Modified half-normal distribution

In probability theory and statistics, the modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. It can be viewed as a generalization of multiple families, including the half-normal distribution, truncated normal distribution, gamma distribution, and square root of the gamma distribution, all of which are special cases of the MHN distribution. Therefore, it is a flexible probability model for analyzing real-valued positive data. The name of the distribution is motivated by the similarities of its density function with that of the half-normal distribution.

Modified half-normal distribution

Notation	${\displaystyle {\text{MHN}}\left(\alpha ,\beta ,\gamma \right)}$
Parameters	${\displaystyle \alpha >0,\beta >0,{\text{ and }}\gamma \in \mathbb {R} }$
Support	${\displaystyle x\geq 0}$
PDF	${\displaystyle f_{_{\text{MHN}}}(x)={\frac {2\beta ^{\alpha /2}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi {\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}}}$
CDF	${\displaystyle {\begin{aligned}F_{_{\text{MHN}}}(x\mid \alpha ,\beta ,\gamma )={}&{\frac {2\beta ^{\alpha /2}}{\Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}\\[4pt]&{}\times \sum _{i=0}^{\infty }{\frac {\gamma ^{i}}{2i!}}\beta ^{-(\alpha +i)/2}\gamma \left({\frac {\alpha +i}{2}},\beta x^{2}\right),\end{aligned}}}$ where ${\displaystyle \gamma (s,y)}$ denotes the lower incomplete gamma function.
Mean	${\displaystyle E(X)={\frac {\Psi \left({\frac {\alpha +1}{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}{\beta ^{1/2}\Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}}$
Mode	${\displaystyle {\frac {\gamma +{\sqrt {\gamma ^{2}+8\beta (\alpha -1)}}}{4\beta }}{\text{ if }}\alpha >1}$
Variance	${\displaystyle \operatorname {Var} (X)={\frac {\Psi \left({\frac {\alpha +2}{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}{\beta \Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}-\left[{\frac {\Psi \left({\frac {\alpha +1}{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}{\beta ^{1/2}\Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}\right]^{2}}$

In addition to being used as a probability model, MHN distribution also appears in Markov chain Monte Carlo (MCMC)-based Bayesian procedures, including Bayesian modeling of the directional data, Bayesian binary regression, and Bayesian graphical modeling.

In Bayesian analysis, new distributions often appear as a conditional posterior distribution; usage for many such probability distributions are too contextual, and they may not carry significance in a broader perspective. Additionally, many such distributions lack a tractable representation of its distributional aspects, such as the known functional form of the normalizing constant. However, the MHN distribution occurs in diverse areas of research, signifying its relevance to contemporary Bayesian statistical modeling and the associated computation.

The moments (including variance and skewness) of the MHN distribution can be represented via the Fox–Wright Psi functions. There exists a recursive relation between the three consecutive moments of the distribution; this is helpful in developing an efficient approximation for the mean of the distribution, as well as constructing a moment-based estimation of its parameters.