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| {{Unreferenced|date=December 2009}}
| | She is known by the title of Myrtle Shryock. The preferred pastime for my kids and me is to perform baseball but I haven't produced a dime with it. California is our birth location. He used to be unemployed but now he is a pc operator but his marketing by no means arrives.<br><br>My web-site ... [http://www.fuguporn.com/blog/36031 www.fuguporn.com] |
| {{Probability distribution |
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| name =generalised hyperbolic|
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| type =density|
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| pdf_image =|
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| cdf_image =|
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| parameters = <math>\lambda</math> <!--to do--> (real)<br /> <math>\alpha</math> <!--to do--> (real)<br /> <math>\beta</math> asymmetry parameter (real)<br /> <math>\delta</math> [[scale parameter]] (real)<br /> <math>\mu</math> [[location parameter|location]] ([[real number|real]])<br /> <math>\gamma = \sqrt{\alpha^2 - \beta^2}</math>|
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| support =<math>x \in (-\infty; +\infty)\!</math>|
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| pdf =<math>\frac{(\gamma/\delta)^\lambda}{\sqrt{2\pi}K_\lambda(\delta \gamma)} \; e^{\beta (x - \mu)} \!</math><br /> <math>\times \frac{K_{\lambda - 1/2}\left(\alpha \sqrt{\delta^2 + (x - \mu)^2}\right)}{\left(\sqrt{\delta^2 + (x - \mu)^2} / \alpha\right)^{1/2 - \lambda}} \!</math>|
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| cdf =<!-- to do -->|
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| mean =<math>\mu + \frac{\delta \beta K_{\lambda+1}(\delta \gamma)}{\gamma K_\lambda(\delta\gamma)}</math>|
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| median =<!-- to do -->|
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| mode =<!-- to do -->|
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| variance =<math>\frac{\delta K_{\lambda+1}(\delta \gamma)}{\gamma K_\lambda(\delta\gamma)} + \frac{\beta^2\delta^2}{\gamma^2}\left( \frac{K_{\lambda+2}(\delta\gamma)}{K_{\lambda}(\delta\gamma)} -
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| \frac{K_{\lambda+1}^2(\delta\gamma)}{K_{\lambda}^2(\delta\gamma)} \right)</math>|
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| skewness =<!-- to do -->|
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| kurtosis =<!-- to do -->|
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| entropy =<!-- to do -->|
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| mgf =<math>\frac{e^{\mu z}\gamma^\lambda}{(\sqrt{\alpha^2 -(\beta +z)^2})^\lambda} \frac{K_\lambda(\delta \sqrt{\alpha^2 -(\beta +z)^2})}{K_\lambda (\delta \gamma)}</math>|
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| char =<!-- to do -->|
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| }}
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| 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 [[Bessel function#Modified Bessel functions|modified Bessel function of the second kind]], denoted by <math>K_\lambda</math>.
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| As the name suggests it is of a very general form, being the superclass of, among others, the [[Student's t-distribution|Student's ''t''-distribution]], the [[Laplace distribution]], the [[hyperbolic distribution]], the [[normal-inverse Gaussian distribution]] and the [[variance-gamma distribution]].
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| It is mainly applied to areas that require sufficient probability of far-field behaviour, which it can model due to its semi-heavy tails—a property the [[normal distribution]] does not possess. The '''generalised hyperbolic distribution''' is often used in economics, with particular application in the fields of [[statistical analysis of financial markets|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]].
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| == Related distributions ==
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| * <math>X \sim \mathrm{GH}(-\frac{\nu}{2}, 0, 0, \sqrt{\nu}, \mu)\,</math> has a [[Student's t-distribution|Student's ''t''-distribution]] with <math>\nu</math> degrees of freedom.
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| * <math>X \sim \mathrm{GH}(1, \alpha, \beta, \delta, \mu)\,</math> has a [[hyperbolic distribution]].
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| <!-- what do alpha, beta, delta, mu mean in terms of the derived distribution? -->
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| * <math>X \sim \mathrm{GH}(-1/2, \alpha, \beta, \delta, \mu)\,</math> has a [[normal-inverse Gaussian distribution]] (NIG).<!-- what do alpha, beta, delta, mu mean in terms of the derived distribution? -->
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| * <math>X \sim \mathrm{GH}(?, ?, ?, ?, ?)\,</math> [[normal-inverse chi-squared distribution]]
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| * <math>X \sim \mathrm{GH}(?, ?, ?, ?, ?)\,</math> [[normal-inverse gamma distribution]] (NI)
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| * <math>X \sim \mathrm{GH}(\lambda, \alpha, \beta, 0, \mu)\,</math> has a [[variance-gamma distribution]].
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| {{ProbDistributions|continuous-infinite}}
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| {{DEFAULTSORT:Generalised Hyperbolic Distribution}}
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| [[Category:Generalized hyperbolic distributions|*]]
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| [[Category:Continuous distributions]]
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| [[Category:Probability distributions]]
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She is known by the title of Myrtle Shryock. The preferred pastime for my kids and me is to perform baseball but I haven't produced a dime with it. California is our birth location. He used to be unemployed but now he is a pc operator but his marketing by no means arrives.
My web-site ... www.fuguporn.com