beta4dist: A Python package for the four-parameter Beta distribution and likelihood-based estimation

Abstract

We present beta4dist, the first open-source Python package that implements a likelihood-based estimation framework for the four-parameter Beta distribution. This flexible distribution is widely used to model bounded, continuous data with diverse shapes, including skewed and heavy-tailed patterns. Such datasets are common in fields such as hydrology, environmental science, and reliability engineering. The software estimates location parameters via order statistics and computes shape parameters using marginal likelihood optimization, ensuring that all estimates adhere to natural parameter constraints. In addition to core estimation routines, beta4dist includes utilities for density evaluation, random sampling, cumulative distribution, quantiles, and model diagnostics. The package is fully tested, easy to integrate into standard Python workflows, and supports both research reproducibility and practical applications requiring shape-robust modeling tools. © 2025 Elsevier B.V., All rights reserved.