An Algorithm for Computing the Distribution Function of the Generalized Poisson-Binomial Distribution

TL;DR

This paper introduces a new algorithm to efficiently compute the distribution function of the generalized Poisson-binomial distribution, extending the classical model to variables taking arbitrary two values, with applications in various scientific fields.

Contribution

We derive the distribution using Fourier transforms and develop an efficient FFT-based algorithm for computation, addressing the lack of existing methods for the generalized case.

Findings

The algorithm accurately computes the distribution compared to enumeration methods.

It is computationally efficient across various parameter settings.

The software implementation demonstrates practical applicability.

Abstract

The Poisson-binomial distribution is useful in many applied problems in engineering, actuarial science, and data mining. The Poisson-binomial distribution models the distribution of the sum of independent but not identically distributed Bernoulli random variables whose success probabilities vary. In this paper, we extend the Poisson-binomial distribution to the generalized Poisson-binomial (GPB) distribution. The GPB distribution is defined in cases where the Bernoulli variables can take any two arbitrary values instead of 0 and~1. The GPB distribution is useful in many areas such as voting theory, actuarial science, warranty prediction, and probability theory. With few previous works studying the GPB distribution, we derive the probability distribution via the discrete Fourier transform of the characteristic function of the distribution. We develop an efficient algorithm for computing the distribution function, which uses the fast Fourier transform. We test the accuracy of the developed algorithm upon comparing it with enumeration-based exact method and the results from the binomial distribution. We also study the computational time of the algorithm in various parameter settings. Finally, we discus the factors affecting the computational efficiency of this algorithm, and illustrate the use of the software package.