We now turn our interest to the derivation of the asymptotic distribution of
.
We do this by discretizing
as
where
are the mid-points of the original bins in formulating
.
If we choose
such that
for all
,
then
are independent and each
.
This means that under the alternative
We may also establish the asymptotic normality of
by applying the central limit theorem for
a triangular array, which together with (12.28) and (12.29) means that
We see from the above that the binning based on the bandwidth value provides a key role in the derivation of
the asymptotic distributions.
However, the binning discretizes the null hypothesis and
unavoidably leads to some loss of power as shown in the simulation reported in the next section.
From the point of view of retaining power, we would like to have the size of the bins smaller than
that prescribed by the smoothing bandwidth in order to
increase the resolution of the discretized null hypothesis to the original
.
However, this will create dependence between
the empirical likelihood evaluated at neighbouring bins and make the above asymptotic distributions invalid.
One possibility is to evaluate the distribution of
by using the approach of Wood and Chan (1994)
by simulating the normal process
under
.
However, this is not our focus here and hence is not considered in this chapter.