Half-normal distribution

Probability density function ${\displaystyle \sigma =1}$
Cumulative distribution function ${\displaystyle \sigma =1}$
Parameters	${\displaystyle \sigma >0}$ — (scale)
Support	${\displaystyle x\in [0,\infty )}$
PDF	${\displaystyle f(x;\sigma )={\frac {\sqrt {2)){\sigma {\sqrt {\pi ))))\exp \left(-{\frac {x^{2)){2\sigma ^{2))}\right)\quad x>0}$
CDF	${\displaystyle F(x;\sigma )=\operatorname {erf} \left({\frac {x}{\sigma {\sqrt {2))))\right)}$
Quantile	${\displaystyle Q(F;\sigma )=\sigma {\sqrt {2))\operatorname {erf} ^{-1}(F)}$
Mean	${\displaystyle {\frac {\sigma {\sqrt {2))}{\sqrt {\pi ))))$
Median	${\displaystyle \sigma {\sqrt {2))\operatorname {erf} ^{-1}(1/2)}$
Mode	${\displaystyle 0}$
Variance	${\displaystyle \sigma ^{2}\left(1-{\frac {2}{\pi ))\right)}$
Skewness	${\displaystyle {\frac ((\sqrt {2))(4-\pi )}{(\pi -2)^{3/2))}\approx 0.9952717}$
Ex. kurtosis	${\displaystyle {\frac {8(\pi -3)}{(\pi -2)^{2))}\approx 0.869177}$
Entropy	${\displaystyle {\frac {1}{2))\log _{2}\left(2\pi e\sigma ^{2}\right)-1}$

In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.

Let ${\displaystyle X}$ follow an ordinary normal distribution, ${\displaystyle N(0,\sigma ^{2})}$. Then, ${\displaystyle Y=|X|}$ follows a half-normal distribution. Thus, the half-normal distribution is a fold at the mean of an ordinary normal distribution with mean zero.

Properties

Using the ${\displaystyle \sigma }$ parametrization of the normal distribution, the probability density function (PDF) of the half-normal is given by

${\displaystyle f_{Y}(y;\sigma )={\frac {\sqrt {2)){\sigma {\sqrt {\pi ))))\exp \left(-{\frac {y^{2)){2\sigma ^{2))}\right)\quad y\geq 0,}$

where ${\displaystyle E[Y]=\mu ={\frac {\sigma {\sqrt {2))}{\sqrt {\pi ))))$.

Alternatively using a scaled precision (inverse of the variance) parametrization (to avoid issues if ${\displaystyle \sigma }$ is near zero), obtained by setting ${\displaystyle \theta ={\frac {\sqrt {\pi )){\sigma {\sqrt {2))))}$, the probability density function is given by

${\displaystyle f_{Y}(y;\theta )={\frac {2\theta }{\pi ))\exp \left(-{\frac {y^{2}\theta ^{2)){\pi ))\right)\quad y\geq 0,}$

where ${\displaystyle E[Y]=\mu ={\frac {1}{\theta ))}$.

The cumulative distribution function (CDF) is given by

${\displaystyle F_{Y}(y;\sigma )=\int _{0}^{y}{\frac {1}{\sigma )){\sqrt {\frac {2}{\pi ))}\,\exp \left(-{\frac {x^{2)){2\sigma ^{2))}\right)\,dx}$

Using the change-of-variables ${\displaystyle z=x/({\sqrt {2))\sigma )}$, the CDF can be written as

${\displaystyle F_{Y}(y;\sigma )={\frac {2}{\sqrt {\pi ))}\,\int _{0}^{y/({\sqrt {2))\sigma )}\exp \left(-z^{2}\right)dz=\operatorname {erf} \left({\frac {y}((\sqrt {2))\sigma ))\right),}$

where erf is the error function, a standard function in many mathematical software packages.

The quantile function (or inverse CDF) is written:

${\displaystyle Q(F;\sigma )=\sigma {\sqrt {2))\operatorname {erf} ^{-1}(F)}$

where ${\displaystyle 0\leq F\leq 1}$ and ${\displaystyle \operatorname {erf} ^{-1))$ is the inverse error function

The expectation is then given by

${\displaystyle E[Y]=\sigma {\sqrt {2/\pi )),}$

The variance is given by

${\displaystyle \operatorname {var} (Y)=\sigma ^{2}\left(1-{\frac {2}{\pi ))\right).}$

Since this is proportional to the variance σ² of X, σ can be seen as a scale parameter of the new distribution.

The differential entropy of the half-normal distribution is exactly one bit less the differential entropy of a zero-mean normal distribution with the same second moment about 0. This can be understood intuitively since the magnitude operator reduces information by one bit (if the probability distribution at its input is even). Alternatively, since a half-normal distribution is always positive, the one bit it would take to record whether a standard normal random variable were positive (say, a 1) or negative (say, a 0) is no longer necessary. Thus,

${\displaystyle h(Y)={\frac {1}{2))\log _{2}\left({\frac {\pi e\sigma ^{2)){2))\right)={\frac {1}{2))\log _{2}\left(2\pi e\sigma ^{2}\right)-1.}$

Applications

The half-normal distribution is commonly utilized as a prior probability distribution for variance parameters in Bayesian inference applications.^[1][2]

Parameter estimation

Given numbers ${\displaystyle \{x_{i}\}_{i=1}^{n))$ drawn from a half-normal distribution, the unknown parameter ${\displaystyle \sigma }$ of that distribution can be estimated by the method of maximum likelihood, giving

${\displaystyle {\hat {\sigma ))={\sqrt ((\frac {1}{n))\sum _{i=1}^{n}x_{i}^{2))))$

The bias is equal to

${\displaystyle b\equiv \operatorname {E} {\bigg [}\;({\hat {\sigma ))_{\mathrm {mle} }-\sigma )\;{\bigg ]}=-{\frac {\sigma }{4n))}$

which yields the bias-corrected maximum likelihood estimator

${\displaystyle {\hat {\sigma \,))_{\text{mle))^{*}={\hat {\sigma \,))_{\text{mle))-{\hat {b\,)).}$

Related distributions

• The distribution is a special case of the folded normal distribution with μ = 0.
• It also coincides with a zero-mean normal distribution truncated from below at zero (see truncated normal distribution)
• If Y has a half-normal distribution, then (Y/σ)² has a chi square distribution with 1 degree of freedom, i.e. Y/σ has a chi distribution with 1 degree of freedom.
• The half-normal distribution is a special case of the generalized gamma distribution with d = 1, p = 2, a = ${\displaystyle {\sqrt {2))\sigma }$.
• If Y has a half-normal distribution, Y ^-2 has a Levy distribution
• The Rayleigh distribution is a moment-tilted and scaled generalization of the half-normal distribution.

Modification

The 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	${\displaystyle x>0}$
PDF	${\displaystyle f(x)={\frac {2\beta ^{\frac {\alpha }{2))x^{\alpha -1}\exp(-\beta x^{2}+\gamma x)}{\Psi {\left({\frac {\alpha }{2)),{\frac {\gamma }{\sqrt {\beta ))}\right)))))$, where ${\displaystyle \Psi (\alpha ,z)={}_{1}\Psi _{1}\left({\begin{matrix}\left(\alpha ,{\frac {1}{2))\right)\\(1,0)\end{matrix));z\right)}$ denotes the Fox-Wright Psi function.
CDF	${\displaystyle F_{_{\text{MHN))}(x\mid \alpha ,\beta ,\gamma )={\frac {2\beta ^{\frac {\alpha }{2))}{\Psi {\left({\frac {\alpha }{2)),{\frac {\gamma }{\sqrt {\beta ))}\right)))}\sum _{i=0}^{\infty }{\frac {\gamma ^{i)){2i!))\beta ^{-{\frac {\alpha +i}{2))}\gamma ({\frac {\alpha +i}{2)),\beta x^{2})}$, where ${\displaystyle \gamma (s,y)}$ denotes the lower incomplete gamma function.
Mean	${\displaystyle E(X)={\frac {\Psi \left({\frac {\alpha +1}{2)),{\frac {\gamma }{\sqrt {\beta ))}\right)}{\beta ^{\frac {1}{2))\Psi \left({\frac {\alpha }{2)),{\frac {\gamma }{\sqrt {\beta ))}\right)))}$ ${\displaystyle X_{\text{mode))\leq E(X)\leq {\frac {\gamma +{\sqrt {\gamma ^{2}+8\alpha \beta ))}{4\beta )){\text{ if ))\alpha >1.}$
Mode	${\displaystyle X_{\text{mode))={\frac {\gamma +{\sqrt {\gamma ^{2}+8\beta (\alpha -1)))}{4\beta )){\text{ if ))\alpha >1}$.
Variance	${\displaystyle {\text{Var))(X)={\frac {\Psi \left({\frac {\alpha +2}{2)),{\frac {\gamma }{\sqrt {\beta ))}\right)}{\beta \Psi \left({\frac {\alpha }{2)),{\frac {\gamma }{\sqrt {\beta ))}\right)))-\left[{\frac {\Psi \left({\frac {\alpha +1}{2)),{\frac {\gamma }{\sqrt {\beta ))}\right)}{\beta ^{\frac {1}{2))\Psi \left({\frac {\alpha }{2)),{\frac {\gamma }{\sqrt {\beta ))}\right)))\right]^{2))$.

The modified half-normal distribution (MHN)^[3] 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 MHN distribution.

The MHN distribution is used a probability model, additionally it appears in a number of Markov Chain Monte Carlo (MCMC) based Bayesian procedures including the Bayesian modeling of the Directional Data,^[4][5][6] Bayesian Binary regression,^[7] Bayesian Graphical model.^[8] The MHN distribution occurs in the diverse areas of research ^{[9][10][11] [12]} signifying its relevance to the 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, denoted by ${\displaystyle \Psi (\cdot ,\cdot )}$. There exists a recursive relation between the three consecutive moments of the distribution.

Moments

• 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 ${\displaystyle E(X^{k})={\frac {\Psi \left({\frac {\alpha +k}{2)),{\frac {\gamma }{\sqrt {\beta ))}\right)}{\beta ^{\frac {k}{2))\Psi \left({\frac {\alpha }{2)),{\frac {\gamma }{\sqrt {\beta ))}\right)))}$

• If ${\displaystyle \alpha +k>0}$, then ${\displaystyle E(X^{k+2})={\frac {\alpha +k}{2\beta ))E(X^{k})+{\frac {\gamma }{2\beta ))E(X^{k+1})}$^[3]

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

Modal characterization of MHN

Consider the MHN${\displaystyle (\alpha ,\beta ,\gamma )}$ with ${\displaystyle \alpha >0}$, ${\displaystyle \beta >0}$ and ${\displaystyle \gamma \in \mathbb {R} }$.^[3]

• 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 maxima at

${\displaystyle {\frac {\gamma +{\sqrt {\gamma ^{2}+8\beta (\alpha -1)))}{4\beta ))}$ and a local minima 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 doesn't exist, if either ${\displaystyle \gamma >0}$, ${\displaystyle 0<\alpha <1-{\frac {\gamma ^{2)){8\beta ))}$ or ${\displaystyle \gamma <0,\alpha \leq 1}$.^[3]

Additional properties involving mode and Expected values

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 the above inequality approaches to zero as ${\displaystyle \alpha }$ gets larger. Therefore, it 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 ${\displaystyle \alpha >0,\beta >0{\text{ and ))\gamma \in \mathbb {R} }$, ${\displaystyle {\text{Var))(X)\leq {\frac {1}{2\beta ))}$. An implication of the fact ${\displaystyle E(X)\geq X_{\text{mode))}$ is that the distribution is positively skewed.^[3]