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Modified half-normal distribution

Probability distribution

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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}}

Inprobability theory and statistics, the modified half-normal distribution (MHN)^{[1][2][3][4][5][6][7][8]} 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.

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,^[4] 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.^{[clarification needed]}

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.

Definitions[edit]

The probability density function of the modified half-normal distribution is

${\displaystyle \Psi \left({\frac {\alpha }{2}},{\frac {\gamma }{\sqrt {\beta }}}\right)={}_{1}\Psi _{1}\left[{\begin{matrix}({\frac {\alpha }{2}},{\frac {1}{2}})\\(1,0)\end{matrix}};{\frac {\gamma }{\sqrt {\beta }}}\right]}$

The cumulative distribution function (CDF) is

${\displaystyle \gamma (s,y)=\int _{0}^{y}t^{s-1}e^{-t}\,dt}$

Properties[edit]

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

Moments[edit]

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 ${\displaystyle k}$thmomentis

${\displaystyle \operatorname {Var} ($X)={\frac {\alpha }{2\beta }}+E(X)\left({\frac {\gamma }{2\beta }}-E(X)\right).}

Modal characterization[edit]

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${\displaystyle \gamma <0,\alpha \leq 1}$.

Additional properties involving mode and expected values[edit]

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

${\displaystyle \gamma \in \mathbb {R} }$

${\displaystyle \alpha }$

${\displaystyle E($X)}

${\displaystyle \alpha }$

On the other hand, if ${\displaystyle \gamma >0}$ and ${\displaystyle \alpha \geq 4}$, then

${\displaystyle \alpha >0}$

${\displaystyle \beta >0}$

${\displaystyle \gamma \in \mathbb {R} }$

${\displaystyle {\text{Var}}($X)\leq {\frac {1}{2\beta }}}

${\displaystyle \alpha \geq 4}$

${\displaystyle X_{\text{mode}}\leq E($X)}

Mixture representation[edit]

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.

References[edit]