Abstract

We introduce a new family of probability distributions that we call the Modified-Half-Normal distributions. It is supported on the positive part of the real line with its probability density proportional to the function $x\mapsto x^{\alpha -1}\hspace{0.17em}\text{exp}\hspace{0.17em}(-\beta x^2+\gamma x)\mathbb{I}(x>0),\alpha >0,\beta >0,\gamma \in \mathbb{R}.$ We explore a number of its properties including showing the fact that the normalizing constant and moments of the distribution can be represented in terms of the Fox-Wright function. We demonstrate its relevance by showing its connection to a number of Bayesian statistical methods appearing from multiple areas of research such as Bayesian Binary regression, Analysis of Directional data, and Bayesian graphical model. The availability of its efficient sampling scheme is important to the success of such Bayesian procedures. Therefore, a major focus of this article is the development of methods for generating random samples from the Modified-Half-Normal distribution. To ensure efficiency, we prove that the constructed accept reject algorithms are “uniformly efficient” with high acceptance probability irrespective of the choice of the parameter specifications. Finally, we provide basic inference procedure for its parameters when analyzing data assuming the Modified-Half-Normal probability model.

摘要

我们引入了一个新的概率分布族，我们称之为修正半正态分布。它支持在实线的正部分，其概率密度与函数成正比$X\mapsto X^{\alpha -\mathrm{1个}}\hspace{0.17em}\text{exp}\hspace{0.17em}(-\beta X^{\mathrm{2个}}+\gamma X)\mathbb{我}(X>0),\alpha >0,\beta >0,\gamma \in \mathbb{R}.$我们探讨了它的许多属性，包括表明分布的归一化常数和矩可以用 Fox-Wright 函数来表示这一事实。我们通过展示它与来自多个研究领域的许多贝叶斯统计方法的联系来证明它的相关性，例如贝叶斯二元回归、方向数据分析和贝叶斯图形模型。其有效抽样方案的可用性对于此类贝叶斯程序的成功非常重要。因此，本文的一个主要重点是开发从修正半正态分布生成随机样本的方法。为保证效率，我们证明，无论参数规范的选择如何，所构建的接受拒绝算法都是“一致有效”的，具有高接受概率。最后，我们在假设修正半正态概率模型分析数据时提供了其参数的基本推理过程。