Half-normal distribution – Wikipedia

Probability distribution

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[edit]

Using the

${displaystyle sigma }$

parametrization of the normal distribution, the probability density function (PDF) of the half-normal is given by

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

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

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

where

${displaystyle E[Y]=mu ={frac {1}{theta }}}$

.

The cumulative distribution function (CDF) is given by

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

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

${displaystyle 0leq Fleq 1}$

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,

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

Applications[edit]

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

Parameter estimation[edit]

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

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

which yields the bias-corrected maximum likelihood estimator

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

Related distributions[edit]

• 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.
• Modified half-normal distribution^[3] with the pdf on
  ${displaystyle (0,infty )}$

  is given as

  ${displaystyle f(x)={frac {2beta ^{frac {alpha }{2}}x^{alpha -1}exp(-beta x^{2}+gamma x)}{Psi {left({frac {alpha }{2}},{frac {gamma }{sqrt {beta }}}right)}}}}$

  , where

  ${displaystyle Psi (alpha ,z)={}_{1}Psi _{1}left({begin{matrix}left(alpha ,{frac {1}{2}}right)\(1,0)end{matrix}};zright)}$

  denotes the Fox-Wright Psi function.

See also[edit]

References[edit]

Further reading[edit]

External links[edit]

(note that MathWorld uses the parameter

${displaystyle theta ={frac {1}{sigma }}{sqrt {pi /2}}}$