On the Finite Sample Behavior of Fixed Bandwidth Bickel–Rosenblatt Test for Univariate and Multivariate Uniformity

Abstract

The Bickel–Rosenblatt (BR) goodness-of-fit test with fixed bandwidth was introduced by Fan in 1998. Although its asymptotic properties have been studied by several authors, little is known about its finite sample performance. Restricting our attention to the test of uniformity in the d-unit cube for d ≥ 1, we present in this article a description of the finite sample behavior of the BR test as a function of the bandwidth h. For d = 1 our analysis is based not only on empirical power results but also on the Bahadur's concept of efficiency. The numerical evaluation of the Bahadur local slopes of the BR test statistic for different values of h for a set of Legendre and trigonometric alternatives give us some additional insight about the role played by the smoothing parameter in the detection of departures from the null hypothesis. For d > 1 we develop a Monte-Carlo study based on a set of meta-type uniforme alternative distributions and a rule-of-thumb for the practical choice of the bandwidth is proposed. For both univariate and multivariate cases, comparisons with existing uniformity tests are presented. The BR test reveals an overall good comparative performance, being clearly superior to the considered competiting tests for bivariate data.

Keywords:

• Bahadur efficiency
• Bickel–Rosenblatt test
• Goodness-of-fit test
• Kernel density estimator
• Uniformity test

Mathematics Subject Classification:

• 62G10
• 62G20

Acknowledgments

This work was partially supported by CMUC (Centre for Mathematics, University of Coimbra)/FCT.