Differentiable fit of empirical distributions by Bernstein phase types

Abstract: The empirical cumulative distribution function (ECDF) is an efficient estimator of the distribution of sampled data. Yet, it has discontinuities at each step (where probability mass is concentrated) which result in a probability density function (PDF) with discrete spikes, preventing its use in models that require continuous PDFs. Bernstein approximants bridge this gap, yielding smooth estimators for bounded ECDFs and their PDFs. In this paper, we consider a strategy from the literature to select the order of Bernstein polynomials (BP) over bounded support, and we extend the approach to Bernstein phase types (BPH) in order to approximate distributions with unbounded support. Preliminary experiments confirm the effectiveness of this approach in balancing the bias-variance tradeoff, giving insight into the relation between the sample size and the Bernstein order. Comparative experiments with Gaussian kernel density estimators show that the BPH approximant achieves better or comparable performance on a benchmark of Erlang PDFs with unit mean, while guaranteeing a low memory footprint thanks to its closed form and allowing integration into continuous-time Markov chains. We provide a replication package to support reproducibility of experiments.

Proceedings of EPEW 2026, pp. 115-130, Springer, 2026

Stochastic ProcessesPerformance ModelsApplications



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