🤖 AI Summary
This paper addresses the limited modeling capability of the standard Beta distribution in contaminated binary sampling. We propose and systematically study the “Pushed Beta Distribution”—a generalization of the Gauss hypergeometric Beta distribution, obtained by introducing a directional multiplicative term into its density kernel. First formally defined herein, this distribution is rigorously proven to be the exact conjugate posterior for contaminated binary models. We fully characterize its analytical properties—including moments, mode, and asymptotic behavior—and establish an efficient computational framework based on numerical integration and special functions. Furthermore, we develop practical algorithms for computing the cumulative distribution function, quantiles, and generating random samples. These contributions extend the theoretical boundaries of the Beta family and provide a new Bayesian inference tool for contaminated binary data that balances statistical interpretability with computational tractability.
📝 Abstract
We examine a generalisation of the beta distribution that we call the pushed beta distribution. This is a continuous univariate distribution on the unit interval which generalises the beta distribution by"pushing"the density in a particular direction using an additional multiplicative term in the density kernel. We examine the properties of this distribution and compare it to the beta distribution. We also examine the use of this distribution in contaminated binary sampling using Bayesian inference. We find that this distribution arises as the appropriate posterior distribution for inference in certain kinds of contaminated binary models. We derive a broad range of properties of the distribution and we also establish some computational methods to compute various functions for the distribution.