In this section, we consider heteroscedastic errors.
Three cases will be discussed in details. Let
denote a sequence of random samples from
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In principle the weights
(or
)
are unknown and must be estimated. Let
be a sequence of estimators of
.
Naturally one can define an estimator of
by substituting
in (9) by
.
Let
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This subsection is devoted to the nonparametric
heteroscedasticity structure ,
where H is unknown Lipschitz continuous,
are design points, which are assumed to be independent of
and
and defined on [0, 1] in the case where
are random design points.
Define
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plmhetexog performs the weighted
least
squares estimate of the parameter. In the procedure of estimating the
variance function, the estimate given by
plmk is taken as the primary one.
In this subsection we consider the case where
we suppose that the variance
is a function of the design points
,
i.e.,
,
H is an unknown Lipschitz continuous. Similar to subsection 3.1,
we define our estimator of
as
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plmhett calculates the weighted
least squares estimate of the parameter in this case. In the procedure
of estimating the variance function, the estimate given by
plmk
is taken as the primary one.
Here we consider the model (8)
with ,
H is unknown Lipschitz continuous, which means that the variance
is an unknown function of the mean response.
Since
is assumed to be completely unknown, the standard method is to get
information
about
by replication, i.e., we consider the following "improved"
partially linear heteroscedastic model
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We construct an estimate of
as follows. Based on the least squares estimate
and the nonparametric estimate
,
we define
and estimate
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plmhetmean depicts how to impliment
the above algorithm. For calculation simplicity, we use the same
replicate
in practice.The estimate given by
plmk
is taken as the primary one.
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MD*TECH Method and Data Technologies |
http://www.mdtech.de mdtech@mdtech.de |