Modified half-normal distribution

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The Modified Half Normal Distribution

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

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. The Truncated normal distribution, Half-normal distribution, and square root of the Gamma distribution are special cases of the Modified Half Normal distribution. The name of the distribution is motivated by the similarities of its density function with that of the Half Normal distribution.

The MHN distribution is used as a probability model; additionally, it appears in a number of Markov Chain Monte Carlo (MCMC) based Bayesian procedures including the Bayesian modeling of Directional Data., Bayesian Binary regression, Bayesian Graphical model. The MHN distribution occurs in diverse areas of research. signifying its relevance to contemporary statistical modeling and associated computation. Additionally, the moments and its other moment-based statistics (including variance, skewness) can be represented via the Fox-Wright Psi functions. There exists a recursive relation between three consecutive moments of the distribution. It is helpful in developing an efficient approximation for the mean of the distribution but also beneficial to construct moment-based estimation of its parameters. Note that the family of MHN distributions can be viewed as a generalization of multiple families including Half Normal, Truncated Normal, square root of a Gamma, and Gamma distributions. Therefore, it is a flexible probability model for analyzing real-valued positive data.

Definitions

The probability density function of the distribution is given as

$f(x)=\frac{2{\beta }^{\frac{\alpha }{2}}{x}^{\alpha -1}\exp (-\beta x^2+\gamma x)}{\mathrm{\Psi }(\frac{\alpha }{2},\frac{\gamma }{\sqrt{\beta }})}$

where $\mathrm{\Psi }(\frac{\alpha }{2},\frac{\gamma }{\sqrt{\beta }})={}_1{\mathrm{\Psi }}_1[\begin{array}{c}(\frac{\alpha }{2},\frac{1}{2})\\ (1,0)\end{array};\frac{\gamma }{\sqrt{\beta }}]$ denotes the Fox-Wright Psi function. The connection between the normalizing constant of the distribution and the Fox-Wright function is provided in Sun, Kong, Pal. The Cumulative Distribution Function (CDF) is given as follows:

$F_{{}_{\text{MHN}}}(x\mid \alpha ,\beta ,\gamma )=\frac{2{\beta }^{\frac{\alpha }{2}}}{\mathrm{\Psi }(\frac{\alpha }{2},\frac{\gamma }{\sqrt{\beta }})}\sum _{i=0}^{\mathrm{\infty }}\frac{{\gamma }^i}{2i!}{\beta }^{-\frac{\alpha +i}{2}}\gamma (\frac{\alpha +i}{2},\beta x^2),$

where $\gamma (s,y)=\underset{0}{\overset{y}{\int }}{t}^{s-1}{e}^{-t}dt$, denotes the lower incomplete gamma function.

Properties

The Modified Half Normal distribution is an exponential family of distributions. Therefore, the properties of the exponential family of distributions are automatically applicable to the MHN distribution.

Moments

• Let $X\sim MHN(\alpha ,\beta ,\gamma )$ then for $k\ge 0$, then assuming $\alpha +k$ to be a positive real number, $E(X^k)=\frac{\mathrm{\Psi }(\frac{\alpha +k}{2},\frac{\gamma }{\sqrt{\beta }})}{{\beta }^{\frac{k}{2}}\mathrm{\Psi }(\frac{\alpha }{2},\frac{\gamma }{\sqrt{\beta }})}$

• If $\alpha +k>0$, then $E(X^{k+2})=\frac{\alpha +k}{2\beta }E(X^k)+\frac{\gamma }{2\beta }E(X^{k+1})$

• The variance of the distribution $\text{Var}(X)=\frac{\alpha }{2\beta }+E(X)(\frac{\gamma }{2\beta }-E(X))$

Moment Generating Function

• The moment generating function of the distribution is given as $M_X(t)=\frac{\mathrm{\Psi }(\frac{\alpha }{2},\frac{\gamma +t}{\sqrt{\beta }})}{\mathrm{\Psi }(\frac{\alpha }{2},\frac{\gamma }{\sqrt{\beta }})}.$

Modal characterization of MHN

Consider the MHN$(\alpha ,\beta ,\gamma )$ with $\alpha >0$, $\beta >0$ and $\gamma \in \mathbb{ℝ}$.

• The probability density function of the distribution is log-concave if $\alpha \ge 1$.
• The mode of the distribution is located at $\frac{\gamma +\sqrt{{\gamma }^2+8\beta (\alpha -1)}}{4\beta }\text{ if }\alpha >1$.
• If $\gamma >0$ and $1-\frac{{\gamma }^2}{8\beta }\le \alpha <1$ then the density has a local maximum at

$\frac{\gamma +\sqrt{{\gamma }^2+8\beta (\alpha -1)}}{4\beta }$ and a local minimum at $\frac{\gamma -\sqrt{{\gamma }^2+8\beta (\alpha -1)}}{4\beta }$.

• The density function is gradually decreasing on ${\mathbb{ℝ}}_{+}$ and the mode of the distribution does not exist if either $\gamma >0$, $0<\alpha <1-\frac{{\gamma }^2}{8\beta }$ or $\gamma <0,\alpha \le 1$.

Additional properties involving mode and expected values

Let $X\sim \text{MHN}(\alpha ,\beta ,\gamma )$ for $\alpha \ge 1$, $\beta >0$ and $\gamma \in \mathbb{ℝ}$. Let $X_{\text{mode}}=\frac{\gamma +\sqrt{{\gamma }^2+8\beta (\alpha -1)}}{4\beta }$ denote the mode of the distribution. For all $\gamma \in \mathbb{ℝ}$ if $\alpha >1$ then, $X_{\text{mode}}\le E(X)\le \frac{\gamma +\sqrt{{\gamma }^2+8\alpha \beta }}{4\beta }.$ The difference between the upper and lower bound provided in the above inequality approaches zero as $\alpha$ gets larger. Therefore, it also provides a high-precision approximation of $E(X)$ when $\alpha$ is large. On the other hand, if $\gamma >0$ and $\alpha \ge 4$, $\log (X_{\text{mode}})\le E(\log (X))\le \log \left(\frac{\gamma +\sqrt{{\gamma }^2+8\alpha \beta }}{4\beta }\right)$. For all $\alpha >0,\beta >0\text{ and }\gamma \in \mathbb{ℝ}$, $\text{Var}(X)\le \frac{1}{2\beta }$. An implication of the fact $E(X)\ge X_{\text{mode}}$ is that the distribution is positively skewed.

Mixture representation

Let $X\sim MHN(\alpha ,\beta ,\gamma )$. If $\gamma >0$ then there exists a random variable $V$ such that $V\mid X\sim \text{Poisson}(\gamma X)\text{ and }{X}^2\mid V\sim Gamma(\frac{\alpha +V}{2},\beta )$. On the contrary, if $\gamma <0$ then there exists a random variable $U$ such that $U\mid X\sim \text{GIG}(\frac{1}{2},1,{\gamma }^2{X}^2)\text{ and }{X}^2\mid U\sim \text{Gamma}(\frac{\alpha }{2},(\beta +\frac{{\gamma }^2}{U}))$. Here the GIG denotes the Generalized inverse Gaussian distribution.

References

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