Polynomial Distributions and Transformations

TL;DR

This paper introduces polynomial distributions as a flexible and mathematically tractable way to model stochastic systems, including methods for their transformation, fitting, and parameter estimation.

Contribution

It defines polynomial distributions explicitly, explores their properties, and develops methods for their transformation, fitting, and sampling, expanding their application beyond approximation.

Findings

Polynomial distributions can be non-negative over finite support intervals.

Closed-form expressions for key properties of polynomial distributions are derived.

A piecewise construction ensures non-negativity of polynomial distributions.

Abstract

Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of systems rather than to assume polynomials for only approximating known or empirically estimated distributions. Polynomial distributions offer a great modeling flexibility, and often, also mathematical tractability. However, unlike canonical distributions, polynomial functions may have non-negative values in the interval of support for some parameter values, the number of their parameters is usually much larger than for canonical distributions, and the interval of support must be finite. In particular, polynomial distributions are defined here assuming three forms of polynomial function. The transformation of polynomial distributions and fitting a histogram to a polynomial distribution are considered. The key properties of polynomial distributions are derived in closed-form. A piecewise polynomial distribution construction is devised to ensure that it is non-negative over the support interval. Finally, the problems of estimating parameters of polynomial distributions and generating polynomially distributed samples are also studied.