Generalised hyperbolic distribution

Probability distribution
name =generalised hyperbolic
type =density
pdf_

cdf_

parameters = $mu$ location (real) $lambda$ (real) $alpha$ (real) $eta$ asymmetry parameter (real) $delta$ scale parameter (real) $gamma = sqrt{alpha^2 - eta^2}$
support = $x in (-infty; +infty)!$
pdf = $frac{(gamma/delta)^lambda}{sqrt{2pi}K_lambda(delta gamma)} ; e^{eta (x - mu)} !$ $imes frac{K_{lambda - 1/2}left(alpha sqrt{delta^2 + (x - mu)^2}
ight)}{left(sqrt{delta^2 + (x - mu)^2} / alpha
ight)^{1/2 - lambda$ !
cdf =
mean = $mu + frac{delta eta K_{lambda+1}(delta gamma)}{gamma K_lambda(deltagamma)}$
median =
mode =
variance = $frac{delta K_{lambda+1}(delta gamma)}{gamma K_lambda(deltagamma)} + frac{eta^2delta^2}{gamma^2}left( frac{K_{lambda+2}(deltagamma)}{K_{lambda}(deltagamma)} - frac{K_{lambda+1}^2(deltagamma)}{K_{lambda}^2(deltagamma)}
ight)$
skewness =
kurtosis =
entropy =
mgf = $frac{e^{mu z}gamma^lambda}{(sqrt{alpha^2 -(eta +z)^2})^lambda} frac{K_lambda(delta sqrt{alpha^2 -(eta +z)^2})}{K_lambda (delta gamma)}$
char =

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 $K_lambda$ .

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

* $X sim mathrm{GH}(-frac{
u}{2}, 0, 0, sqrt{
u}, mu),$ has a Student's "t"-distribution with $u$ degrees of freedom.
* $X sim mathrm{GH}(1, alpha, eta, delta, mu),$ has a hyperbolic distribution.

* $X sim mathrm{GH}(-1/2, alpha, eta, delta, mu),$ has a normal-inverse Gaussian distribution (NIG).
* $X sim mathrm{GH}(?, ?, ?, ?, ?),$ normal-inverse chi-square distribution
* $X sim mathrm{GH}(?, ?, ?, ?, ?),$ normal-inverse gamma distribution (NI)

* $X sim mathrm{GH}(lambda, alpha, eta, 0, mu),$ has a variance-gamma distribution.

Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

Hyperbolic distribution — Probability distribution name =hyperbolic type =density pdf cdf parameters =mu location (real) alpha (real) eta asymmetry parameter (real) delta scale parameter (real) gamma = sqrt{alpha^2 eta^2} support =x in ( infty; +infty)! pdf… … Wikipedia
Student's t-distribution — Probability distribution name =Student s t type =density pdf cdf parameters = u > 0 degrees of freedom (real) support =x in ( infty; +infty)! pdf =frac{Gamma(frac{ u+1}{2})} {sqrt{ upi},Gamma(frac{ u}{2})} left(1+frac{x^2}{ u} ight)^{ (frac{… … Wikipedia
Variance-gamma distribution — Probability distribution name =variance gamma distribution type =density pdf cdf parameters =mu location (real) alpha (real) eta asymmetry parameter (real) lambda > 0 gamma = sqrt{alpha^2 eta^2} > 0 support =x in ( infty; +infty)! pdf… … Wikipedia
Normal-inverse Gaussian distribution — Normal inverse Gaussian (NIG) parameters: μ location (real) α tail heavyness (real) β asymmetry parameter (real) δ scale parameter (real) support … Wikipedia
Ole Barndorff-Nielsen — Ole Eiler Barndorff Nielsen Ole Barndorff Nielsen at the Levy conference in Copenhagen in August 2007. Born … Wikipedia
List of mathematics articles (G) — NOTOC G G₂ G delta space G networks Gδ set G structure G test G127 G2 manifold G2 structure Gabor atom Gabor filter Gabor transform Gabor Wigner transform Gabow s algorithm Gabriel graph Gabriel s Horn Gain graph Gain group Galerkin method… … Wikipedia
List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… … Wikipedia
Normal variance-mean mixture — In probability theory and statistics, a normal variance mean mixture with mixing probability density g is the continuous probability distribution of a random variable Y of the form where α and β are real numbers and σ > 0 and random variables… … Wikipedia
Barndorff-Nielsen — Ole Eiler Barndorff Nielsen (* 18. März 1935 in Kopenhagen) ist ein dänischer Mathematiker, dessen Spezialgebiet im Bereich der Statistik liegt. Er ist der Namensgeber der Barndorff Nielsen Formel für Maximum Likelihood Schätzer und des nach ihm… … Deutsch Wikipedia
Ole Barndorff-Nielsen — Ole Eiler Barndorff Nielsen (* 18. März 1935 in Kopenhagen) ist ein dänischer Mathematiker, dessen Spezialgebiet im Bereich der Statistik liegt. Er ist der Namensgeber der Barndorff Nielsen Formel für Maximum Likelihood Schätzer und des nach ihm… … Deutsch Wikipedia

Academic Dictionaries and Encyclopedias

Generalised hyperbolic distribution

Look at other dictionaries:

Share the article and excerpts

Academic Dictionaries and Encyclopedias

Wikipedia

Generalised hyperbolic distribution

Look at other dictionaries:

Share the article and excerpts

Direct link