Discrete distributions from a Markov chain

TL;DR

This paper introduces a method to generalize discrete distributions using a Markov chain model, allowing for flexible dispersion and straightforward inference, with practical computation and random number generation.

Contribution

It presents a novel approach to derive generalized discrete distributions from a Markov chain, extending classical distributions with additional parameters for dispersion control.

Findings

Generalized distributions can be underdispersed or overdispersed.

Mean can be expressed simply in terms of model parameters.

Probabilities and random number generation are computationally efficient.

Abstract

A discrete-time stochastic process derived from a model of basketball is used to generalize any discrete distribution. The generalized distributions can have one or two more parameters than the parent distribution. Those derived from binomial, Poisson and negative binomial distributions can be underdispersed or overdispersed. The mean can be simply expressed in terms of model parameters, thus making inference for the mean straightforward. Probabilities can be quickly computed, enabling likelihood-based inference. Random number generation is also straightforward. The properties of some of the new distributions are described and their use is illustrated with examples.