On automatic kernel density estimate-based tests for goodness-of-fit

Although estimation and testing are different statistical problems, if we want to use a test statistic based on the Parzen–Rosenblatt estimator to test the hypothesis that the underlying density function f is a member of a location-scale family of probability density functions, it may be found reasonable to choose the smoothing parameter in such a way that the kernel density estimator is an effective estimator of f irrespective of which of the null or the alternative hypothesis is true. In this paper we address this question by considering the well-known Bickel–Rosenblatt test statistics which are based on the quadratic distance between the nonparametric kernel estimator and two parametric estimators of f under the null hypothesis. For each one of these test statistics we describe their asymptotic behaviours for a general data-dependent smoothing parameter, and we state their limiting Gaussian null distribution and the consistency of the associated goodness-of-fit test procedures for location-scale families. In order to compare the finite sample power performance of the Bickel–Rosenblatt tests based on a null hypothesis-based bandwidth selector with other bandwidth selector methods existing in the literature, a simulation study for the normal, logistic and Gumbel null location-scale models is included in this work.