Generalised Hyperbolic Distribution

Information about Generalised Hyperbolic Distribution

generalised hyperbolic
Probability density function
Cumulative distribution function
Parameters	$\text{[math]}$ location (real) $\text{[math]}$ (real) $\text{[math]}$ (real) $\text{[math]}$ asymmetry parameter (real) $\text{[math]}$ scale parameter (real) $\text{[math]}$
Support	$\text{[math]}$
Probability density function (pdf)	$\text{[math]}$ $\text{[math]}$
Cumulative distribution function (cdf)
Mean	$\text{[math]}$
Median
Mode
Variance	$\text{[math]}$
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)	$\text{[math]}$
Characteristic function

The generalised hyperbolic distribution (GH) is a continuous probability distribution defined as the normal variance-mean mixture where the mixing distribution is the generalized inverse Gaussian distribution. Its probability density function (see the box) is given in terms of modified Bessel function of the third kind, denoted by $\text{[math]}$ .

As the name suggests it is of a very general form, being the superclass of, among others, the Student's t-distribution, the Laplace distribution, the hyperbolic distribution, the normal-inverse Gaussian distribution and the variance-gamma distribution.

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. It was introduced by Ole Barndorff-Nielsen.

Related distributions

$\text{[math]}$ has a Student's t-distribution with $\text{[math]}$ degrees of freedom.
$\text{[math]}$ has a hyperbolic distribution.
$\text{[math]}$ has a normal-inverse Gaussian distribution (NIG).
$\text{[math]}$ normal-inverse chi-square distribution
$\text{[math]}$ normal-inverse gamma distribution (NI)
$\text{[math]}$ has a variance-gamma distribution.

	Probability distributions [ • • [ edit] ]
	Univariate	Multivariate
Discrete:	Benford • Bernoulli • binomial • Boltzmann • categorical • compound Poisson • discrete phase-type • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot	Ewens • multinomial • multivariate Polya
Continuous:	Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Coxian • Erlang • exponential • exponential power • F • fading • Fermi-Dirac • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square (scaled inverse chi-square) • inverse Gaussian • inverse gamma (scaled inverse gamma) • Kumaraswamy • Landau • Laplace • Lvy • Lvy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • Nakagami • normal (Gaussian) • normal-gamma • normal inverse Gaussian • Pareto • Pearson • phase-type • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • shifted Gompertz • Student's t • triangular • truncated normal • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda	Dirichlet • Generalized Dirichlet distribution . inverse-Wishart • Kent • matrix normal • multivariate normal • multivariate Student • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous:	bimodal • Cantor • conditional • equilibrium • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • posterior • prior • quasi • sampling • singular

location parameter, since its value determines the "location" of the probability distribution.

In other words, when you graph the function, the location parameter determines where the origin will be located.
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In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
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In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions.

Definition

If a family of probability densities with parameter s is of the form

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In mathematics, a support of a function f from a set X to the real numbers R is a subset Y of X such that f (x) is zero for all x in X and outside Y.
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In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals.

Formally, a probability distribution has density f, if f
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In probability theory, the cumulative distribution function (CDF), also called probability distribution function or just distribution function,^[1] completely describes the probability distribution of a real-valued random variable X.
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expected value (or mathematical expectation, or mean) of a discrete random variable is the sum of the probability of each possible outcome of the experiment multiplied by the outcome value (or payoff).
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median is described as the number separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to highest value and picking
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In statistics, mode means the most frequent value assumed by a random variable, or occurring in a sampling of a random variable. The term is applied both to probability distributions and to collections of experimental data.
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variance of a random variable (or somewhat more precisely, of a probability distribution) is one measure of statistical dispersion, averaging the squared distance of its possible values from the expected value.
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skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable.

Introduction

Consider the distribution in the figure. The bars on the right side of the distribution taper differently than the bars on the left side.
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kurtosis (from the Greek word kurtos, meaning bulging) is a measure of the "peakedness" of the probability distribution of a real-valued random variable. Higher kurtosis means more of the variance is due to infrequent extreme deviations, as opposed to frequent
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Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable.

Shannon entropy quantifies the information contained in a piece of data: it is the minimum average message length, in bits (if using base-2 logarithms), that must
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In probability theory and statistics, the moment-generating function of a random variable X is

wherever this expectation exists. The moment-generating function generates the moments of the probability distribution.
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In probability theory, the characteristic function of any random variable completely defines its probability distribution. On the real line it is given by the following formula, where X is any random variable with the distribution in question:

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In probability theory, a probability distribution is called continuous if its cumulative distribution function is continuous. That is equivalent to saying that for random variables X with the distribution in question, Pr[X = a
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In probability theory a normal variance-mean mixture with mixing probability density is the continuous probability distribution of a random variable of the form

where and are real numbers and .
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generalized inverse Gaussian distribution (GIG) is a continuous probability distribution with probability density function

where x > 0, K_p is a modified Bessel function of the third kind, a >
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t-distribution or Student's 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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Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also known as the double exponential distribution, because it can be thought of as two exponential distributions (with an additional location parameter) spliced together
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hyperbolic distribution is a continuous probability distribution that is characterized by the fact that the logarithm of the probability density function is a hyperbola. Thus the distribution decreases exponentially, which is more slowly than the normal distribution.
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normal-inverse Gaussian distribution (NIG) is continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution. The tails of the distribution decrease more slowly than the normal distribution.
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variance-gamma distribution is continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the gamma distribution. The tails of the distribution decrease more slowly than the normal distribution.
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normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. Each member of the family may be defined by two parameters, location and scale: the mean ("average",
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Much effort has gone into the study of financial markets and how prices vary with time. Charles Dow, one of the founders of Dow Jones & Company and The Wall Street Journal, enunciated a set of ideas on the subject which are now called Dow Theory.
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Ole Eiler Barndorff-Nielsen

Born March 18 1935
Copenhagen
Residence Denmark
Nationality Danish
Field Mathematics
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