Frequentist nonparametric goodness-of-fit tests via marginal likelihood ratios Jeffrey D. Hart a , Taeryon Choi b,∗ , Seongbaek Yi c a Department of Statistics, Texas A&M University, U.S.A. b Department of Statistics, Korea University, Republic of Korea c Department of Statistics, Pukyoung National University, Republic of Korea Abstract A nonparametric procedure for testing the goodness of fit of a paramet- ric density is investigated. The test statistic is the ratio of two marginal likelihoods corresponding to a kernel estimate and the parametric model. The marginal likelihood for the kernel estimate is obtained by proposing a prior for the estimate’s bandwidth, and then integrating the product of this prior and a leave-one-out kernel likelihood. Properties of the kernel-based marginal likelihood depend importantly on the kernel used. In particular, a specific, somewhat heavy-tailed, kernel K 0 yields better performing marginal likelihood ratios than does the popular Gaussian kernel. Monte Carlo is used to compare the power of the new test with that of the Shapiro-Wilk test, the Kolomogorov-Smirnov test, and a recently proposed goodness-of-fit test based on empirical likelihood ratios. Properties of these tests are considered when testing the fit of normal and double exponential distributions. The new test is used to establish a claim made in the astronomy literature con- cerning the distribution of nebulae brightnesses in the Andromeda galaxy. Generalizations to the multivariate case are also described. Keywords: Bandwidth parameter, Empirical null distribution, Goodness-of-fit tests, Kernel density estimation, Marginal likelihoods 1. Introduction We consider the classical problem of testing the goodness-of-fit of a para- metric model for a distribution. Our approach is nonparametric in that our * Corresponding Author Email : trchoi@gmail.com, Tel : +82-2-3290-2245, Fax : +82-2-924-9895 Preprint submitted to CSDA November 1, 2015 © 2015. This manuscript version is made available under the Elsevier user license http://www.elsevier.com/open-access/userlicense/1.0/