<DffIGUIlY IN Na.'WAlWIETRIC ESTIllATIaf OF A <DIDITIafAL FUNCI'IafAL.
BY ASHIS K. GANGOPADHYAY AND PRANAB K. SEN
University of North Carolina at Chapel Hill
Induced order statistics play a vi tal role in the estimation of a
functional of a conditional distribution. The popular notion of contiguity
of probability measures is extended to such a conditional setup and
incorporated in the study of the asymptotic theory of the estimators.
1.
Introduction.
~
Let {(Xi ,Zi)' i
identically distributed random vectors
~
function (d.f.) rr(x.z). (x.z)·€
R2.
1} be a· sequence of independent and
(i.i.d.r.v.)
the real plane.
with a
Let F(x) =
distribution
rr(x.~). x €
ffi
be the marginal d.f. of X and let G(zlx) be the conditional d.f. of Z given
At a given point xO' consider a functional of the
X=x. for z € ffi. x € ffi.
conditional d.f. G(·IXO):
(1.1)
»'
O(xO) = T(G(·lxo
and consider the problem of estimating O(x ) from the sample observations
O
(XI.Z I ) .... • (Xn·Zn )·
In this setup. the functional T(·) is of mown form.
AMS (1980) Subject Classifications:
Key words and phrases:
derivative.
conditional
Primary 62G05. Secondary 62G20. 62030.
Asymptotic normali ty. bias. contiguity. Hadamard
functional.
order
statistics.
induced
statistics. nearest neighbor estimator. kernel estimator. U-statistics.
ABBREVIATED TITLE:
roNTIGUITY AND INDUCED ORDER STATISTICS
order
-2-
while G( -Ix ) is not mown.
Our task is to construct suitable nonparametric
o
estimators of G( -Ix ) and incorporate them in the formulation of sui table
o
estimators of 9(xO).
In this context. based on a suitable norm II-II (viz ..
the Euclidean norm). we may set Yi = IIX i -xO"' i=l, ...• n. so that we have the
(Xi.Z i )
data transformation:
~
(Yi.Z i ). i=l •.... n.
Let 0
~
Yn1
~
~
...
Ynn
be the order statistics corresponding to Y1 •. ·· .Yn • and let Zn1.· .. ,Znn be
the induced order statistics (i.e .• Zni = Zj if Yni = Yj • for i.j=l •...• n).
For every positive integer k
xo)
d.f. of Z (With respect to
A
(1.2)
Gnk(z) = k
-1 -Ie
ri=l
(~
n). the k-nearest neighbor (k-NN) empirical
is defined as
I(Zni ~ z). z € ~.
where I(A) stands for the indicator function of the set A.
Then. for a
suitably chosen k (relative to n). we consider
A
(1.3)
Tnk = T(Gnk )
(k
~
~
n)
».
as an estimator of 9(XO) = T(G(-Ixo
Although typically T(-) may not be a
linear functional. under sui table differentiabil ity condi tions. T
may be
nk
A
approximated adequately by a linear one.
(1.2)
is not based on i.i.d.r.v.
However. the empirical d. f. G in
nk
(but on conditionally independent and
non-i.d.r.v. ·s). and hence. the usual treatment of i.i.d.r.v.·s may not hold
here.
Although. for k(=k ) increasing with n (as is usually the case). the
n
A
asymptotic properties of G may be studied by some direct analysis [viz ..
nk
Bhat tacharya and Gangopadhyay (1988)]. we shall find i t convenient to extend
the notion of contiguity of probability measures to such a conditional setup.
and to incorporate the same in a novel and considerably simpler approach to
the desired asymptotic theory.
Along wi th the preliminary notions.
contiguity is established in Section 2.
this
Asymptotic distribution of Tnk in
-3-
(1.3)
is considered in Section 3.
and.
in this context.
Section 2 are incorporated in a unified manner.
k
n
4/5 ).
is O(n
advocate
•
the resul ts of
Al though the optimal rate of
there is a bias term of comparable magnitude.
the use of k
5
= O(n4/ -T1).
n
for some TJ
>
O.
Hence. we
where the bias is
asymptotically negligible.
2.
A contiguity
theorem for
induced order statistics.
The
following
regularity conditions are assumed:
[AI]
The d. f.
F admits an absolutely continuous density function f.
such
that
(2.1)
f(x )
O
> O.
and f' (x) = (d/dx)f(x) exists in a
posi tive numbers e.
(2.2)
•
ko
neighborhood of x • and there exist
o
and a. such that Ix-"o I
<e
implies that
If'(x) - fl(xo)1 ~ kolx-xol.
[A2]
The d. f. G(z Ix ) has (a.a. z € R) a continuous density g(z Ix ) such
o
o
that
the
partial
derivatives
~(zlx)
=
(8/ox)g(zlx)
and
~(zlx)
=
2
(a2/ox )g(zlx) exist in a neighborhood of xO(a.a. z).
As in Section 1. let Y = Ixi-xol. 1 ~ i ~ n. and let Fy(y). y ~ O. be
i
the marginal d. f. of Yi .
(2.3)
fy(y) = f(xO+Y) + f(xO-y). Y ~ O.
so that by an appeal to
too.
Then Fy admits a densi ty function
(2.1)
G*(zly) (and g*(zly»
y=y. are given by
•
Note that
and (2.2). we conclude that [AI] holds for f y
the conditional d.f. (and density) of Z. given
-4-
(2.6)
g*{zIO) = g{zIXO)
and G*{zIO) = G{zlxo )' z
€
R.
For notational simplicity. in the sequel. g{zIXO) and G{zlxo ) will be denoted
by g{z) and G{z). respectively.
[A3]
There exist
~
> O. a > 0 and some Lebesgue measurable functions u 1{z)
and u {z). such that for all y : y ~ ~.
2
(2.7) 1{8/8y2) g*{zly)I ~ u 1{z) a.a. z .•
and
•
1{{8/8y2) g*{zly)} - {{8/8y2) g*{zly)IY=Q} I ~ u2{z)ya.
There exists ~ > O. such that
(2.S)
[A4]
(2.9)
I{zly) = E[{{8/8y2) log g*{zly)}2 I Y=y]
uniformly in y : 0
~
y
< ~.
~ ~.
In passing. we may remark that for [AJ] to hold the following condi tions
suffice:
There exist
~
> O. a > 0 and Lebesgue measurable functions (on m)
(2.10)
I~{z Ix) I ~ u~{z). I~{z Ix) I ~ u;{z):
(2.11)
I~(zlx) - ~(zlxo)1 ~ u;(z)lx-xol .
•
a
However. expressing in terms of (8/8y2)g*(zly). we are able to bring the
analogy with the classical (location/regression/scale) case treated in Hajek
and Sidak (1967).
Let us now denote by
g*{zIYni ) = ~i{z) and G*(zIYni ) = Gni{z). 1 ~ i ~ n.
Also. for every k (~n). let ~ and ~ denote the joint conditional density
(2.12)
and distribution of Zn1 •.... Znk.
given Yn1 •...• Ynk .
Note that [viZ ..
Bhattacharya (1974)] given (Yn 1.....Y).
Zn 1•...• Znn are conditionally
nn
independent. so that
•
-5k
k
a. (e) = IT g (e)
"IlK
i=1 ni
(2.13)
and 0 _ (.e)
--nk
= i=1
IT
G ICe).
n
o
0
Let Zn1 •...• Znk be i.i.d.r.v.·s with a density (and d.f.) g(z) and G(z). and
let
k
•
k
Pnk(e) = IT gee) and Pnk(e) = IT G(e).
j=1
j=1
Thus. in (2.13) we conceive of the actual conditional density of the Znj
(2.14)
(given
the Ynj ).
while
in
(conditional) density g(z)
(2.14).
(= g(zlxo
we have an
»'
i.i.d.
model
with
the
In this conditional setup.
we
consider the following.
A
Theorem 2.1.
t(O<a<t<b<~).
Let k = [tn ]
for some A (0
<
A
.~
4/5)
and for
some
Then under the regularity conditions [A1]-[A4]. the densities
CIru< are contiguous to
the dens! ties Pnk'
A detailed proof of the theorem is given in section 4.
"-
Asymptotic distribution of T(G
3.
established in Theorem 2.1,
By the use of the contigui ty
nk ) .
we will be able to study the asymptotic
"-
distribution of T(Gnk ) under the usual regularity conditions on T(e) and some
other conditions on g*(zly); these are therefore introduced first.
Let
~(A.B)
be
a
set of continuous
linear
transformations
from a
topological vector sPaCe A to another B. and let C be a class of compact
subsets of A. such that every subset consisting of a single point belongs to
C.
For G € A and for every H € K € C. we assume that
(3.1)
T(H) = T(G + (H-G»
= T(G) + S T1 (G;z.xO) d[H(z)-G(z)] + R1 (G;H-G).
where G(e) = G(elxo) and
•
(3.2)
IR1(G;
H-G) I = o(IIH-GII). uniformly in H € K.
and II-II stands for the "sup-norm".
This relates to the so called first order
-6-
Hadamard-dtfferenttabtttty of T(e) at G, and T1(G;z,xO) is called the first
order compact (or Hadamard) derivative of T(e) at G; it is so normalized that
(3.3) I T1(G;z,xo ) dG(zlxo )
=0
(a.a. xO)
Similarly, if we assume that
(3.4)
T(H) = T(G) + I T1(G;z,xo ) d[H(z)-G(z)]
+ ~ II T (G;z,z',x ) d[H(z)-G(z)]d[H(z')-G(z')]
o
2
+
~(G;H-G),
V H € K € C,
where
(3.5)
II~(G;H-G)II
= o(IIH-G1i 2 ), uniformlY in H € K,
then T(e) is second order Hadamard differentiable at G and T (G;z,z',x ) is
2
O
the second order compact (or Hadamard) derivative of T(e) at G. We may set
(3.7) I T2 (G;z,z',xo ) dG(z') = 0 = I T2 (G;z,z',xo ) dG(z)
(a.e.)
Further, bearing in mind the contiguity in Theorem 2.1, we make the following
assumption.
[B1]
There exist
~
> 0,
a
>0
and A
<~
such that uniformly in yeO
(3.8)
II T1(G;z,~) {(o/ay2)g*(zly)} dzl ~ A;
(3.9)
II T1(G;z,xo ) {(o/ay2)g*(zly)} dz
~
y
~ ~),
- I T1(G;z,xo ){(o/ay2)g*(zly)I :Q} dzl ~ Aya.
Y
As in (2.10)-(2.11), the terms involving (o/ay2) may be replaced by (O/OX)
and (a2/Ox2 ). Then, we have the following.
Theorem 3.1.
Suppose that (i) T( e) ts ftrst order Hadamard-dtfferenHabte at
G(elxo)' (ii) k=kn=[tn4/5] for some 0
< a < t < b <~, (iii)
the assumpttons
•
-7-
[A1]-[A4] and [B1] hold. and
(3.10)
0 < o2{xO)
Then. as n -+ ~.
•
(3.11)
n
215
=I
~ (G;z.xO) g{zlxO) dz < ~.
A
[T{Gnk ) - T{G)]
~
-+N{~{xO)'
2
a (xO)/t) •
uhere
(3.12)
= t 2{12f 2 (xo)} -1
~(xO)
1mT1{G;z.xO)q{z.xo) dz
and
In passing. we may remark that if in (3.11) • we replace n2l5 by k~. then
we would have
•
~
A
!II
~
A
~
2
~
».
k [T{Gnk ) - T{G)] ~ N{t ~(xO)' a (xO
In this form. the result also extends to the following.
4/5
Corollary 3.1.1. If k = o{n
) and (i). (iii) and (3.10) hold. then.
(3.14)
(3.15)
2
»'
k [T{Gnk ) - T{G)] -+N{O.o (xO
as n
-+~.
The main difference between (3.14) and (3.15) is that for k
= O{n4/5).
the asymptotic normal distribution has a non-vanishing centering constant.
4/5
while for k = o{n
). this bias term is asymptotically insignificant. This
feature plays an important role in the choice of {kn }.
We consider the
following lemmas and incorporate them in the proof of the theorem and
corollary.
Lemma. 3.1.
Let k = [tnA] for some A (O< A<l) and t{O<a<t<b<~).
If the
assumption [A1] holds. then
(a)
for every B such that Bf{xO)
> b > a > O.
there exists an nO (<
~)
such
-8-
that
-l+X
-1+2A
2
X >B n
] ~ exp[-2 n
(Bf(xo)-b)]. V n ~ nO.
n[bn ]
If Un1 < U < ... < Unn denote the order statistics in a random sample
(3.16)
(b)
pry
n2
U1 •...• Un of size n from uniform distribution (0.1). then for B > b. there
exists an nO ««») such that
Hence
(3.18)
U
X
n[bn ]
are O(n-l+X ).
and Y
X
n[bn ]
For a proof of lemma 3.1. we may refer to Bhattacharya and Mack (1987).
Let k = [tn4l5 ]
Lemma 3.2.
for O<a<t<b<<<».. Suppose that the assumption [AI]
holds. then
(3.19)
k-
1
~~ r_-2
i=l
ni
= {12
2
O)}
f (x
-1
(kin)
2
+ Rnk •
where
(3.20)
max
k~[n~~]
IRnk I =
-215 )
o(n
a.s.
A similar result was proved by Bhattacharya and Mack (1987) assuming
f"(x) exists and is continuous at xO. However. that proof can be easily
modified so that the result will remain true under [AI].
Lemma 3.3.
(3.21)
(a)
under [AI]. [A2] and [Bl]. we have
xo)
g* (z Iy) = g(z I
+ y2 q(z.xO) + y2 r(z.y.xO)'
where
(3.22)
q(z.xO) = (8/ay2) g*(zIY)IY=o
= ~[~(zIXO) + 2 f'(xo)~(zlxo)/f(xO)]'
•
-9-
and
(3.23)
for some y*
•
= (8/ay2)
g*(zly)I * - (8/ay2) g*(zly)I
'
y
y=O
between 0 and y. Moreover, there exists ~ > 0 and M (0
r(z,y,xO)
such that uniformly for all yeO
< y < ~),
and for some a
II T1 (G:z,xo) q(z,xo) dzl ~ M
(3.25) II T1 (G:z,xo) r(z,y,xo) dzl ~ Mya.
(b) If in addition, we assume [AJ], then for 0
<M<
00)
>0
(3.24)
(3.26)
00
1-00 q(z,xO) dz
<y <~
=0
and
00
(3.27)
1-00 r(z,y,xO) dz = o.
Proof:
The expansion in (3.21) follows from a Taylor's series expansion of
± y) and
O
g(z Ixo ± y) about xo' substituting the expansions in (2.5), taking derivative
with respect to y2 and evaluating the derivative at y=O. (3.24) and (3.25)
g*(z Iy) about y=O.
(3.22) can be established by expanding f(x
are easy consequences of [B1].
(3.26) and (3.27) follow from the fact that
[AJ] allows us to interchange the integral and the derivative.
.
Proof of Theorem 3.1
A
•
Considering the expansion in (3.1), we have
A
=
A
T(Gnk ) - T(G)
I T1(G:z,~) d[Gnk{z)-G(z)] + o{IIGnk-GII).
Since by Theorem 2.1, the densities ~ are contiguous to the densities
(3.28)
Pnk' and under Pnk
~
A
~
A
k IIGnk-G1I = 0p{l),
it follows that under. ~
(3.29)
(3.30)
k IIGnk-G1I = 0p{l).
So, we can rewrite (3.28) as
A
(3.31)
T(Gnk ) - T{G)
=k
-1
k
-215
1:1 T1{G:Zni ,xO) + open
),
-10-
or.
~
A
(3.32) k [T(Gnk ) - T(G)]
= o(xO)
Snk + 0p(l).
where
(3.33) Snk
=k
~
k
I ¢<Z i)
i=l
n
•
and
Note that for all sequences {M } of positive numbers tending to infinity
n
(3.35)
I
Izl>Mn
~(z) dG(z) ~ 0
which in turn ensures that under Pnk
(3.36) Snk !N(O.l).
nk } of real
Our objective is to show that there exists a sequence {a
~
numbers such that under
(3.31) Snk - ank
!)
~
N(O.l).
Define
(3.38) ~rik(z)
I¢<z)
= 10
otherwise.
and
(3.39)
~nk(z)
=~rik(z)
- I
~rik(z)g(z)
It can be shown easily that
(3.40)
I
~nk(z)g(z)
dz
= O.
(3.41) I(~nk(z) - ¢<z»2 g(z) dz ~ O.
2
(3.42) k ~ sup "'nk(z)
z
Now. if we define
~
O.
dz.
-11-
k
(3.44)
Snk = I a i h i(Z i)'
i=1 n n n
then the proof of (3.37) will follow from the proof of the central limit
•
theorem under contiguous alternatives in Behnen and Neuhaus (1975).
we omit the details.
that the sequence a
nk
Hence.
However. note that the proof of their theorem shows
in (3.37) may have the form
Finally. we show that as n -. (1)
2
-1 5/2 (1) tht
(3.46) a nk -. {12 f (xO)}
J-OI) ~z) q(z.xa) dz.
t
Using the expansion in lemma 3.3 .• along with (3.45) and lemma. 3.2. we have
(3.47)
a nk = k
~
k
_..2
i:1 Y~i J ~nk(z)q(z.xO) dz
k
+
k~ I
= k
+ k
i=1
~
y
2
J
ni
~nk(z)r(z'Yni'xO) dz
2
-1
[{12 f (xO)}
(kin) 2 + o(n-215 )] J
~nk(z)q(z.xO)
dz
~ k _..2
i~1 Y~i J ~nk(z) r(z.Yni.xa) dz.
By (3.39) and lemma 3.3; we have
(3.48)
J ~nk q(z.xO)dz = J ~rik(z) q(z.xO) dz - (J ~rik(z)g(z)dz)(J q(z.xO)dz)
=J
.
l<I>(z) I~k
1/6 <l>(z) q(z.xO) dz
-. J <l>(z) q(z'xa)dz
•
and
<
(1).
-12-
=I ~rik(z)
= 0(1).
r(z.Yni.xO) dz -
(I
~rik(z)dz)(I r(z.Yni.xO) dZ)
Hence by lemma. 3.1. we have
(3.50)
k
~
k
__2
i:1 ~i I ~nk(z) r(z.Yni.xO) dz
= 0(1)
a.s.
•
Combining (3.47). (3.48) and (3.50). we have
~
2
-1 5/2 co
t
{12 f (xO)}
I-co
(3.51)
a nk
where k
= [tn4/5 ]
for some t (0
N
~z)q(z.xO)
dz. as n
~
co.
< a < t < b < co).
Now (3.32). (3.37) and (3.51) together imply
(3.52)
n
2l5
[T(Gnk ) - T(G)]
2
-1,.co
2
{12 f (xO)}
J-co T1(G:z.xO)q(z.xO) dz. a (xO)/t].
This completes the proof of theorem 3.1.
!)
~H[t
2
•
The proof of Corollary 3.1.1. follows on the same line and hence is omitted.
"-
Next we consider the 11mi ting distribution of T(Gnk ) under degenerate
case: i.e .• when the functional T is second order compact differentiable. and
satisfies the following conditions
2
(3.53) a (xO) = I ~(G:z.XO) g(z) dz
and
=0
< II ~(G:z1.z2'xO)
g(zl) g(z2) dZ 1 dZ 2 < co.
Let {Al(xO)' l = 0.1.2 •... } denote the finite or infinite collection of
eigenvalues of T (G: •••• x ) corresponding to orthonormal eigenfunctions
2
O
(3.54)
0
{Tl(·.XO): l
(3.55)
= 0.1.2 •... }.
such that
I T2 (G:z 1 .z2 .xO) Tl (zl' xO) dG(zl)
= Al(xO)
Tl(z2'xO)
a.e. (G)
for all l
~
o.
-13-
where
6em = 1 if e = m; = 0 if e ~ m.
In addition, let [B2] relate to the following conditions:
(3.57)
00
(3.58)
~
e=o
Ae(Xo)
and there exist e
< 00;
> 0,
a
>0
and A <
00
such that for O<y<e and for all
e~
0,
(3.59)
II Te{Z,Xo ) {{8/8y2) g*{zly)ly=o} dzl ~ A,
(3.60)
II Te{Z,Xo ) {{8/8y2) g*{zly)} dz
- I Te{Z,Xo ) {(8/8y2) g*(zly)ly=o} dzl ~ Aya.
Remark:
A sufficient condition for (3.59) and (3.60) to hold is given by the
following:
and
(3.63)
ITe{Z,Xo ) ~(zlx) dz - I Te{Z,Xo ) ~(zlxo) dzl ~ Alx-xola,
for Ix-~I < e.
The following lemma can be proved using an argument similar to the one
used in lemma 3.3.
Lemma 3.4.
Under the assumptions [A1], [A2] and [B2] , the eXPansion (3.21)
for g*(zly) holds, and there exists e ) 0 and M (0 < M < 00) such that for
Ix-xol < e,
(3.64)
.
and
ITe{Z,Xo ) q{z,xO) dzl ~ M
-14-
If, in addition, we assume [AJ], then for Ix-xol < e
(3.60)
m
I-m q(z,x) dz = 0
Theorem 3.2.
Suppose that (i) T(e) is second order Hardamard-differentiable
4/5
(ii) k=kn=[tn
] for some O<a<t<b<m, (iii) the assumptions
a G(e IxO)'
[Al]-[A4] and
(3.53)-(3.56).
(3.67)
m
and I-m r(z,y,x) dz = O.
[B2]
hold,
Then as n -+
and
(iv)
the
functional
T( e)
satisfies
m,
4/5 A
1 k
~ mOO 2
2n [T(Gnk ) - T(G) - k- I T1(G;Zni'xO)] -+ I Ae(XO) (Ze+ae) ,
i=1
e=o
o
0
uhere ZO' ZI' ... are C Cd. normal ran.d.om variables wHh. JRean 0 and variance
(l/t) and
o 2
2
-1
a e = t {12 f (xO)}
I R Te(Z,Xo ) q(z,xO) dz.
Note that we can express (3.67) also in the following form
(3.68)
0*0*
where ZOo ZI •... are i.i.d. standard normal random variables.
In this form. the result also extends to the following.
Corollary 3.2.1.
If k = o(n4/5) and (i). (iii) and (iv) hold. then as n -+m.
As in the Theorem 3.1. we note that the main difference between (3.69)
and (3.70) is that for k = O(n4/5). the asymptotic distribution has a non
vanishing centering constant. while for k = o(n4/5).
insignificant.
this bias term is
-15Proof.
Using the expansion given in (3.4), we have
(3.71)
T(Gnk ) - T(G) - k
A
-1
k
i:1 T1(G;Zni'xO)
As we have seen in the proof of Theorem 3.1, under
~,
A
2
-415
o(IIGnk-GlI) = open
)
(3.72)
so that
(3.73)
2k[T(Gnk ) - T(G) - k-
=k
-1
1 k
i:1 T1(G;Zni'xO)]
k
1 1 T2 (G;Z i'Z j'xO) + k
i~j
n
-1
n
k
T (G;Z i'Z i'xO) + 0 (1).
i=l ' 2
n n
p
1
Define
~
T'*
2 (G) = I T2(G;z,z,~) g(z) dz,
then by Dunford and Schwartz (1963) page 1087,
(3.74)
(3.75)
=
T*
2 (G)
1
,~O
A,(XO) < ~.
Under Pnk' by SLLN
(3.76)
k
-1
*
k
i:1 T2 (G;Zni,Zni'xO) ~T2(G) =
Since the densities
that under
(3.77)
k
-1
k
1
i=l
a.s.
are contiguous to the densities Pnk' it follows
~
Now define
and
~
,io A,(XO)'
T2 (G;Z i'Z i'xO)
n n
~
1
,~O
A.(XO), in probability.
~
-16-
where
(3.S1)
Cnkl
=k
-1
k
2
i:1 Tl{Zni)·
Using (3.56). the fact that the densi ties
densities Pnk and SLLN. we have under
~
are contiguous to the
~
Cnkl ~ 1 in probability.
By lemma 3.1. lemma 3.2 and lemma 3.4. we have under
(3.S2)
(3.S3)
~ -1
E(Enkl ) = k [k
k
1
i=l
I Tl(Z.XQ)
~i(z)
~
dz]
= a *l (say). a.s .• V l = O.l •...• R.
Now using an argument similar to the one used in proving Theorem 3.1. we
have (under
~)
* are standard normal random variables.
where Zl* •...• ZR
So. it follows from (3.79). that for fixed R. under
~
-17-
From Gregory (1977). we have that (3.SS) implies that under
<Ink
•
Finally. (3.67) follows from (3. 73). (3. 77). (3.86) and the fact that
k = [tn4/5] for fixed t(O<a<t<b<~}.
•
This completes the proof of theorem 3.2.
The proof of Corollary 3.2.1. follows the same line of argument and
hence is omitted.
We end this section with the following' observation.
We define a kernel
estimator (With uniform kernel) of the conditional distribution G(zlx } as
o
follows:
(3.S7)
~
Gnh(z} = {K (h}}
n
-1
K (h)
n
I
i=l
l(Zni
~
z}.
where h is the bandwidth and
(3.88) K (h)
n
=
n
I
i=l
l(Y
n
i ~
h/2);
and. we may consider a kernel estimator of T(G} as
A
(3.89)
Tnh = T(Gnh }.
This situation is same as the k-nearest neighbor case. except that here
k=Kn (h) is a random variable.
However. as Bhat tacharya and GangoPadhyay
choose h = O(n- 1/ 5 }. then the difference
(1988) have noted. if we
-4/5
n
[Kn(h)-nhf(xO}] is asymptotically negligible.
So. using a technique
similar to theirs. the asymptotic distribution results of T(Gnh ) will follow
from Theorem 2.1. Theorem 3.1 and Theorem 3.2.
~
-18-
4.
Proof' of' the Theorem 2.1.
Lemma 4.1
First, we establish the following lelllDa:
Under the assumptions [AI], [A2] and [A3], the expansion (3.21)
Also there exist ~
for g*(zly} hold.
> 0 and Lebesgue integrable functions
·(on IR) u 1 (z) and u2 (z} such that for o<y<~, Iq(z,xo } I and Ir(z,y,~} I are
bounded by u (z} and u (z}ya respectively, and
1
2
(4.1) I q(z,x ) dz = 0, I r(z,y,~) dz = O.
o
The proof of the lemma is exactly the same as the proof of lemma 3.3.
Now, define
if Pnk
if
>0
Pnk= ~ = 0
if Pnk = 0
=00
< ~.
Then along the fashion of LeCam's first lelllDa (see, e.g., Hajek and Sidak
(1967), page 203-204), we shall prove the following:
2
log Lnk is asymptotically normal (~ d2 , d),
densities ~ are contiguous to the densities Pnk'
LelllDa 4.2
so that the
..
For this purpose, first note that
k
(4.4)
log Lrik
= i~1
log[~i(Zni)/g(Zni)]'
In the next lelllDa we will show that the summands in (4.4) are uniformly
asymptotically negligible (UAN).
Lemma 4.3 Assume [Al]-[A4].
(4.5)
Proof:
(4.6)
lim max
n~ l~i~k
Then under Pnk'
p[I{2 i (Z i)/g(Z i)} - 11
-n
n
n
> ~)
= O.
By Chebyehev' s inequal i ty
P[I{~i(Zni)/g(Zni)} -
11
> ~] ~ E[{~i(Zni)/g(Zni)}-1]2/~2.
But by lelllDa 4.1, lelllDa 3.1 and [A4], under P
nk
•
-19-
4 2 2
I {r (z,Yni,xO)/g{z)} dz
= Yni[I {q (z,xO)/g{z)} dz +
+ 2
I {q{z,xo)r{z,Yni,xO)/g{z)} dz]
= o{I).
•
From this point on, we restrict ourselves to the case A=4I5 (i.e., k =
[tn4l5 ] for O<a<t<b<w). At the end of this section we will indicate how the
proof can be modified when A < 415.
Lemma 4.4 Let
k
(4.S)
Wnk
= i:l
~
{[~i{Zni)/g{Zni)] - I}.
Assume [Al]-[A4] hold and the statistics Wnk are asymptotically normal
2 2
(~A d ,d ) under Pnk' then under P
nk
2
(4.9) lim P{llog Lnk-Wnk + ~ d 1 > ~) = 0, for every ~ > 0,
~
".
and log L is asymptotically normal (~ d2 ,d2 ).
nk
Proof:
Since the UAN condition is satisfied (by lemma. 4.3), the resul t
follows from LeCam's second lemma. (see, e.g., Hajek and Sidak (1967) page
•
205).
(~
Thus, it is enough to show that under P , Wnk is asymptotically normal
nk
2 2
d ,d). For this, first we prove a series of results.
Lemma 4.5 Assume [Al]-[A4] hold.
(4.10)
E{Wnk ) ~
Then under Pnk
2
-(1/4)d ,
where
2
= {SO
(4.11)
d
Proof:
Denote
(4.12)
s{zly)
4
f (xO)}
-1
= [g{zly)]~
t
5
w
2
I-w{q (z,xO)/g{z)}dz.
-20-
and
(4.13)
s{z)
=s{zIO) = [g{z)]~.
So. under Pnk
(4.14)
E{'nk) = 2
k
I
i=1
2
I [{s(zIY i)/s{z)} - l]s (z) dz
n
k
4
k
4
_~
= - I Y i I {{s{zIY i)-s{z»~i}
i=1 n
n
n
2
= - I Yni I {(8/By ) S{Zly)1 *}
i=1
Yni
*
where 0 < Yni
2
2
dz
dz.
Yni : i ~ 1.2•...• k.
Note that for some e > O. by assumption [A4]
(4.15)
~
I{{8/By2) s{z ly)}2dz =
=
~ I{{8/By2) log g{z ly)}2 g{z Iy)dz
! I{z Iy) <
co.
uniformly for all y (O<y<e).
Also. from lemma 4.1 it is seen easily that
(4.16)
(8/By2) s{zly)ly=o = (1/2) {q{z.xO)/{g{z»~}.
So. (4.15) (4.16). along with lemma 3.1 give
(4.l7)
uniformly i (=1.2 •...• k).
Finally let Un1 < ... < U denote the order statistics of a random
nn
sample (Ul •...• Un ) of size n from the uniform distribution on (O.l). Then
-1
Yni = h{Uni ). where h = Fy .
[AI] the following is true:
It can be seen easily that under assumption
-21-1
h{u) = Fy (u) is defined for 0 < u <
(i)
~
for some
~
> 0 as a unique
solution of Fy(u) = u.
(ii)
h'(u) is continuous at O.
(iii) h(O) = h"(O) = O. h'(O) = {2 f(XO)}-1.
Let Fn denote the empirical cdf of U1 •...• Un ; then
(4.1S)
max
l~j~n
Iu j-(j/n)I = max Iu j-F (U j)1 ~ sup IF (u)-ul = 0 (n~).
n
l~j~n
n n n
O<u<l
n
p
So.
(4.19)
4
4
-4/5
Un j = (j/n) + 0p (n
).
for all j;
k
4
5 -4
l U i = O(k n
i=1 n
By I elllll8. 3. 1
(4.20)
k
4
k
4
Y i = I {h{Uni )}i=l n
i=l
I
k
= I {h'(O)U i + {h'('YU i) - h'(O)}U i}4
i=l
n
n
n
k
4
= lUi {h'(O) + 0 (1)}
i=l n
p
= (h'(O»
4
k
4
4
l U i + 0 (1).
i=l n
p
Substituting (4.19) in (4.20). we have
(4.21)
~
i=l
y4
-4 k (j/n}4 + 0 (1)
nl = (2 f(xO})
j:1
p
= (2 f(xO»
-4
4/5
-4/5 [t n ]
4/5 4
[n
I
(j/n) ] + 0p(l}
j=l
).
-22~
(2 f{XQ»
-4
5
4
-1 5
(t /5) = {SO f (xO)}
t.
From (4.14). (4.17) and (4.21). we have
(4.22) E{ink ) ~ -{1/4)d2 •
where
(4.23) d2 = {SO f4{XQ)}-1 t 5 I~ {q2{z.xO}/g{Z)}dZ.
Lemma 4.6 Define
k
(4.24)
Vnk = i:1
Assume [A1]-[A4].
~i{q{Zni·xO}/g{Zni}}·
Then under Pnk
Proof:
Under Pnk
(4.26)
_~ q{Zni'XQ)
k
s{ZniIYni}
Var{ink-Vnk ) = 4 I Var[ s{Z )
- 1 - (1/2)Y~i
(Z) ]
i=1
ni
g ni
~ 4
k
4
~
*
Yni
~ 2
__2
Y i I[{{s{zIY i} - s{z)}~i} - ~ {q{z.xO}/{g{z}) }] dz
i=1 n
n
n
I
k
4
2
~ 4 I Yni I[{8/8y } S{Zly)1 *
yn i
. i=1
where 0
•
-
~ 2
~ {q{z.xO}/{g{z}} }] dz
~
Yni . i=1.2 ..... k.
So. in view of {4.15}. {4.16}. {4.21} and Theorem V.1.3. of Hajek and
SidBk {1967}. we can conclude that the last integral converges to 0 uniformly
for all i=1.2 •...• k.
Proof of the theorem:
•
Note that by lemma. 4.1. under Pnk
-23-
(4.27)
E(Vnk )
k
~ y2i I{q(z,xO)/g(z)} g(z) dz
=
i=l
n
k
=
2
Y i I q(z,xo) dz
~
i=l
n
= 0,
and similarly under Pnk
(4.28)
Var(Vnk )
k
= i:1
4
2
Yni I{q (z,xO)/g(z)} dz
= {[SO
4
-1 5
2
f (xO)] t + 0p(l)} I{q (z,xO)/g(z)} dz
2
-+d.
So,
it
follows from Theorem V.1.2 of Hajek and Sidak (1967) that under Pnk
~
2
Vnk -+H(O,d
).
Now. lenuna. 4.5 and leDIIIB. 4.6 together imply that under Pnk
(4.29)
E[Wnk - Vnk + (1/4)d2 ] -+ O.
We combine (4.30) with lenuna. 4.4 to obtain that under P
nk
2
(4.31) E[log Lnk - Vnk + (1/2)d ] -+ O.
(4.30)
•
Since (4.29) and (4.31) together imply that under Pnk
(4.32)
log L ~
-+
nk
H(~
2 2
d ,d ),
from leDIIIB. 4.2. we conclude that the densi ties ~ are contiguous to the
4/5
densities Pnk' when k = [tn
] for some O<a<t<b<~.
Now, we consider the case when k
= [tnA]
for
0
<
A
< 4/5.
By lenuna. 3.1,
we have
k
..
(4.33)
~
i=1
y
4
ni
= 0(1).
So, in leDIIIB. 4.4, E(Wnk ) -+ 0 (i.e., d2
= 0).
So, it follows easily that
-24-
under Pnk
(4.34)
log L
nk
Hence.
the LeCam
~
0
in probability.
first
,
lelllllB. holds
through
the degenerate normal
law.
•
This completes the proof of Theorem 2.1.
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1)
Behnen. K. and Neuhaus.
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2)
Bhattacharya. P.K. (1914). Convergence of sample paths of normalized sums
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Bhattacharya. P.K. and Gangopadhyay. A.K. (1988).
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Bhattacharya. P.K. and Mack. Y.P. (1981).
Weak convergence of k-NN
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G.
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limit
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Stattst. 15. 916-994.
Linear operators
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5)
Dunford. N. and Schwartz. J.T. (1963).
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Fernholz. L.T. (1983).
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Gregory. G.G. (1911). Large sample theory for U-statistics and tests of
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Hajek. J. and Sidak. Z. (1961).
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Sen. P .K. (1988).
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•