Generalized Score Distribution

A class of discrete probability distributions contains distributions with limited support, i.e. possible argument values are limited to a set of numbers (typically consecutive). Examples of such data are results from subjective experiments utilizing the Absolute Category Rating (ACR) technique, where possible answers (argument values) are $\{1, 2, \cdots, 5\}$ or typical Likert scale $\{-3, -2, \cdots, 3\}$. An interesting subclass of those distributions are distributions limited to two parameters: describing the mean value and the spread of the answers, and having no more than one change in the probability monotonicity. In this paper we propose a general distribution passing those limitations called Generalized Score Distribution (GSD). The proposed GSD covers all spreads of the answers, from very small, given by the Bernoulli distribution, to the maximum given by a Beta Binomial distribution. We also show that GSD correctly describes subjective experiments scores from video quality evaluations with probability of 99.7\%. A Google Collaboratory website with implementation of the GSD estimation, simulation, and visualization is provided.