3. HETEROSKEDASTIC CASES

In this section, we consider heteroscedastic errors. Three cases will be discussed in details. Let image denote a sequence of random samples from

 image
where image are the same as those in model (1). image are i.i.d. with mean 0 and variance 1, and image are some functions of other variables. The specific form of image will be discussed in later subsections. The least squares estimator image is modified to a weighted least squares estimator
 image
for some weight image, image. In our model (8) we take image

In principle the weights image (or image) are unknown and must be estimated. Let image be a sequence of estimators of image. Naturally one can define an estimator of image by substituting image in (9) by image. Let

image
be the estimator of image.

3.1. VARIANCE IS A FUNCTION OF EXOGENOUS VARIABLES

This subsection is devoted to the nonparametric heteroscedasticity structure image, where H is unknown Lipschitz continuous, image are design points, which are assumed to be independent of image and image and defined on [0, 1] in the case where image are random design points.

Define

image
as the estimator of H(w). Where image is a sequence of weight functions satisfying with W in the place of T. image.

XploRe plmhetexog performs the weighted least squares estimate of the parameter. In the procedure of estimating the variance function, the estimate given by XploReplmk is taken as the primary one.

3.2. VARIANCE IS AN UNKNOWN FUNCTION OF T

In this subsection we consider the case where we suppose that the variance image is a function of the design points image, i.e., image, H is an unknown Lipschitz continuous. Similar to subsection 3.1, we define our estimator of image as

image

XploRe 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 XploReplmk is taken as the primary one.

3.3. VARIANCE IS A FUNCTION OF THE MEAN

Here we consider the model (8) with image, H is unknown Lipschitz continuous, which means that the variance is an unknown function of the mean response.

Since image is assumed to be completely unknown, the standard method is to get information about image by replication, i.e., we consider the following "improved" partially linear heteroscedastic model

image
where image is the response of the jth replicate at the design point image, image are i.i.d. with mean 0 and variance 1, image, image and image are the same as before.

We construct an estimate of image as follows. Based on the least squares estimate image and the nonparametric estimate image, we define image and estimate

image
where each image is unbounded.

XploRe plmhetmean depicts how to impliment the above algorithm. For calculation simplicity, we use the same replicate in practice.The estimate given by XploReplmk is taken as the primary one.


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