Generalised Score Distribution: Underdispersed Continuation of the Beta-Binomial Distribution

TL;DR

This paper introduces the Generalised Score Distribution (GSD) class, a flexible family of discrete distributions for finite support data, extending the beta-binomial distribution to better model Likert-scale responses with intuitive parameters.

Contribution

The paper proposes the GSD class, a new family of distributions with comprehensive mean-variance coverage, easy parameter estimation, and a novel interpretation as a sum of dichotomous variables.

Findings

GSD accurately models multimedia quality responses.

Parameters can be estimated via method of moments and MLE.

GSD generalizes the beta-binomial distribution for underdispersed data.

Abstract

A class of discrete probability distributions contains distributions with limited support. A typical example is some variant of a Likert scale, with response mapped to either the $\{1, 2, \ldots, 5\}$ or $\{-3, -2, \ldots, 2, 3\}$ set. An interesting subclass of discrete distributions with finite support are distributions limited to two parameters and having no more than one change in probability monotonicity. The main contribution of this paper is to propose a family of distributions fitting the above description, which we call the Generalised Score Distribution (GSD) class. The proposed GSD class covers the whole set of possible mean and variances, for any fixed and finite support. Furthermore, the GSD class can be treated as an underdispersed continuation of a reparametrized beta-binomial distribution. The GSD class parameters are intuitive and can be easily estimated by the method of moments. We also offer a Maximum Likelihood Estimation (MLE) algorithm for the GSD class and evidence that the class properly describes response distributions coming from 24 Multimedia Quality Assessment experiments. At last, we show that the GSD class can be represented as a sum of dichotomous zero-one random variables, which points to an interesting interpretation of the class.