Frequentist nonparametric goodness-of-fit tests via marginal likelihood ratios

A nonparametric procedure for testing the goodness of fit of a parametric 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 K0K0 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 Kolmogorov–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 concerning the distribution of nebulae brightnesses in the Andromeda galaxy. Generalizations to the multivariate case are also described.