Modified half-normal distribution

Modified half-normal distribution

The modified half-normal distribution

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

In probability theory and statistics, the family of modified half-normal distributions (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 can not only be used a probability model but it appears in a a number of Markov chain Monte Carlo (MCMC) based Bayesian procedures including the Bayesian modeling of the directional data, Bayesian binary regression, Bayesian graphical model. 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 tractable representation of its distributional aspects, such as the known functional form of the normalizing constant. However, the MHN distribution occurs in the diverse areas of research signifying its relevance to the contemporary Bayesian 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 the three consecutive moments of the distribution. It is not only be helpful in developing a efficient approximation for the mean of the distribution but also beneficial to construct moment based estimation of its the parameters. Note that the family of MHN distributions can be viewed as a generalizations of multiple families including half normal, truncated normal, square root of a gamma and gamma distributions. Therefore, it is flexible probability model to analyzing real valued positive data.

Definitions

The probability density function of the distribution is

$f(x)={\frac {2\beta ^{\alpha /2}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}{\text{ for }}x>0$

where $\Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)={}_{1}\Psi _{1}\left[{\begin{matrix}({\frac {\alpha }{2}},{\frac {1}{2}})\\(1,0)\end{matrix}};{\frac {\gamma }{\sqrt {\beta }}}\right]$ denotes the Fox–Wright Psi function. The connection between the normalizing constant of the distribution and the Fox–Wright function in provided in Sun, Kong, Pal. The cumulative distribution function (CDF) is

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

where $\gamma (s,y)=\int _{0}^{y}t^{s-1}e^{-t}\,dt$, 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 $k\geq 0$, then assuming $\alpha +k$ to be a positive real number, $E(X^{k})={\frac {\Psi \left({\frac {\alpha +k}{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}{\beta ^{k/2}\Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}$

• 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 ${\text{Var}}(X)={\frac {\alpha }{2\beta }}+E(X)\left({\frac {\gamma }{2\beta }}-E(X)\right)$

Moment generating function

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

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 $\alpha \geq 1$.
• 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 $1-{\frac {\gamma ^{2}}{8\beta }}\leq \alpha <1$ 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 decresing on $\mathbb {R} _{+}$ and mode of the distribution does not exist, if either $\gamma >0$, $0<\alpha <1-{\frac {\gamma ^{2}}{8\beta }}$ or $\gamma <0,\alpha \leq 1$.

Additional properties involving mode and expected values

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

Mixture representation

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