Model Checking of Dimension-reduction Type for Regression

Abstract

In this paper, a covariate marked empirical process is defined. Stute's (1997) method of nonparametric principal component decomposition, as a basis of creating tests, can be applied componentwise. The tests based on the process only involve one-dimensional projections of the covariate and do not need local smoothing techniques. Hence the curse of dimensionality is largely avoided. Furthermore,in contrast with the case that Stute, Manteiga and Quindimil (1998) investigated,it is shown that in our case, the classical bootstrap is consistent but the wild bootstrap is inconsistent. A variant of the wild bootstrap is suggested, which is consistent. A simulation study is performed to show how the test works. In addition to formal testing, residuals versus covariates along the most interesting projection direction are plotted to demonstrate how the plots can be used for model checking in multivariate cases.