108 P. Coretto and M.L. Parrella
The Nadaraya-Watson estimator of the regression functionm(x)is given bymˆh(x)=∑S
j= 1 Rj,p+^1 Kh(x−Xj,p)
∑S
j= 1 Kh(x−Xj,p), (9)
whereKh(u)=h−dK(h−^1 u)andKis ad-dimensional product kernel, as defined in
[8]. The parameterhis the bandwidth of the estimator, which regulates the smoothing
of the estimated function with respect to all regressors. We use a common bandwidth
because we assume that all the regressors have been standardised.
When applied to kernel estimators, empirical likelihood can be defined as follows.
For a givenx,letpj(x)be nonnegative weights assigned to the pairs(Xj,p,Rj,p+ 1 ),
forj= 1 ,...,S. The empirical likelihood for a smoothed version ofmγˆ(x)is defined
as
L{ ̃mγˆ(x)}=max⎧
⎨
⎩
∏S
j= 1pj(x)⎫
⎬
⎭
, (10)
where the maximisation is subject to the following constraints
∑Sj= 1pj(x)= 1 ;∑S
j= 1pj(x)K(
x−Xj,p
h)
[
Rj,p+ 1 − ̃mγˆ(x)]
= 0. (11)
As is clear from equation (11), the comparison is based on a smoothed version of the
estimated parametric functionmγˆ(x)(see [8] for a discussion), given by
m ̃γˆ(x)=∑S
j= 1 mγˆ(Xj,p)Kh(x−Xj,p)
∑S
j= 1 Kh(x−Xj,p). (12)
By using Lagrange’s method, the empirical log-likelihood ratio is given by
l{ ̃mγˆ(x)}=−2log[L{ ̃mγˆ(x)}SS]. (13)Note thatSScomes from the maximisation in (10), since the maximum is achieved
atpj(x)=S−^1.
Theorem 1.Under H 0 and the assumptions A.1 in [8], we have
l{
m ̃γˆ(x)} d
−→χ 12. (14)Proof (sketch).The proof of the theorem is based on the following asymptotic equiv-
alence (see [4] and [8])
l{
m ̃γˆ(x)}
≈
[
(Shd)^1 /^2{
mˆh(x)− ̃mγ(x)}
V^1 /^2 (x;h)] 2
, (15)
whereV(x;h)is the conditional variance ofRj,p+ 1 givenXj,p=x.Fortheorem3.4
of [2], the quantity in brackets is asymptoticallyN( 0 , 1 ).