Adequacy checking for the variance function in nonparametric regression

Abstract

Correctly specifying the parametric form of the variance function in regression models can help us make more efficient statistical inferences. Many existing Lack-of-fit testing procedures have already been proposed to decide the proper forms of the variance function, however, most of them are either checking the homoscedasticity, that is, to see if the variance function is a constant, or checking a pre-specified parametric forms of the variance function under the assumption of the mean regression function being known. In this report, we would like to construct some formal testing procedure to check the appropriateness of certain parametric forms for the variance function when the mean regression function is unknown. The report consists of two parts. In the first part, we propose a minimum distance-based test to check the forms of the variance function. The test statistics is a modified L2-distance between a nonparametric estimate and a parametric estimate of the variance function under the null hypothesis. The Nadaraya-Watson kernel regression function estimator is used to estimate the regression function. The large sample properties, including the consistency and asymptotic normality, of the minimum distance estimate for the parameters in the variance function are discussed; the asymptotic distribution of the test statistics under the null hypothesis is established, as well as the consistency of the test and the power under local alternative hypotheses. Simulation studies, comparison studies, as well as some applications to the real data sets, are carried out to evaluate the finite sample performance of the proposed test. In the second part, we proposed a computationally efficient test procedure for checking the parametric forms of the variance function. The test is based on an empirical smoothing of the fitted residuals by replacing the mean regression function with the Nadaraya-Watson estimator and a pre-obtained root-n consistent estimate of the parameter in the variance function. By multiplying the kernel density estimate at each individual sample points to the fitted residual, we successfully remove the constraint of compact support for design variables assumed in some existing work. Large sample properties of the proposed test under the null hypothesis is discussed alongside with consistency of the test and the power under local alternatives. Finally, some simulation studies are carried out showing the performance of the test under finite population.