Half-normal distribution


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 of the half-normal is given by
where.
Alternatively using a scaled precision parametrization, obtained by setting, the probability density function is given by
where.
The cumulative distribution function is given by
Using the change-of-variables, the CDF can be written as
where erf is the error function, a standard function in many mathematical software packages.
The quantile function is written:
where and is the inverse error function
The expectation is then given by
The variance 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 entropy of the half-normal distribution is exactly one bit less the 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. 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 or negative is no longer necessary. Thus,

Parameter estimation

Given numbers drawn from a half-normal distribution, the unknown parameter of that distribution can be estimated by the method of maximum likelihood, giving
The bias is equal to
which yields the bias-corrected maximum likelihood estimator

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