In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution.
Let follow an ordinary normal distribution, . Then, 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 parametrization of the normal distribution, the probability density function (PDF) of the half-normal is given by
where .
Alternatively using a scaled precision (inverse of the variance) parametrization (to avoid issues if is near zero), obtained by setting , the probability density function is given by
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,
Given numbers drawn from a half-normal distribution, the unknown parameter of that distribution can be estimated by the method of maximum likelihood, giving
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 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 . There exists a recursive relation between the three consecutive moments of the distribution.
Moments
Let then for , then assuming to be a positive real number
The probability density function of the distribution is log-concave if .
The mode of the distribution is located at .
If and then the density has a local maxima at
and a local minima at .
The density function is gradually decresing on and mode of the distribution doesn't exist, if either , or .[3]
Additional properties involving mode and Expected values
Let for , and . Let denotes the mode of the distribution. For all if then,
The difference between the upper and lower bound provided in the above inequality approaches to zero as gets larger. Therefore, it also provides high precision approximation of when is large. On the other hand, if and , . For all , . An implication of the fact is that the distribution is positively skewed.[3]