Nonparametric goodness-of-fit testing for a continuous multivariate parametric model

Abstract

A novel goodness-of-fit test of a continuous parametric model in the multivariate setting, based on aggregating local discrepancies between a nonparametric estimate of the density and the parametrically estimated density under the null model, is introduced. The theoretical results of the article include analytic quantification of the test statistic’s asymptotic distribution under both the null and alternative hypotheses, including closed-form expressions for its asymptotic power under fixed and local alternatives. Motivated by a Berry–Esseen type bound that we derive, a bandwidth selector which optimizes a measure of the test statistic’s rate of convergence to normality is introduced. A bootstrap size function approximation yields cut-off points suitable for finite sample implementations of the test. An extensive simulation study under Pitman and Kullback–Leibler alternatives compares the new test to well-established tests in the literature and demonstrates the strong and competitive performance of the former in the majority of the examples considered. Finally, the practical usefulness of the new test is demonstrated in the analysis of a real dataset involving stock market returns.