Английская Википедия:Generalized inverse Gaussian distribution

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Шаблон:Probability distribution{2 K_p(\sqrt{ab})} x^{(p-1)} e^{-(ax + b/x)/2}</math>|

  cdf        =|
  mean       =<math>\operatorname{E}[x]=\frac{\sqrt{b}\ K_{p+1}(\sqrt{a b}) }{ \sqrt{a}\ K_{p}(\sqrt{a b})}</math>
<math>\operatorname{E}[x^{-1}]=\frac{\sqrt{a}\ K_{p+1}(\sqrt{a b}) }{ \sqrt{b}\ K_{p}(\sqrt{a b})}-\frac{2p}{b}</math>
<math>\operatorname{E}[\ln x]=\ln \frac{\sqrt{b}}{\sqrt{a}}+\frac{\partial}{\partial p} \ln K_{p}(\sqrt{a b})</math>|
  median     =|
  mode       =<math>\frac{(p-1)+\sqrt{(p-1)^2+ab}}{a}</math>|
  variance   =<math>\left(\frac{b}{a}\right)\left[\frac{K_{p+2}(\sqrt{ab})}{K_p(\sqrt{ab})}-\left(\frac{K_{p+1}(\sqrt{ab})}{K_p(\sqrt{ab})}\right)^2\right]</math>|
  skewness   =|
  kurtosis   =|
  entropy    =|
  mgf        =<math>\left(\frac{a}{a-2t}\right)^{\frac{p}{2}}\frac{K_p(\sqrt{b(a-2t)})}{K_p(\sqrt{ab})}</math>|
  char       =<math>\left(\frac{a}{a-2it}\right)^{\frac{p}{2}}\frac{K_p(\sqrt{b(a-2it)})}{K_p(\sqrt{ab})}</math>|
}}

In probability theory and statistics, the generalized inverse Gaussian distribution (GIG) is a three-parameter family of continuous probability distributions with probability density function

<math>f(x) = \frac{(a/b)^{p/2}}{2 K_p(\sqrt{ab})} x^{(p-1)} e^{-(ax + b/x)/2},\qquad x>0,</math>

where K_p is a modified Bessel function of the second kind, a > 0, b > 0 and p a real parameter. It is used extensively in geostatistics, statistical linguistics, finance, etc. This distribution was first proposed by Étienne Halphen. It was rediscovered and popularised by Ole Barndorff-Nielsen, who called it the generalized inverse Gaussian distribution. Its statistical properties are discussed in Bent Jørgensen's lecture notes.

Properties

Alternative parametrization

By setting <math>\theta = \sqrt{ab}</math> and <math>\eta = \sqrt{b/a}</math>, we can alternatively express the GIG distribution as

<math>f(x) = \frac{1}{2\eta K_p(\theta)} \left(\frac{x}{\eta}\right)^{p-1} e^{-\theta(x/\eta + \eta/x)/2}, </math>

where <math>\theta</math> is the concentration parameter while <math>\eta</math> is the scaling parameter.

Summation

Barndorff-Nielsen and Halgreen proved that the GIG distribution is infinitely divisible.

Entropy

The entropy of the generalized inverse Gaussian distribution is given asШаблон:Citation needed

<math>

\begin{align} H = \frac{1}{2} \log \left( \frac b a \right) & {} +\log \left(2 K_p\left(\sqrt{ab} \right)\right) - (p-1) \frac{\left[\frac{d}{d\nu}K_\nu\left(\sqrt{ab}\right)\right]_{\nu=p}}{K_p\left(\sqrt{a b}\right)} \\ & {} + \frac{\sqrt{a b}}{2 K_p\left(\sqrt{a b}\right)}\left( K_{p+1}\left(\sqrt{ab}\right) + K_{p-1}\left(\sqrt{a b}\right)\right) \end{align} </math>

where <math>\left[\frac{d}{d\nu}K_\nu\left(\sqrt{a b}\right)\right]_{\nu=p}</math> is a derivative of the modified Bessel function of the second kind with respect to the order <math>\nu</math> evaluated at <math>\nu=p</math>

Characteristic Function

The characteristic of a random variable <math> X\sim GIG(p, a, b) </math> is given as(for a derivation of the characteristic function, see supplementary materials of )

<math> E(e^{itX}) = \left(\frac{a }{a-2it }\right)^{\frac{p}{2}} \frac{K_{p}\left( \sqrt{(a-2it)b} \right)}{ K_{p}\left( \sqrt{ab} \right) } </math>

for <math> t \in \mathbb{R}</math> where <math> i </math> denotes the imaginary number.

Related distributions

Special cases

The inverse Gaussian and gamma distributions are special cases of the generalized inverse Gaussian distribution for p = −1/2 and b = 0, respectively. Specifically, an inverse Gaussian distribution of the form

<math> f(x;\mu,\lambda) = \left[\frac{\lambda}{2 \pi x^3}\right]^{1/2} \exp{ \left( \frac{-\lambda (x-\mu)^2}{2 \mu^2 x} \right)}</math>

is a GIG with <math>a = \lambda/\mu^2</math>, <math>b = \lambda</math>, and <math>p=-1/2</math>. A Gamma distribution of the form

<math>

g(x;\alpha,\beta) = \beta^\alpha \frac 1 {\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x} </math> is a GIG with <math>a = 2 \beta</math>, <math>b = 0</math>, and <math>p = \alpha</math>.

Other special cases include the inverse-gamma distribution, for a = 0.

Conjugate prior for Gaussian

The GIG distribution is conjugate to the normal distribution when serving as the mixing distribution in a normal variance-mean mixture. Let the prior distribution for some hidden variable, say <math>z</math>, be GIG:

<math>

P(z\mid a,b,p) = \operatorname{GIG}(z\mid a,b,p) </math> and let there be <math>T</math> observed data points, <math>X=x_1,\ldots,x_T</math>, with normal likelihood function, conditioned on <math>z:</math>

<math>

P(X\mid z,\alpha,\beta) = \prod_{i=1}^T N(x_i\mid\alpha+\beta z,z) </math>

where <math>N(x\mid\mu,v)</math> is the normal distribution, with mean <math>\mu</math> and variance <math>v</math>. Then the posterior for <math>z</math>, given the data is also GIG:

<math>

P(z\mid X,a,b,p,\alpha,\beta) = \text{GIG}\left(z\mid a+T\beta^2,b+S,p-\frac T 2 \right) </math> where <math>\textstyle S = \sum_{i=1}^T (x_i-\alpha)^2</math>.

Sichel distribution

The Sichel distribution results when the GIG is used as the mixing distribution for the Poisson parameter <math>\lambda</math>.

See also

• Inverse Gaussian distribution
• Gamma distribution

Шаблон:ProbDistributions