NationMaster - Encyclopedia: Generalised hyperbolic distribution

**generalised hyperbolic**
Probability density function
Cumulative distribution function
Parameters	$μ$ location (real) $λ$ (real) $α$ (real) $β$ skewness (real) $δ$ scale (real) $gamma = sqrt{alpha^2 - beta^2}$
Support	$x in (-infty; +infty)!$
pdf	$frac{(gamma/delta)^lambda}{sqrt{2pi}K_lambda(delta gamma)} ; e^{beta (x - mu)} !$ $times frac{K_{lambda - 1/2}left(alpha sqrt{delta^2 + (x - mu)^2}right)}{left(sqrt{delta^2 + (x - mu)^2} / alpharight)^{1/2 - lambda}} !$
cdf
Mean	$μ$
Median
Mode
Variance
Skewness
Kurtosis
Entropy
mgf	$e^{mu z} frac{gamma^lambda}{gamma_z^lambda} frac{K_lambda(delta gamma_z)}{K_lambda (delta gamma)}$
Char. func.

The generalised hyperbolic distribution is a continuous probability distribution defined by the probability density function In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line. ... In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... In statistics, if a family of probabiblity densities parametrized by a parameter s is of the form fs(x) = f(sx)/s then s is called a scale parameter, since its value determines the scale of the probability distribution. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical... In probability theory and statistics, the median is a number that separates the higher half of a sample, a population, or a probability distribution from the lower half. ... In statistics, the mode is the value that has the largest number of observations, namely the most frequent value or values. ... In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ... In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ... Entropy of a Bernoulli trial as a function of success probability. ... In probability theory and statistics, the moment-generating function of a random variable X is wherever this expectation exists. ... In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question: Here t is a real number, E denotes the expected value, and F is the cumulative distribution function. ... By one convention, a random variable X is called continuous if its cumulative distribution function is continuous. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

$f(x) = frac{(gamma/delta)^lambda}{sqrt{2pi}K_lambda(delta gamma)}; frac{K_{lambda - 1/2}left(alpha sqrt{delta^2 + (x - mu)^2}right)}{left(sqrt{delta^2 + (x - mu)^2} / alpharight)^{1/2 - lambda}}; e^{beta (x - mu)}$

where $K ν$ is the modified Bessel function of the second kind. In mathematics, Bessel functions, first defined by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real number Î± (the order). ...

As the name suggests it is of a very general form, being the superclass of, among others, the Student's t-distribution, the hyperbolic distribution and the normal-inverse Gaussian distribution. In probability and statistics, the t-distribution or Students t-distribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. ...

Its main areas of application are those which require sufficient probability of far-field behaviour, which it can model due to its semi-heavy tails, a property that the normal distribution does not possess. The generalised hyperbolic distribution is well-used in economics, with particular application in the fields of modelling financial markets and risk management, due to its semi-heavy tails. This class is closed under linear operations. The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ...

Related distributions

$X sim mathrm{GH}(-frac{nu}{2}, 0, 0, sqrt{nu}, mu)$ has a Student's t-distribution with $ν$ degrees of freedom.
$X sim mathrm{GH}(1, alpha, beta, delta, mu)$ has a hyperbolic distribution.
$X sim mathrm{GH}(-1/2, alpha, beta, delta, mu)$ has a normal-inverse Gaussian distribution.

Category: Continuous distributions In probability and statistics, the t-distribution or Students t-distribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. ...

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