Testing Parametric Distribution Family Assumptions via Differences in Differential Entropy

We introduce a broadly applicable statistical procedure for testing which parametric distribution family generated a random sample of data. The method, termed the Difference in Differential Entropy (DDE) test, provides a unified framework applicable to a wide range of distributional families, with asymptotic validity grounded in established maximum likelihood, bootstrap, and kernel density estimation principles. The test is straightforward to implement, computationally efficient, and requires no user-defined tuning parameters or complex specialized regularity conditions. It compares an MLE-based estimate of differential entropy under the null hypothesis with a nonparametric bootstrapped kernel density estimate, using their divergence as an information-theoretic measure of model fit. The test procedure is constructive in the sense of being informative regardless of whether the null hypothesis is rejected or not, where in the latter case the outcome suggests that the hypothesized distribution can be close to the actual distribution of the data in shape and probability implications. Monte Carlo experiments demonstrate its notable size accuracy and power even in relatively small samples, and three empirical applications using classical datasets from distinct domains illustrate the method’s practical utility.