Random approximations to some measures of accuracy in nonparametric curve estimation

Abstract

This paper deals with a quite general nonparametric statistical curve estimation setting. Special cases include estimation or probability density functions, regression functions, and hazard functions. The class of “fractional delta sequence estimators” is defined and treated here. This class includes the familiar kernel, orthogonal series, and histogram methods. It is seen that, under some mild assumptions, both the average square error and integrated square error provide reasonable (random) approximations to the mean integrated square error. This is important for two reasons. First, it provides theoretical backing to a practice that has been employed in several simulation studies. Second, it provides a vital tool for proving theorems about selecting smoothing parameters for several different nonparametric curve estimators.