A Bayesian Motivated Two-Sample Test Based on Kernel Density Estimates

• Published in: Entropy (Basel, Switzerland) Vol. 24; no. 8; p. 1071
• Main Authors: Merchant, Naveed, Hart, Jeffrey D.
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 03.08.2022; MDPI
• Subjects: Analysis; Bandwidths; Bayes factors; Bayesian statistical decision theory; Bivariate analysis; consistent tests; cross-validation; Density; Density functionals; Estimates; Estimation theory; Hypotheses; Kernel functions; Kernels; Kolmogorov–Smirnov test; Methods; Multivariate analysis; permutation tests; Permutations; Statistical sampling; Statistics; United States
• ISSN: 1099-4300; 1099-4300
• DOI: 10.3390/e24081071

A new nonparametric test of equality of two densities is investigated. The test statistic is an average of log-Bayes factors, each of which is constructed from a kernel density estimate. Prior densities for the bandwidths of the kernel estimates are required, and it is shown how to choose priors so that the log-Bayes factors can be calculated exactly. Critical values of the test statistic are determined by a permutation distribution, conditional on the data. An attractive property of the methodology is that a critical value of 0 leads to a test for which both type I and II error probabilities tend to 0 as sample sizes tend to ∞. Existing results on Kullback–Leibler loss of kernel estimates are crucial to obtaining these asymptotic results, and also imply that the proposed test works best with heavy-tailed kernels. Finite sample characteristics of the test are studied via simulation, and extensions to multivariate data are straightforward, as illustrated by an application to bivariate connectionist data.