The Modified-Half-Normal distribution: Properties and an efficient sampling scheme

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.

Keywords:

• Modified-Half-Normal distribution
• accept reject algorithm
• Fox-Wright Psi function
• distribution theory
• random variate generation

Acknowledgements

We thank the editor and the anonymous referee for their detailed and constructive comments, which have led to a considerable improvement of the article.