Modified half-normal distribution

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

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

• 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 distribution is given as ${\displaystyle M_{X}(t)={\frac {\Psi \left({\frac {\alpha }{2}},{\frac {\gamma +t}{\sqrt {\beta }}}\right)}{\left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)}}.}$

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

• The probability density function of the distribution is log-concave if ${\displaystyle \alpha \geq 1}$.
• The mode of the distribution is located at ${\displaystyle {\frac {\gamma +{\sqrt {\gamma ^{2}+8\beta (\alpha -1)}}}{4\beta }}{\text{ if }}\alpha >1}$.
• 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 decresing 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 ${\displaystyle \gamma \in \mathbb {R} {}}$. Let ${\displaystyle X_{\text{mode}}={\frac {\gamma +{\sqrt {\gamma ^{2}+8\beta (\alpha -1)}}}{4\beta }}}$ denotes the mode of the distribution. For all ${\displaystyle \gamma \in \mathbb {R} }$ if ${\displaystyle \alpha >1}$ then, ${\displaystyle X_{\text{mode}}\leq E(X)\leq {\frac {\gamma +{\sqrt {\gamma ^{2}+8\alpha \beta }}}{4\beta }}.}$ 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 ${\displaystyle E(X)}$ when ${\displaystyle \alpha }$ is large. On the other hand, if ${\displaystyle \gamma >0}$ and ${\displaystyle \alpha \geq 4}$, ${\displaystyle \log(X_{\text{mode}})\leq E(\log(X))\leq \log \left({\frac {\gamma +{\sqrt {\gamma ^{2}+8\alpha \beta }}}{4\beta }}\right)}$. For all ${\displaystyle \alpha >0,\beta >0{\text{ and }}\gamma \in \mathbb {R} }$, ${\displaystyle {\text{Var}}(X)\leq {\frac {1}{2\beta }}}$. Also, the condition ${\displaystyle \alpha \geq 4}$ is a sufficient condition for its validity. An implication of the fact ${\displaystyle E(X)\geq X_{\text{mode}}}$ is that 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 ${\displaystyle V\mid X\sim \operatorname {Poisson} (\gamma X){\text{ and }}X^{2}\mid V\sim \operatorname {Gamma} \left({\frac {\alpha +V}{2}},\beta \right)}$. On the contarary, if ${\displaystyle \gamma <0}$ then there exists a random variable ${\displaystyle U}$ such that ${\displaystyle U\mid X\sim {\text{GIG}}\left({\frac {1}{2}},1,\gamma ^{2}X^{2}\right){\text{ and }}X^{2}\mid U\sim {\text{Gamma}}\left({\frac {\alpha }{2}},\left(\beta +{\frac {\gamma ^{2}}{U}}\right)\right)}$. Here the GIG denotes the generalized inverse Gaussian distribution.