The SaddlePoint Divergence Metric (SPDM): a non-parametric probability density-based framework for quantifying distributional differences

We propose the SaddlePoint Divergence Metric (SPDM), a non-parametric framework for comparing probability distributions through smooth density estimates obtained from Extended Empirical Saddlepoint Approximations (EESA). SPDM evaluates the estimated densities on a shared domain, summarizes their separation with Hellinger and total variation (TV) distances, and calibrates two-sample inference by permutation. The study evaluates SPDM across simulated distributional departures, grid and decay sensitivity analyses, computational benchmarks, targeted comparisons with KDE-based divergences and Energy distance, an Old Faithful real-data example, and an implemented bivariate extension. The results show that SPDM provides stable, interpretable density-based effect sizes and can be sensitive to shape differences that are muted in empirical cumulative distribution function (ECDF)-based comparisons, especially in small and moderate samples. The method is computationally more expensive than ECDF, KDE, and Energy-distance alternatives, so we report timing benchmarks and discuss parallelization, bootstrap, and asymptotic approximations as practical accelerations. Together, the simulations and examples position SPDM as a complementary diagnostic and inferential tool for distributional comparison when density shape and bounded divergence measures are of substantive interest.