Half-normal Distribution
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In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution. Let X 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 ...
, N(0,\sigma^2). Then, 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 \sigma 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 : f_Y(y; \sigma) = \frac\exp \left( -\frac \right) \quad y \geq 0, where E = \mu = \frac. Alternatively using a scaled precision (inverse of the variance) parametrization (to avoid issues if \sigma is near zero), obtained by setting \theta=\frac, 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 : f_Y(y; \theta) = \frac\exp \left( -\frac \right) \quad y \geq 0, where E = \mu = \frac. 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 : F_Y(y; \sigma) = \int_0^y \frac\sqrt \, \exp \left( -\frac \right)\, dx Using the change-of-variables z = x/(\sqrt\sigma), the CDF can be written as : F_Y(y; \sigma) = \frac \,\int_0^\exp \left(-z^2\right)dz = \operatorname\left(\frac\right), where erf is the error function, a standard function in many mathematical software packages. The quantile function (or inverse CDF) is written: :Q(F;\sigma)=\sigma\sqrt \operatorname^(F) where 0\le F \le 1 and \operatorname^ 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 : E = \sigma \sqrt, The variance is given by : \operatorname(Y) = \sigma^2\left(1 - \frac\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, : h(Y) = \frac \log_2 \left( \frac \right) = \frac \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 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 \_^n drawn from a half-normal distribution, the unknown parameter \sigma of that distribution can be estimated by the method of maximum likelihood, giving : \hat \sigma = \sqrt The bias is equal to : b \equiv \operatorname\bigg ;(\hat\sigma_\mathrm - \sigma)\;\bigg = - \frac which yields the bias-corrected maximum likelihood estimator : \hat^*_\text = \hat_\text - \hat.


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'' = \sqrt\sigma. * 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 \Psi(\cdot,\cdot). There exists a recursive relation between the three consecutive moments of the distribution.


Moments

* Let X\sim MHN(\alpha, \beta, \gamma)then for k\geq 0, then assuming \alpha+k to be a positive real number E(X^k)= \frac * If \alpha+k>0, then E(X^) =\frac E(X^) +\frac E(X^) * The variance of the distribution \text(X)= \frac+E(X)\left( \frac-E(X)\right)


Modal characterization of MHN

Consider the MHN(\alpha, \beta, \gamma) with \alpha>0, \beta >0 and \gamma \in \mathbb. * The probability density function of the distribution is log-concave if \alpha\geq 1. * The mode of the distribution is located at \frac \text \alpha> 1. * If \gamma>0 and 1- \frac \leq \alpha < 1 then the density has a local maxima at \frac and a local minima at \frac. * The density function is gradually decresing on \mathbb_ and mode of the distribution doesn't exist, if either \gamma>0, 0 <\alpha <1-\frac or \gamma<0, \alpha\leq 1.


Additional properties involving mode and Expected values

Let X\sim \text(\alpha,\beta,\gamma) for \alpha \geq 1, \beta>0 and \gamma\in \R. Let X_=\frac denotes the mode of the distribution. For all \gamma\in \mathbb if \alpha>1 then, X_ \leq E(X)\leq \frac. The difference between the upper and lower bound provided in the above inequality approaches to zero as \alpha gets larger. Therefore, it also provides high precision approximation of E(X) when \alpha is large. On the other hand, if \gamma>0 and \alpha\geq 4, \log(X_) \leq E(\log(X))\leq \log\left( \frac \right) . For all \alpha>0, \beta>0 \text \gamma\in \mathbb, \text(X)\leq \frac. An implication of the fact E(X)\geq X_ 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 Distribution
at MathWorld :(note that MathWorld uses the parameter \theta = \frac\sqrt {{ProbDistributions, continuous-semi-infinite Continuous distributions Normal distribution