To develop a test about we first
introduce a nonparametric kernel estimator for
.
For an introduction into kernel estimation see
Härdle (1990), Wand and Jones (1995)
and
(Härdle et al.; 2000)
.
Without loss of generality we assume that we are only interested
in
for
and that
with a positive constant
.
If
in a particular problem
the data are supported by another closed interval,
this problem
can be transformed by rescaling into an
equivalent problem with data support
.
Let be a bounded probability density function with a
compact support on
that satisfies the moment
conditions:
The nonparametric estimator considered is the Nadaraya-Watson (NW) estimator
The parameter estimation of depends on the null hypothesis. We
assume here, that the parameter
is estimated by
a
-consistent estimator.
Let
The local linear estimator can be used to replace the NW estimator in estimating .
However, as we compare
with
in formulating the Goodness-of-Fit test, the possible bias
associated with the NW estimator is not an issue here. In addition,
the NW estimator has a simpler analytic form.