{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:03:30Z","timestamp":1760238210573,"version":"build-2065373602"},"reference-count":8,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,3]],"date-time":"2022-08-03T00:00:00Z","timestamp":1659484800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Department of Statistics, Texas A&amp;M University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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 \u221e. Existing results on Kullback\u2013Leibler 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.<\/jats:p>","DOI":"10.3390\/e24081071","type":"journal-article","created":{"date-parts":[[2022,8,3]],"date-time":"2022-08-03T20:52:01Z","timestamp":1659559921000},"page":"1071","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Bayesian Motivated Two-Sample Test Based on Kernel Density Estimates"],"prefix":"10.3390","volume":"24","author":[{"given":"Naveed","family":"Merchant","sequence":"first","affiliation":[{"name":"Department of Statistics, Texas A&M University, College Station, TX 77840, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jeffrey D.","family":"Hart","sequence":"additional","affiliation":[{"name":"Department of Statistics, Texas A&M University, College Station, TX 77840, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,3]]},"reference":[{"key":"ref_1","unstructured":"Merchant, N., Hart, J., and Choi, T. (2020). Use of cross-validation Bayes factors to test equality of two densities. arXiv."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Bowman, A.W., and Azzalini, A. (1997). Applied Smoothing Techniques for Data Analysis: The Kernel Approach with S-Plus Illustrations, OUP Oxford.","DOI":"10.1093\/oso\/9780198523963.001.0001"},{"key":"ref_3","unstructured":"Baranzano, R. (2011). Non-Parametric Kernel Density Estimation-Based Permutation Test: Implementation and Comparisons. [Ph.D. Thesis, Uppsala University]."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.csda.2015.10.013","article-title":"Frequentist nonparametric goodness-of-fit tests via marginal likelihood ratios","volume":"96","author":"Hart","year":"2016","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1491","DOI":"10.1214\/aos\/1176350606","article-title":"On Kullback-Leibler loss and density estimation","volume":"15","author":"Hall","year":"1987","journal-title":"Ann. Stat."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"920","DOI":"10.2307\/2532993","article-title":"Non-parametric analysis of covariance","volume":"51","author":"Young","year":"1995","journal-title":"Biometrics"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"153","DOI":"10.4236\/ojs.2017.71012","article-title":"Use of BayesSim and smoothing to enhance simulation studies","volume":"7","author":"Hart","year":"2017","journal-title":"Open J. Stat."},{"key":"ref_8","unstructured":"Dua, D., and Graff, C. (2022, March 15). UCI Machine Learning Repository. School of Information and Computer Sciences, University of California, Irvine. Available online: http:\/\/archive.ics.uci.edu\/ml."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/8\/1071\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:01:41Z","timestamp":1760140901000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/8\/1071"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,3]]},"references-count":8,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["e24081071"],"URL":"https:\/\/doi.org\/10.3390\/e24081071","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,8,3]]}}}