In probability theory and statistics, the half-normal distribution is a special case of the
folded normal distribution.
Let
follow an ordinary
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu i ...
,
. 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
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
(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
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) c ...
is given by
:
where
.
The
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...
(CDF) 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 (or inverse CDF) is written:
:
where
and
is the
inverse error function
In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as:
:\operatorname z = \frac\int_0^z e^\,\mathrm dt.
This integral is a special function, speci ...
The expectation is then given by
:
The variance is given by
:
Since this is proportional to the variance σ
2 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,
:
Applications
The half-normal distribution is commonly utilized as a
prior probability distribution for
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...
parameters in
Bayesian inference applications.
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
:
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
In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-square ...
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'' =
.
* If ''Y'' has a half-normal distribution, ''Y''
-2 has a
Levy distribution
Levy, Lévy or Levies may refer to:
People
* Levy (surname), people with the surname Levy or Lévy
* Levy Adcock (born 1988), American football player
* Levy Barent Cohen (1747–1808), Dutch-born British financier and community worker
* Levy F ...
* The
Rayleigh distribution
In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distrib ...
is a moment-tilted and scaled generalization of the half-normal distribution.
Modification
The modified half-normal distribution (MHN)
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
In probability theory and statistics, the gamma distribution is a two- parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma dis ...
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, Bayesian
Binary regression, Bayesian
Graphical model.
The MHN distribution occurs in the diverse areas of research
signifying its relevance to the contemporary statistical modeling and associated computation. Additionally, the
moments and its other moment based statistics (including
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of number ...
,
skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.
For a unimo ...
) 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
* If
, then
* The variance of the distribution
Modal characterization of MHN
Consider the MHN
with
,
and
.
* 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
.
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.
See also
*
Half-''t'' distribution
*
Truncated normal distribution
*
Folded normal distribution
*
Rectified Gaussian distribution
In probability theory, the rectified Gaussian distribution is a modification of the Gaussian distribution when its negative elements are reset to 0 (analogous to an electronic rectifier). It is essentially a mixture of a discrete distribution ( ...
References
Further reading
*
External links
Half-Normal Distributionat
MathWorld
:(note that MathWorld uses the parameter
{{ProbDistributions, continuous-semi-infinite
Continuous distributions
Normal distribution