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.
翻译:一种离散概率分布等级包含有限支持的分布。 一个典型的例子就是Altrt 比例表的一些变体, 其响应范围为$1、 2、 eldots、 5 ⁇ $ 或$ 3、 -2、 eldots、 2、 3 ⁇ 设置。 一个有趣的离散分布分类, 有限支持的分解分布小类是限于两个参数的分布, 概率单一度变化不超过一个。 本文的主要贡献是提出一个符合上述描述的批次, 我们称之为通用分分布( GSD ) 类。 拟议的 GSD 类别覆盖了所有可能的平均值和差异, 用于任何固定和有限支持。 此外, GSD 类可以被视为重新平衡的乙型- 双胞胎分布的不完全分散的延续。 GSD 等级参数是直观的, 并且很容易通过时间法来估计。 我们还为 GSD 类提供了一种最接近的 Emliimation (MLE) 算法, 并证明该类正确描述从24个多介质质量到等级的响应的分布, 度的分级 。