A simple test for the parametric form of the variance function in nonparametric regression

Abstract

In this paper a new test for the parametric form of the variance function in the common nonparametric regression model is proposed which is applicable under very weak assumptions. The new test is based on an empirical process formed from pseudo residuals, for which weak convergence to a Gaussian process can be established. In the special case of testing for homoscedasticity the limiting process is essentially a Brownian bridge, such that critical values are easily available. The new procedure has three main advantages. First, in contrast to many other methods proposed in the literature, it does not depend directly on a smoothing parameter. Secondly, it can detect local alternatives converging to the null hypothesis at a rate n −1/2 . Thirdly, -in contrast to most of the currently available tests -it does not require strong smoothness assumptions regarding the regression and variance function. We also present a simulation study and compare the tests with the procedures which are currently available for this problem and require the same minimal assumptions.

Empirical likelihood estimators for the error distribution in nonparametric regression models

“…Further the empirical distribution function of estimated errors recently turned out to be valuable for goodness-of-fit tests concerning the regression or variance function, see Van Keilegom, González Manteiga and Sánchez Sellero (2004) and Dette and Van Keilegom (2005), or for testing the equality of regression functions in a two-sample problem, see PardoFernández, Van Keilegom and González-Manteiga (2004) and Neumeyer and Dette (2005). We denote by F n the (not available) empirical distribution function of unobserved errors.…”

A test for the parametric form of the variance function in a partial linear regression model

“…We note that the processesŜ * t andŜ * * t exhibit the same asymptotic behaviour as the corresponding process considered by Dette and Hetzler (2006) …”

“…. , Z n ) The weak convergence of the stochastic process defined on the right hand side of (5.7) has been established by Dette and Hetzler (2006) ) by Markov's inequality. For the remaining part of the proof we establish the estimatê…”

On the Choice of Difference Sequence in a Unified Framework for Variance Estimation in Nonparametric Regression

“…We note that the choice of the difference sequence is still rather arbitrary in the recent literature. For instance, Hall and Heckman (2000) and Shen and Brown (2006) used the Rice estimator; Munk et al (2005), Einmahl and Van Keilegom (2008), and Dette and Hetzler (2009) used the ordinary estimators; Brown and Levine (2007), Benko, Härdle andKneip (2009), andPendakur andSperlich (2010) used the optimal estimators.…”