Goodness-of-Fit Tests via Kernel Density Estimation: Univariate and Multivariate Approaches

Abstract

This paper introduces goodness-of-fit tests based on leave-one-out kernel density estimation (KDE) for both univariate and multivariate data. The proposed test statistic takes the form of a log-likelihood ratio comparing a parametric null model with a nonparametric KDE-based estimator, where the leave-one-out construction removes the in-sample bias inherent in standard kernel density estimators. For univariate densities with bounded support, boundary bias is mitigated via reflection techniques. We establish consistency of the proposed tests and show that, under fixed alternatives, the normalized log-statistic converges to the Kullback--Leibler divergence between the true density and the closest parametric model under the null, providing an information-theoretic interpretation of the procedure. A parametric bootstrap is used for finite-sample calibration. Extensive simulation studies and real data applications demonstrate strong empirical performance across a range of univariate and multivariate settings, including heavy-tailed, skewed, and multimodal alternatives.