the smoothed statistical functional may be obtainedby smoothing the inuence function of the original functional. We will prove the next
lemma under the slightly weaker conditions of Lemma 4 of Fernfolz (1993), which issuitable for our purpose.
8

Lemma 1.

Suppose is Gateau dierentiable at the uniform distribution function

U with derivative U0 : If the inuence function IF = IFTFNS is Lebesgue-Stieltjes integrablewith respect to functions of bounded variation, we haven0
0 (Ue ) = 1 X IF(fnUn i=1 Xi ) + U (U )

(2.10);1 and IF
f = IF kn.
where Uen = Fen FNS

Note that when a function g is right continuous having left limit, and a function h isa (right-)continuous monotone nondecreasing function having left limit, it is easily seenthat the composition g h is a right continuous function having left limits.
Proof. Let ti , i = 1 : : :  m be ;1 < t < < tm < 1 and t!i , i = 1 : : :  m ; 1
are middle points ti  !ti  ti . Then as t ! ;1, tm ! 1, maxi(ti ; ti) ! 0,Knx = x kn is approximated uniformly on R by the sum
1

+1

S=

1

+1

m
X
i=1

(Kn (ti ) ; Kn (ti))x;t :+1i; may be approximated by the function of the form S F ; in
Therefore, Knx FNS
NS
the space D0 1].
It follows from the linearity of U0 and (2.7) that
1

0
U (S

1

; )=
FNS

(2.11)

1

=

m
X

; )
(Kn (ti ) ; Kn (ti)) U0 (x;t FNS

i=1
m
X
i=1

+1

i

1

(Kn (ti ) ; Kn (ti))(IF(x ; !ti) + U0 (U )):+1

Since IF is Lebesgue-Stieltjes integrable with respect to Kn , the sum (2.11) converges tof x) + 0 (U ) for each given x.
IF(
UHence from the continuity of U0 , we have0
U (Knx

Since,; ) = IF(f x) + 0 (U ):
FNS
U
1

n
; = 1 XK;Uen = Fen FNS
nX FNS n1

it follows that

i=1

i

1

n
0
0 (Uen ) = 1 X IF(fU
n i=1 Xi ) + U (U ):3. Remainder terms

In this section we will show the convergence in probability of the remainder terms
from the linear approximations.
9Suppose that : D0 1] ! R is dierentiable at U . Also let the remainder term beRem(tH ) = (U + tH ) ; (U ) ; U0 (tH ):It follows from the denition of the Hadamard dierentiability that for any compact
K  D0 1], we haveRem(tH) = 0
limt!
t
uniformly in H 2 K .
In evaluating the remainder term, choosing the norm or metric which give topologies plays an essential role. Usually, D0 1] is equipped with the Skorohod topologies(Billingsley (1999)). However, in Skorohod topologies, pointwise addition of functions is
not a continuous operation and D0 1] is not a topological vector space (Billingsley (1999)
p.137 Problem12.2). Hence in order to utilize von Mises's dierentiation theory, we adopt
the uniform norm topologies.
Nevertheless, problem still remains after adopting the uniform norm. It is well known
that under the uniform norm the empirical distribution function may not be a random
element of D0 1] under some circumstances (cf. Billingsley(1999) pp.157-158, Fernholz (1983) pp.34-35). We show the result along the line of Fernholz. Consider thesituation that n = 1 and values of the population characteristic are distributed (independently of the sampling structures) uniformly on 0 1]. Dene F as the corresponding
empirical distribution function, i.e.
01

F (x) = X1 :1

The random variable X induces a probability measure on 0 1] by (B ) = P(X 2 B )for any Borel set B  0 1]. Since the value of X is uniformly distributed, and from thefact n = 1, coincides with the Lebesgue measure on 0 1].
Now, we dene the open ball Ox in D0 1] with center x(x 2 0 1]) and radius
1=2 as Ox = fG 2 D0 1] : kG ; xk < 1=2g. Ox is open in D0 1], so for any subset
B  0 1], OB = x2B Ox is also open.
For any x 2 0 1], X = x if and only if X1 2 Ox, so if F is a measurable elementthen
P(X 2 B ) = P(F 2 OB )for any set B  0 1]. However, then all subsets of 0 1] are Lebesgue measurable, which
is false.
We will overcome this diculty by using the method of Reeds (1976) and Fernholz(1983). (Note: Dudley (1992, 1994) and Dudley and Norvai"sa (1999), and the references
therein proposed the use of p-variation norm with high feasibility of Frechet dierentiability. However, we adopt Hadamard dierentiation because the usefulness of Dudley's
method to the nite population asymptotics is yet unknown to us.)
1

1

1

1

1

1

1

10

To start with, we will dene the distance between a function H 2 D0 1] and a set
K  D0 1] bydist(H K ) = Ginf
kH ; Gk
2K
where
kH ; Gk = sup jH (x) ; G(x)j:x10

We include the next lemma for completeness.

Lemma 2.Fernholz (1983), pp.35-36, Lemma 4.3.1] Let Q : D0 1] R ! R

be a function and for any compact set K  D0 1], we supposelim
Q(H t) = 0t!
0holds uniformly in H 2 K . Let  > 0 and let n be a sequence such that n # 0.
Then for any compact set K  D0 1], there exists n0  1 for which if for all n > n0 ,dist(H K)  n implies
jQ(H n)j < :

Since, pn(Un ; U ) is not a random element of D0 1], it may not be measurable andno probabilistic statement may be made on this term. In order to overcome the diculty,
Reeds uses the inner probability (Reeds (1976), pp.80-83, Fernholz (1983), p.37). Using
his method, we will obtain the next lemma which is a modication of Fernholz (1983),
p.37 Lemma 4.3.2 for the nite population.
; . Then, for any  > 0, there is a compact set
Lemma 3. Let U~n = F~n FNS
K  D0 1] and a positive sequence Nn # 0 such thatqP(dist( Nn=(N ; n)(U~ ; U) K)    ) > 1 ; :
1

n

Nn

Proof. Let (Un ; U ) be the continuous version of Un ; U dened by Rosen (1964):

it is a constant 0 function in the intervals 0 ) and   1] where  and are the smallest
and the largest jump points respectively, and for all t 2  ], it is obtained by thelinear interpolation between the left endpoints of constancy intervals for Un ; U . Under
Assumption A, the dierence between the original function and the continuous version
is of the order O(n; ), hence we have
1

(3.1)

k(Un ; U ) ; (Un ; U )k = O(n; ) a.s.
1

Next, by Proposition 1,; (t)) ; F (F ; (t))jkU~n ; Un k  supj
F~n(FNSn NSt= sup jF~n(x) ; Fn(x)j1x= o(n; = ) a.s.
1 2

11

1

Combining above inequalities, we havek(U~n ; U ) ; (Un ; U )k  k(U~n ; U ) ; (Un ; U )k + k(Un ; U ) ; (Un ; U )k= o(n; = ) a.s.
1 2

The random element (Nn=(N ; n)) = (Un ; U ) converges weakly to Brownian bridge
W  in C 0 1] as min(n N ; n) ! 1 (Rosen (1964)). It follows that the set of probability
measures P = fP Pnn  1g is relatively compact, where Pn is a probability measureof (Nn=(N ; n)) = (Un ; U ) and P denotes the probability measure for Brownian
bridge in the space C 0 1]. We also note that the space C 0 1] is a completely separable
metric space under the uniform norm. It follows from the converse part of the Prohorov's
Theorem that P is tight and therefore for any  > 0 there exists a compact set K  C 0 1]
for which, for any integer n  1,
1 2

0

1 2

0

Pn (K ) > 1 ; :From the denition of Pn, the above equation is tantamount to

P (Nn=(N ; n)) = (Un ; U ) 2 K > 1 ; :
1 2

We note here that remembering the sample Xi = x  1  i  n where (  : : :  N )take all possible permutations of (1 : : :  N ). Denoting \non"-sample Xi  i = 1 : : : N ;nas Xi = x + and dening the analogue of the empirical distribution function FN ;n as:
i

1

n i

NX
;n
FN ;n(x) = N 1; n I ;1x (Xi):
i=1

Using the fact
we haveor equivalently(]n(Fn(x) ; FN (x)) = ;(N ; n)(FN ;n (x) ; FN (x))
Fn(x) ; FN (x) = ; N n; n (FN ;n(x) ; FN (x))

FN ;n(x) ; FN (x) = ; N n; n (Fn(x) ; FN (x)):
So we may assume here and in the sequel that n=N  1=2. (This is essentially the same
argument of Erdos and Renyi (1959).)
Since, C 0 1]  D0q 1], K is also compact in q
D0 1]. Since k(Un ; U ) ; (U~n ; U )k =o(n; 12 ) and by writing Nn=(N ; n)(U~n ; U ) = qNn=(N ; n)f((U~n ; U ) ; (Un ; U ))+(Un ; U )g, it can be seen easily that whenever Nn=(N ; n)(Un ; U ) 2 K holds wehave (using n=N  1=2)qdist( Nn=(N ; n)(U~n ; U ) K )  Nn:
12

It follows thatqP(dist( Nn=(N ; n)(U~n ; U ) K )  Nn) > 1 ; :It is interesting to note that the empirical distribution function may not be a measurable function in D0 1] under the uniform norm, while the remainder term is a measurable
function. This can be proved by showing that both the statistical functional and the inuence function are measurable, then the remainder term, as a dierence of these terms,
becomes measurable.
It is now possible to evaluate the error under the probability measure.
Lemma 4. If K  D0 1] is an arbitrary compact set for which
Rem(tH) ! 0 as t ! 0tuniformly in H 2 K , thenq
Nn=(N ; n)Rem(U~ ; U) ! 0:pn

Proof. Let  > 0. Then by Lemma 3, there exists a compact set K  D0 1] anda positive sequence Nn # 0 for whichqP(dist( Nn=(N ; n)(U~n ; U ) K )  Nn) > 1 ; =2:
It follows that there is a measurable set En for whichq
En  fdist( Nn=(N ; n)(U~n ; U ) K )  Nng
andP(En) > 1 ; 
for all n.
We now apply Lemma 2 to Q(H t) = Rem(tH )=t. We nd a constant N n # 0 anda positive integer n , such that if n N ; n > n and dist(H K )  N n ,qq
j Nn=(N ; n) Rem( (N ; n)=NnH )j < qfollows. Hence, for all n N ; n > n and H = Nn=(N ; n)(U~n ; U ), we haveqP(j Nn=(N ; n) Rem(U~n ; U )j < )  P(En) > 1 ; :000Hence the lemma follows.
Remark 1. Under the same conditions of Lemma 3 and Lemma 4, we can provefor the unsmoothed caseq
Nn=(N ; n) Rem(U ; U) ! 0:np

Proof. The proof is similar to and simpler than those of Lemma 3 and Lemma 4.

13

4. Asymptotic Normality

We will present and prove our main results of this paper. We will give the asymptotic
normality of the smoothed statistical functionals under the Hadamard dierentiability in
Theorem 1. We will also prove in Theorem 2 and Theorem 3 the asymptotic normality
for smoothed and non-smoothed functionals respectively, under the condition which is
given in terms of the original non-smoothed inuence function. Combining Theorem
2 and Theorem 3, we claim that the asymptotic distributions of smoothed statistical
functionals are the same as the those of non-smoothed functionals.
For small samples Fernholz (1997) proved that smoothed functionals are more ecientthan non-smoothed ones when some regularity conditions are placed on the inuencefunction in I.I.D. settings. For related results for the cases of smoothed bootstrap, we referreaders to Silverman and Young (1987), Hall, DiCiccio and Romano (1989), Polansky and
Schucany (1997), and the references therein.
Theorem 1. Let X  : : :  Xn be the sequence of a random sample chosen without replacement from the distribution function with the Assumption A. And we also let
fkn g be the sequence of regular kernels with nite rst moment. Let T be a statistical functional and is the induced statistical functional on D0 1] by (2:3). Suppose
is Hadamard dierentiable at U with the inuence function IF = IF  N , which isLebesgue-Stieltjes integrable with respect to functions of bounded variations. Suppose
f = EIF]f = P IF(xf )=N and 0 <  = VarIF]
f = P (IF(xf ) ; IF)f =N < 1.
IFN
If the IF satises the Lindeberg condition of Erdos-Renyi
P ff
P (IF(xi ) ; IF) = 0 for anylim> 0nN ;n!1 PN (IF(f xi) ; IF)fiwhere, P is the subset of P = f1 : : :  N g such that
r 

f
f
jIF(xi) ; IFj > n 1 ; Nn N :
1

TF

N
i=1

N
i=1

2

i

i

2



2

=1

Then , as n and N ; n ! 1,

(T (Fen) ; T (FNS ))=Nn !d N (0 1)
where Nn = NN (1 ; n=N )=n(N ; 1) = (N ; n)N =n(N ; 1).
Proof. It follows from Lemma 1 that
(T (Fen) ; T (FNS))=Nn= ( (Uen ) ; (U ))=Nn= U0 (Uen ; U )=Nn + Rem(Uen ; U )=Nnnn Xof Xi ) =Nn + Rem(Ue n ; U )=Nn := 1 IF(
n
2

2

2

i=1

14

2

Since both the inuence function and the functional are measurable, the above equation tells us that Rem(Uen ; U ) is an random element in D0 1]. Also, by the central limit theorem of Erdos and Renyi (1959) and Hajek (1960), the rst term in theright most side of the above equation converges in distribution to N (0 1) as n andeN
q second term Rem(Un ; U )=Nn (=q ; n ! 1. qAlso, we note that by Lemma 4 thenN=(N ; n) (N ; 1)=N Rem(Uen ; U )=N = n(N ; 1)=(N ; n)Rem(Uen ; U )=N )converges to 0 in probability. The theorem follows from Slutzky's lemma.
The next theorem is the central limit theorem for smoothed functionals under theErdos-Renyi condition for the original non-smoothed inuence function, which is theextension of Theorem 1 of Fernholz (1993). In order to prove the theorem, we need thefollowing lemma which is a modication of Lemma 2 of Fernholz (1993).
Lemma 5. Let fkn g be the sequence of regular kernels with nite rst moment,
and X  : : : Xn be the sequence of a random sample chosen without replacement from the
distribution function with Assumption A. Suppose that FNS dened before have bounded
0 . If the function  is bounded function and Lebesgue-Stieltjes
derivatives fNS = FNS
integrable with respect to functions of bounded variation.
n
p1n X(~(Xi) ; (Xi )) !p 0:i
1=1

Proof. First we write

p1n (~(Xi ) ; (Xi)) = pn n1 (~(Xi ) ; (Xi ))iZip= n (~(x) ; (x))dFn(x)p ZZ= n ((x ; t) ; (x))k (t)dt dF (x):
n
X

n
X

=1

=1

n

n

We divide the range of integration of the inner integral,pn Z ((x ; t) ; (x))k (t)dt = pn Z ((x ; t) ; (x))k (t)dtnn
jtjbnp Z+ njtj>bn((x ; t) ; (x))kn(t)dt:

The second term of the above equation is dominated by
pn Z j(x ; t) ; (x)jjk (t)jdt
njtj>bnwhich converges to 0 from the denition of the regular kernels and the fact that  is abounded function.
As for the rst term, we integrate the term with respect to FN
pn ZZ ((x ; t) ; (x))k (t)dtdF (x)nNjtjbnp Z= njtjbnZ((x ; t) ; (x))dFN (x)kn(t)dt:
15

From the assumption, a sequence of functions FNS dened before have bounded derivatives fNS for which
kFN ; FNS k = O( N1 ):From the assumption that function  is bounded, it follows that there is a positive
constant C > 0, such thatZZZ

j ((x ; t) ; (x))dFNS (x)j = j (x ; t)fNS (x)dx ; (x)fNS (x)dxjZ= j (x)(fNS (x) ; fNS (x + t))dxj Cjtj:Hence,Zj ((x ; t) ; (x))dFN (x)jZ
= j ((x ; t) ; (x))d(FN (x) ; FNS (x) + FNS (x))jZ j ((x ; t) ; (x))d(FN (x) ; FNS (x))jZ+j ((x ; t) ; (x))dFNS (x)j 
 O N1 + Cjtj

from the assumption that  is bounded.
By substituting the above into the original equation, we haveZ
 pnbn p C n jtjb jtjjkn(t)jdt + O N  2Cpnbn + o N1 ! 0:n

In addition to the conditions of Theorem 1, we assume that IF =IFTFNS is a bounded function and FNS has bounded derivatives.
Also suppose IF = EIF] = PNi=1 IF(xi)=N and 0 < N2 = VarIF] = PNi=1(IF(xi ) ;IF)2=N < 1.
Replacing the Erdos-Renyi condition of Theorem 1, if the IF satises the Lindeberg condition of Erdos-Renyi
Theorem 2.

P
lim P P (IF(xi)nN ;n!1 N
i=1 (IF(xi); IF) = 0 for any; IF)where, P is the subset of P = f1 : : :  N g such that22

jIF(x ) ; IFj >

r 

i

16


n 1 ; Nn  :N> 0Then, as n and N ; n ! 1,

(T (Fen) ; T (FNS ))=Nn !d N (0 1)
2
where Nn
= NN2 (1 ; n=N )=n(N ; 1) = (N ; n)N2 =n(N ; 1).
Proof. It follows from Lemma 1 that

(T (Fen) ; T (FNS ))=Nn = ( (Uen ) ; (U ))=Nn= U0 (Uen ; U )=Nn + Rem(Uen ; U )=Nnnn Xof Xi ) =Nn + Rem(Uen ; U )=Nn := n1 IF(
i=1

Using the argument of Erdos and Renyi (1959) (the same argument in the proof of
Lemma 3). Remembering the sample Xi = x  1  i  n where (  : : : N ) takeall possible permutations of (1 : : : N ). Denote "non"-sample Xi i = 1 : : : N ; n as
Xi = x + .
Using the fact
n
NX
;n
N
X
X
IF(Xi ) + IF(Xi) = IF(xi)
1

i

n i

i=1

we have

i=1

i=1

n
N
NX
;n
N
X
X
1 (X
1

n i IF(Xi) ; i IF(xi)) = ; N ; n ( i IF(Xi ) ; i IF(xi)):
So we may assume that n=N  1=2.
By Lemma 5, we have (using n=N  1=2)
=1

=1

q

=1

=1

n
X

p
f Xi ) ; IF(Xi )) !
N=n(N ; n) (IF(0i=1

The theorem follows from the central limit theorem for the nite population (Erdos andRenyi (1959) and Hajek (1960)), and Lemma 4.
The last theorem, when compared to the previous Theorem 2, displays that boththe smoothed and non-smoothed statistical functional have the same distribution in thelimit.
Theorem 3. Under the same conditions of Theorem 2, we have, as n and N ; n! 1,(T (Fn) ; T (FNS ))=Nn !d N (0 1)
where Nn = NN (1 ; n=N )=n(N ; 1) = (N ; n)N =n(N ; 1), 0 < N = VarIF] =P(IF(x ) ; IF) =N < 1, and IF = EIF] = P IF(x )=N.
Proof. The proof is performed in a similar manner as those of Theorem 1 andTheorem 2.
2

N
i=1

2

i

2

N
i=1

2

17

2

i

We rst note
(T (Fn) ; T (FNS))=Nn= ( (Un ) ; (U ))=Nn= U0 (Un ; U )=Nn + Rem(Un ; U )=Nnnon X= n1 IF(Xi ) =Nn + Rem(Un ; U )=Nn:i=1

Since both the inuence function and the functional are measurable, the above equation tells us that Rem(Un ; U ) is a random element in D0 1]. Also, by the central limit theorem of Erdos and Renyi (1959) and Hajek (1960), the rst term in theright most side of the above equation converges in distribution to N (0 1) as n N ;nq ! 1. Also,q we note that by Remark 1, theq second term Rem(Un ; U )=Nn (=nN=(N ; n) (N ; 1)=N Rem(Un ; U )=N = n(N ; 1)=(N ; n)Rem(Un ; U )=N )converges to 0 in probability. The theorem follows from Slutzky's lemma.

5. Monte Carlo Simulation

In this section, we present Monte Carlo simulation results for sample medianFn; (0:5)(5.1)
and inter-quartile range
(5.2)

1

Fn; (0:75) ; Fn; (0:25):
1

1

These statistics are often used (ex. Deaton (1997) p.22 Table 1.2. Consumption andincome for panel households, C^ote d'Ivoire, 1985-86) because they are less aected byoutliers which may appear in wide class of data.

5.1. Quasi-populations

We use the following simulated log-normal quasi-populations of dierent sizes whosemean and standard deviation of the distribution on the log scale is 3 and 0:4 respectively.
(i) generated values of log normal random number of size 1 000(ii) generated values of log normal random number of size 5 000(iii) generated values of log normal random number of size 10 000
We use these populations because they approximate many economic variables suchas household income and savings.
18

5.2. Construction of the Condence Intervals

To construct the condence intervals, we calculate the inuence functions of themedian and the inter-quartile range.
As is well-known(cf. Huber (1981)), the inuence function of a non-smoothed medianis ;1 : ; 0:5
IFTF (x) = xf;(FF ; (0
:5))
8
>
;
;
<
;1 : x  F (0:5)
f
F
=>
;
:
f F ;1 : x > F (0:5)
(0 5)
1

2 (
2 (

1
(0 5))

1

1

1(0 5))and the inuence function of a non-smoothed inter-quartile range is;1
1 + 0:75 x;F ;1 : ; 1 + 0:25
IFTF (x) = x;Ff (F :; (0;:75)); f (F ; (0:25)) 
where F is an underlying distribution function and f is its density.
In order to obtain the inuence functions of the smoothed functionals, we utilize thefollowing Lemma of Fernholz (1993).
Lemma 6.Fernholz (1993), Proposition 2] Let k be xed, and dene thesmoothed functional Te for general distribution function G as
(0 75)(0 25)

1

1

Te(G) = T (G k):(5.3)

If T is Gateaux dierentiable in a neighborhood of FNS including Fe = F k , thenIFe = IF e k:

(5.4)

TF

TF

Utilizing this lemma, we calculate the inuence functions of smoothed functionals.
x k (t)dt be the distribution function with respect to kernel k , and let f~
Let K (x) = R;1
be the density function with respect to the smoothed (cumulative) distribution Fe = F k.
Then, the inuence function of the smoothed median is;
IFTeF (x) = K (x ;~ F ; (0:5)) ; 0:5
f (F (0:5))
and the inuence function of the smoothed inter-quartile range is;;
IFTeF (x) = K (x ; F~ (0;:75)) ; 1 + 0:75 ; K (x ; F~ (0;:25)) ; 1 + 0:25 :f (F (0:75))
f (F (0:25))
In our simulations, we smooth two functionals with uniform distribution U ;0:1 0:1],
which satises the conditions of regular kernel and it is easy to calculate the convolutions.
Using these inuence functions，we construct the condence intervals as follows(wedescribe the non-smoothed case, smoothed case is similar):
1

1

1

1

1

1

19

(i) Calculate the variance of the inuence function N as
2

(5.5)

N = VarIF] =(5.6)

IF = EIF] =

2

N
X

i=1
N
X

(IF(xi) ; IF) =N
2

IF(xi)=N:i=1(ii) Approximate(5.7)

T (Fn) ; T (FN )Nnwith standard normal distribution (Nn = NN (1 ; n=N )=n(N ; 1) = (N ;n)N =n(N ; 1)).
2

22(iii) Construct a one-sided 100(1 ; ) condence interval as


(5.8)

; 1 T (Fn) + Nnz

i

and a two-sided 100(1 ; ) condence interval as
(5.9)

h

i

T (Fn) ; Nnz =  T (Fn) + Nnz =2

2

where z is the upper % point of the standard normal distribution．

5.3. Results of Monte Carlo Simulations

In order to evaluate the fruits of theoretical facts, we calculate the empirical coverage
ratio of condence intervals constructed by our normal approximations. The simulated
samples of sampling fractions 10% and 30% are chosen 100 000 times repeatedly, then therelative frequencies that the intervals contain the true value of parameter are evaluated.
We can judge that the intervals are precise when the empirical coverage probability is close
to the nominal condence coecient. Although readers may claim that these sampling
fractions are extraordinarily high in reality, in a stratied population or in a selected unitof population, fractions of these types are not exceptional.
In what follows, the numbers in parentheses in tables for two-sided intervals are thelengths of the intervals.
As a whole, except for the slightest dierence between the smoothed and the unsmoothed cases, we can see the following features:(i) All the intervals, especially the cases with population sizes larger than 5000, display
better features for both the one-sided and two-sided situations.
(ii) The case when the sampling fraction is 10% is better than the case of a sampling
fraction 30% in small sample situations.
20

(iii) The intervals of higher condence coecients perform better than those of lower
condence coecients．
Compared to the case of a sampling fraction of 10%, the case of 30% seems to be
not as good when population and sample sizes are relatively small. However, our normal
approximations show good performance for large size samples, so they are very useful for
application to large scale sample surveys.
Table 1. One-sided condence intervals for sample medians(sampling fraction10%).

sample size 100
population size 1000
sample size 500
population size 5000
sample size 1000
population size 10000

90%(one-sided) 95%(one-sided) 99%(one-sided)
0.95278
0.984410.998390.91295

0.96173

0.99483

0.899810.95590.99309

Table 2. Two-sided condence intervals for sample medians(sampling fraction10%, length ofintervals in parentheses).

90%(two-sided) 95%(two-sided) 99%(two-sided)
sample size 100
0.935120.96572
0.99494
population size 1000
(3.142083)(3.744022)(4.920478)
sample size 500
0.90636
0.95292
0.99251
population size 5000
(1.405430)(1.674672)(2.200892)
sample size 1000
0.89818
0.948310.99091
population size 10000 (0.9937106)(1.184079)(1.556143)

21

Table 3. One-sided condence intervals for sample medians(sampling fraction30%).

sample size 300
population size 1000
sample size 1500
population size 5000
sample size 3000
population size 10000

90%(one-sided) 95%(one-sided) 99%(one-sided)
0.93353
0.982070.99866
0.90336

0.95278

0.992560.93079

0.955120.99066

Table 4. Two-sided condence intervals for sample medians(sampling fraction30%, length ofintervals in parentheses).

90%(two-sided) 95%(two-sided) 99%(two-sided)
sample size 300
0.910390.965480.99424
population size 1000
(1.599870)(1.906363)(2.505385)
sample size 1500
0.88282
0.934820.99091
population size 5000
(0.7156098)(0.8527015)(1.120640)
sample size 3000
0.90292
0.94974
0.98661
population size 10000 (0.5059727)(0.6029035)(0.7923494)

Table 5. One-sided condence intervals for smoothed sample medians(sampling fraction10%).

sample size 100
population size 1000
sample size 500
population size 5000
sample size 1000
population size 10000

90%(one-sided) 95%(one-sided) 99%(one-sided)
0.954450.98435
0.99834
0.91447

0.96367

0.995520.90483

0.95786

0.993

22

Table 6. Two-sided condence intervals for smoothed sample medians(sampling fraction10%,length of intervals in parentheses).

90%(two-sided) 95%(two-sided) 99%(two-sided)
sample size 100
0.935290.96752
0.99521
population size 1000
(3.165274)(3.771655)(4.956795)
sample size 500
0.907140.955810.99304
population size 5000
(1.415939)(1.687195)(2.217350)
sample size 1000
0.89895
0.95001
0.99115
population size 10000(1.000993)(1.192757)(1.567548)

Table 7. One-sided condence intervals for smoothed sample medians(sampling fraction30%).

sample size 300
population size 1000
sample size 1500
population size 5000
sample size 3000
population size 10000

90%(one-sided) 95%(one-sided) 99%(one-sided)
0.93533
0.981130.99870.91001

0.95471

0.993430.93079

0.955120.99066

Table 8. Two-sided condence intervals for smoothed sample medians(sampling fraction30%,
length of intervals in parentheses).

90%(two-sided) 95%(two-sided) 99%(two-sided)
sample size 300
0.905170.96556
0.99476
population size 1000
(1.611678)(1.920433)(2.523876)
sample size 1500
0.88967
0.94752
0.99132
population size 5000
(0.720961)(0.8590779)(1.129020)
sample size 3000
0.909150.95047
0.98905
population size 10000 (0.5096809)(0.6073221)(0.7981564)

23

Table 9. One-sided condence intervals for sample inter-quartile ranges(sampling fraction10%).

sample size 100
population size 1000
sample size 500
population size 5000
sample size 1000
population size 10000

90%(one-sided) 95%(one-sided) 99%(one-sided)
0.877940.94572
0.993290.90063

0.953420.99092

0.89934

0.951810.99202

Table 10. Two-sided condence intervals for sample inter-quartile ranges(sampling fraction10%,length of intervals in parentheses).

90%(two-sided) 95%(two-sided) 99%(two-sided)
sample size 100
0.92177
0.966160.99485
population size 1000
(4.249439)(5.063518)(6.65459)
sample size 500
0.91433
0.95735
0.99087
population size 5000
(1.943841)(2.316229)(3.044041)
sample size 1000
0.919
0.96182
0.99323
population size 10000(1.373149)(1.636208)(2.150341)

Table 11. One-sided condence intervals for sample inter-quartile ranges(sampling fraction30%).

sample size 300
population size 1000
sample size 1500
population size 5000
sample size 3000
population size 10000

90%(one-sided) 95%(one-sided) 99%(one-sided)
0.85486
0.91870.98698
0.91475

0.956810.99182

0.91218

0.95477

0.9900324

Table 12. Two-sided condence intervals for sample inter-quartile ranges(sampling fraction30%,
length of intervals in parentheses).

90%(two-sided) 95%(two-sided) 99%(two-sided)
sample size 300
0.895410.9546
0.99358
population size 1000
(2.163709)(2.578217)(3.388351)
sample size 1500
0.915310.96037
0.99353
population size 5000
(0.9897555)(1.179366)(1.549950)
sample size 3000
0.91720.960210.99273
population size 10000 (0.6991734)(0.8331163)(1.094900)
Table 13. One-sided condence intervals for smoothed sample inter-quartile ranges(sampling
fraction10%).

sample size 100
population size 1000
sample size 500
population size 5000
sample size 1000
population size 10000

90%(one-sided) 95%(one-sided) 99%(one-sided)
0.879
0.94585
0.993310.9024

0.954070.99073

0.89808

0.95104

0.99214

Table 14. Two-sided condence intervals for smoothed sample inter-quartile ranges(sampling
fraction10%, length of intervals in parentheses).

90%(two-sided) 95%(two-sided) 99%(two-sided)
sample size 100
0.92194
0.96566
0.99518
population size 1000
(4.258238)(5.074003)(6.668369)
sample size 500
0.91574
0.95783
0.99081
population size 5000
(1.950190)(2.323794)(3.053983)
sample size 1000
0.91763
0.96177
0.99357
population size 10000(1.376929)(1.640712)(2.156260)

25

Table 15. One-sided condence intervals for smoothed sample inter-quartile ranges(sampling
fraction30%).

sample size 300
population size 1000
sample size 1500
population size 5000
sample size 3000
population size 10000

90%(one-sided) 95%(one-sided) 99%(one-sided)
0.85550.92003
0.987390.91652

0.9582

0.992180.91179

0.954880.99038

Table 16. Two-sided condence intervals for smoothed sample inter-quartile ranges(sampling
fraction30%, length of intervals in parentheses).

90%(two-sided) 95%(two-sided) 99%(two-sided)
sample size 300
0.89756
0.95435
0.99312
population size 1000
(2.168189)(2.583556)(3.395367)
sample size 1500
0.917560.96083
0.99348
population size 5000
(0.9929883)(1.183218)(1.555012)
sample size 3000
0.917820.96035
0.99291
population size 10000 (0.7010979)(0.8354095)(1.097914)

26

27

Appendix

The proof of Proposition 1 will be shown along the lines of Fernholz (1991), and forthis purpose we rst show the following lemma. A covering C of R is a collection ofintervals whose union is R. The intervals need not have nite length.
Lemma 6. Let X  : : :  Xn be a simple random sample without replacement from anite population with distribution function FN . Let fCn g be a sequence of coverings of R
such that the number of intervals in each Cn is O(n ) for some constant . Suppose that
maxI 2C PF (I ) = maxI 2C PF (X 2 I ) = o(n; = ) where PF stands for the probability
generated by FN . If Tn is the maximum number of X 's with values in any I 2 Cn , thenpTnn ! 0
as n N ! 1 a.s..
Proof. For each n  1, dene YI as the number of Xi 's with values in I 2 Cn . ThenYI is a random variable from a hypergeometric distribution with probability function
1

n

n

N

1 2

N

  

N N (1; )p(y) = y N n;y nN

y = 0 1 : : :  min(n N)

where  = PF (I ). Note that  = o(n; = ) as n N ! 1.
p
For any  > 0, and k = b nc, we haveXXpp
P(Tn >  n) P(YI >  n) =P(YI  k):
1 2

N

I 2Cn

I 2Cn

Using the well known identity of hypergeometric distribution,p(y + 1) =(N ; y)(n ; y)p(y)(y + 1)(N ; N ; n + y + 1) max(0 n + N ; N )  y  min(n N) ; 1
we have the following inequalities for k  n (See Feller (1968) for binomial case).
1)((N ; n)(1 ; ) + (n ; k) + 1)
P(YI > k)  P(YI = k) (k + 1)((N(k;+n)(1; ) + (n ; k) + 1) ; (N ; k)(n ; k)
= P(YI = k)O(1)
from  = o(n; = ).
It follows from Stirling's formula that!= )(N ; N )n(N ; n)(NP(YI = k)  2k(N ; k)N (n ; k)(N ; n ; (N ; k))

! 
!N
N (1 ; )(N ; n) NNN (N ; n ; (N ; k))N (1 ; )(N (1 ; ) ; (n ; k))! !
(N ; k)(n ; k) k n(N ; n ; (N ; k)) n:k(N (1 ; ) ; (n ; k))(n ; k)(N ; n)
1 2

1 2

28

Here we have

!1 2



=
(N)(N ; N)n(N ; n)
;=)
=
O
(
n


2k(N ; k)N (n ; k)(N ; n ; (N ; k))!
N (1 ; )(N ; n) N = O(1)N (N ; n ; (N ; k))
!N

N
= O(1):
N (1 ; )(N (1 ; ) ; (n ; k))So for suciently large n N and a constant C < 1,
 pn  n nnpP(YI > k)  C pnn; n :
Recalling the factn p n = e pn O n; nwe have pn pnP(YI > k)  C pn e np C0e; n
1 4

+ (1)

p

for some constants C and C0 and suciently large n, as k = b nc  n for large n.
Thereforep
X ; pnp
P(Tn >  n)  C0e
= O(n )e; n
I 2Cnsince the number of intervals in Cn is O(n ). Hence
1
X

n=1

p

P(Tn >  n) < 1and since  > 0 was arbitrary, the rst Borel-Cantelli Lemma implies thatpTnn ! 0 a.s.
Now we will prove the proposition.
Proof. Dene the function Q on R by+

Q(t) = sup(FN (x + t) ; FN (x)):
x

It is easily seen that Q(0) = 0, limt!1 Q(t) = 1andQ(s + t) = sup(FN (x + s + t) ; FN (x))
x
= sup
(FN (x + s + t) ; FN (x + t) + FN (x + t) ; FN (x))
x sup
(FN (x + s + t) ; FN (x + t)) + sup(FN (x + t) ; FN (x))
xx
= Q(s) + Q(t):29Since fkng is regular, there exists a positive sequence fbng satisfying the condition ofDenition 1. We may assume b;n = o(n) without any restrictions, or else we may replacebn by maxfbn n; = g.
Let Qn = Q(bn) and dene x = ;1 xi = FN; (iQn) for iQn < 1 and xk = 1 fork = inf fi : iQn  1g. The intervals I = (;1 x ] I = (x  x ] : : : Ik = (xk;  1)dene a covering Cn of R.
From Assumption A,
1

3 4

1

0

1

1

2

1

2

1

jQ(t) ; Q(s)j  j sup(FN (x + t) ; FN (x)) ; sup(FN (x + s) ; FN (x))jxx  2M jt ; sj + O N1 :So, we have 
PF (Ij ) = FN (xj ) ; FN (xj; )  2Qn  4Mbn + O N1 = o(n; = ):
1 2

1

N

The number of intervals in Cn satises the relationship k  Q;n +1. Since b;n = o(n)as n ! 1 we have nbn ! 1 and hence Q(nbn) ! 1, so for suciently large n, Q(nbn) >1=2. We see Q;n = O(n) from Q(nbn)  nQ(bn) = nQn . Therefore fCn g satises theassumption of Lemma 1.
For any x 2 R, x 2 Ij for some j , we have (x ; bn) 2 Ij Ij; and (x + bn) 2 Ij Ij .
Hence if jtj  bn , thenpnjF (x ; t) ; F (x)j  2pTn 
n
n
n
1

1

1

1

+1

where Tn is dened as in Lemma 6. Therefore,

pnjFe (x) ; F (x)j  pn Z jF (x ; t) ; F (x)jjk (t)jdt
n
n
n
n
n
Z
p
 n jtjb jFn(x ; t) ; Fn(x)jjkn (t)jdt
p Z jF (x ; t) ; F (x)jjk (t)jdt+ nn

jtj>bnnnn

 2pTnn jtjb jkn (t)jdt + pn jtj>b jkn (t)jdt:
Z

n

Z

n

The rst term of the last inequality converges almost sure to zero by Lemma 1 and thesecond term converges to zero since fkng is a regular sequence. The proposition follows.
Corollary 1. Under the assumption of Proposition 1,p(a) The smoothed and non-smoothed Kolmogorov-Smirnov statistics n supx jFen(x) ;FN (x)j and pn supx jFn(x) ; FN (x)j have the same asymptotic distribution.
(b) For x such that 0 < f  FN (x)  f < 1 in suciently large N , the normalizedsmoothed empirical process has a normal distribution as a limit12pn(Fe (x) ; F (x))= !d N(0 1) n N ; n ! 1
n
N
Nn
30

where Nn = ((N ; n)=(N ; 1))FN (x)(1 ; FN (x)).
Proof. By triangular inequality,

pn sup jF (x) ; F (x)j ; pn sup jFe (x) ; F (x)j
n
N
n
n
x
x
p
 n supjFen(x) ; FN (x)jx
 pn sup jF (x) ; F (x)j + pn sup jFe (x) ; F (x)j:
x

n

N

x

Hence, (a) follows from proposition 1.
Dividingnn

pnfFe (x) ; F (x)g = pnfFe (x) ; F (x)g + pnfF (x) ; F (x)g
n

N

n

n

n

N

(b) follows from the proposition and the asymptotic normality of the hypergeometricpdistribution(Eeden and Runnenburg (1960)): nfFn(x) ; FN (x)g= !d N (0 1) as n and
N ; n ! 1.

Acknowledgements

We would like to thank an anonymous referee for valuable comments and constructive
suggestions that helped to much improve the paper.
References

1] Billingsley, P. (1999). Convergence of Probability Measures, 2nd. ed., John Wiley & Sons.
2] Campbell, C. (1980). A Dierent View of Finite Population Estimation, Proceedings of
the Survey Research Methods Section, ASA, 319{324.
3] Csorgo, M. and L. Horvath (1995). On the Distance between Smoothed Empirical andQuantile Processes, The Annals of Statistics, 23, 113{131.
4] Deaton, A. (1997). The Analysis of Household Surveys: A Microeconometric Approach to
Development Policy , The Johns Hopkins University Press.
5] Dudley, R. (1992). Frechet Dierentiability, p-Variation and Uniform Donsker Classes,
The Annals of Probability, 20, 1968{1982.
6] Dudley, R. (1994). The Order of the Remainder in Derivatives of Composition and Inverse
Operators for p-Variation Norms, The Annals of Statistics, 22, 1{22.
7] Dudley, R. and R. Norvaisa (1999). Lecture Notes in Mathematics # 1703 Di erentiabilityof Six Operators on Nonsmooth Functions and p-Variation, Springer-Verlag.
8] Erdos, P. and A. Renyi (1959). On the Central Limit Theorem for Samples from a Finite Population, Publications of the Mathematical Institute of the Hungarian Academy of
Sciences, 4, 49{61.

319] Eeden, van C. and J.Th. Runnenburg (1960). Conditional Limit-distributions for the
Entries in a 2 2-table, Statistica Neerlandica, 14, 111{126.
10] Falk, M. (1985). Asymptotic Normality of the Kernel Quantile Estimator, The Annals of
Statistics, 13, 428{433.
11] Feller, W. (1968). An Introduction to Probability Theory 3rd ed., John-Wiley & Sons, Inc.
12] Fernholz, L.T. (1983). Lecture Notes in Statistics # 19 von Mises Calculus for Statistical
Functionals, Springer-Verlag.
13] Fernholz, L.T. (1991). Almost Sure Convergence of Smoothed Empirical Distribution
Functions, Scandinavian Journal of Statistics, 18, 255{262.
14] Fernholz, L.T. (1993). Smoothed Versions of Statistical Functionals, in New Directions in
Statistical Data Analysis and Robustness , Edited by Morgenthalter, S. Ronchetti, E. and
W. A. Stahel., Birkhauser Verlag.
15] Fernholz, L.T. (1997). Reducing the Variance by Smoothing, Journal of Statistical Planning and Inference, 57, 29{38.
16] Hajek, J. (1960). Limiting Distributions in Simple Random Sampling from a Finite Population, Publications of the Mathematical Institute of the Hungarian Academy of Sciences,5, 361{374.
17] Hall, P., DiCiccio, T.J. and J.P. Romano (1989). On Smoothing and the Bootstrap, The
Annals of Statistics, 17, 692{704.
18] Huber, P. (1981). Robust Statistics , John Wiley & Sons.
19] Motoyama, H. and H. Takahashi (2003). On Normal Approximation for Statistical Functional in Finite Population, Presented at Workshop in Hitotsubashi University, Tokyo,
Japan.
20] Polansky, A.M. and W.R. Schucany (1997). Kernel Smoothing to Improve BootstrapCondence Intervals, Journal of the Royal Statistical Society, Series B(Statistical Methodology), 59, 821{838.
21] Reeds, J.A. III (1976). On the Denition of von Mises Functionals , Ph.D thesis, Harvard
University, Cambridge, Massachusetts.
22] Rosen, B. (1964). Limit Theorems for Sampling from Finite Populations, Arkiv for Matematik, 5, 383{422.
23] Sering, R.J. (1980). Approximation Theorems of Mathematical Statistics , John Wiley &Sons, Inc.

3224] Shao, J. (1994). L-Statistics in Complex Survey Problem, The Annals of Statistics, 22,946{967.
25] Silverman, B.W. and G.A. Young (1987). The bootstrap: To smooth or not to smooth?,
Biometrika, 74, 469{79.
26] Takahashi, H. (1988). A Note on Edgeworth Expansions for the von Mises Functionals,
Journal of Multivariate Analysis, 24, 54{65.
27] van der Vaart, A (1994). Weak Convergence of Smoothed Empirical Processes, Scandinavian Journal of Statistics, 21, 501{504.
28] von Mises, R. (1947). On the Asymptotic Distribution of Dierentiable Statistical Functions, The Annals of Mathematical Statistics, 18, 309-348.
29] Young, G.A. (1990). Alternative Smoothed Bootstraps, Journal of the Royal StatisticalSociety. Ser. B, 52, 477{484.
30] Yukich, J.E. (1992). Weak Convergence of Smoothed Empirical Processes, Scandinavian
Journal of Statistics, 19, 271{279.

33