Generalised score distribution: underdispersed continuation of the beta-binomial distribution

Abstract

Consider a class of discrete probability distributions with a limited support. A typical example of such support is some variant of a Likert scale, with a response mapped to either the {1,2,…,5}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{1, 2, \\ldots , 5\\}$$\\end{document} or {-3,-2,…,2,3}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{-3, -2, \\ldots , 2, 3\\}$$\\end{document} set. Such type of data is common for Multimedia Quality Assessment but can also be found in many other research fields. For modelling such data a latent variable approach is usually used (e.g., Ordered Probit). In many cases it is convenient or even necessary to avoid latent variable approach (e.g., when dealing with too small sample size). To avoid it the proper class of discrete distributions is needed. The main idea of this paper is to propose a family of discrete probability distributions with only two parameters that play the same role as the parameters of the normal distribution. We call the new class the Generalised Score Distribution (GSD). The proposed GSD class covers the entire set of possible means 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.