Modified half-normal distribution

The probability density function of the modified half-normal distribution is $${\displaystyle f(x)={\frac {2\beta ^{\alpha /2}x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}{\text{ for }}x>0}$$  where ${\displaystyle \Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)={}_{1}\Psi _{1}\left[{\begin{matrix}({\frac {\alpha }{2}},{\frac {1}{2}})\\(1,0)\end{matrix}};{\frac {\gamma }{\sqrt {\beta }}}\right]}$  denotes the Fox–Wright Psi function.^[9][10][11] The connection between the normalizing constant of the distribution and the Fox–Wright function in provided in Sun, Kong, Pal.^[1]

The cumulative distribution function (CDF) is $${\displaystyle F_{_{\text{MHN}}}(x\mid \alpha ,\beta ,\gamma )={\frac {2\beta ^{\alpha /2}}{\Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}\sum _{i=0}^{\infty }{\frac {\gamma ^{i}}{2i!}}\beta ^{-(\alpha +i)/2}\gamma \left({\frac {\alpha +i}{2}},\beta x^{2}\right){\text{ for }}x\geq 0,}$$  where ${\displaystyle \gamma (s,y)=\int _{0}^{y}t^{s-1}e^{-t}\,dt}$  denotes the lower incomplete gamma function.

The modified half-normal distribution is an exponential family of distributions, and thus inherits the properties of exponential families.

Let ${\displaystyle X\sim {\text{MHN}}(\alpha ,\beta ,\gamma )}$ . Choose a real value ${\displaystyle k\geq 0}$  such that ${\displaystyle \alpha +k>0}$ . Then the ${\displaystyle k}$ th moment is$${\displaystyle E(X^{k})={\frac {\Psi \left({\frac {\alpha +k}{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}{\beta ^{k/2}\Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}.}$$ Additionally,$${\displaystyle E(X^{k+2})={\frac {\alpha +k}{2\beta }}E(X^{k})+{\frac {\gamma }{2\beta }}E(X^{k+1}).}$$ The variance of the distribution is ${\displaystyle \operatorname {Var} (X)={\frac {\alpha }{2\beta }}+E(X)\left({\frac {\gamma }{2\beta }}-E(X)\right).}$  The moment generating function of the MHN distribution is given as$${\displaystyle M_{X}(t)={\frac {\Psi \left({\frac {\alpha }{2}},{\frac {\gamma +t}{\sqrt {\beta }}}\right)}{\Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}.}$$ 

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

• If ${\displaystyle \alpha \geq 1}$ , then the probability density function of the distribution is log-concave.
• If ${\displaystyle \alpha >1}$ , then the mode of the distribution is located at ${\displaystyle {\frac {\gamma +{\sqrt {\gamma ^{2}+8\beta (\alpha -1)}}}{4\beta }}.}$ 
• If ${\displaystyle \gamma >0}$  and ${\displaystyle 1-{\frac {\gamma ^{2}}{8\beta }}\leq \alpha <1}$ , then the density has a local maximum at ${\displaystyle {\frac {\gamma +{\sqrt {\gamma ^{2}+8\beta (\alpha -1)}}}{4\beta }}}$  and a local minimum at ${\displaystyle {\frac {\gamma -{\sqrt {\gamma ^{2}+8\beta (\alpha -1)}}}{4\beta }}.}$ 
• The density function is gradually decreasing on ${\displaystyle \mathbb {R} _{+}}$  and mode of the distribution does not exist, if either ${\displaystyle \gamma >0}$ , ${\displaystyle 0<\alpha <1-{\frac {\gamma ^{2}}{8\beta }}}$  or ${\displaystyle \gamma <0,\alpha \leq 1}$ .

Let ${\displaystyle X\sim {\text{MHN}}(\alpha ,\beta ,\gamma )}$  for ${\displaystyle \alpha \geq 1}$ , ${\displaystyle \beta >0}$ , and ${\displaystyle \gamma \in \mathbb {R} {}}$ , and let the mode of the distribution be denoted by ${\displaystyle X_{\text{mode}}={\frac {\gamma +{\sqrt {\gamma ^{2}+8\beta (\alpha -1)}}}{4\beta }}.}$ 

If ${\displaystyle \alpha >1}$ , then $${\displaystyle X_{\text{mode}}\leq E(X)\leq {\frac {\gamma +{\sqrt {\gamma ^{2}+8\alpha \beta }}}{4\beta }}}$$ for all ${\displaystyle \gamma \in \mathbb {R} }$ . As ${\displaystyle \alpha }$  gets larger, the difference between the upper and lower bounds approaches zero. Therefore, this also provides a high precision approximation of ${\displaystyle E(X)}$  when ${\displaystyle \alpha }$  is large.

On the other hand, if ${\displaystyle \gamma >0}$  and ${\displaystyle \alpha \geq 4}$ , then $${\displaystyle \log(X_{\text{mode}})\leq E(\log(X))\leq \log \left({\frac {\gamma +{\sqrt {\gamma ^{2}+8\alpha \beta }}}{4\beta }}\right).}$$ For all ${\displaystyle \alpha >0}$ , ${\displaystyle \beta >0}$ , and ${\displaystyle \gamma \in \mathbb {R} }$ , ${\displaystyle {\text{Var}}(X)\leq {\frac {1}{2\beta }}}$ . Also, the condition ${\displaystyle \alpha \geq 4}$  is a sufficient condition for its validity. The fact that ${\displaystyle X_{\text{mode}}\leq E(X)}$  implies the distribution is positively skewed.

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