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Normal-exponential-gamma distribution

{{Probability distribution |

name =Normal-Exponential-Gamma|

type =density|

pdf_image =|

cdf_image =|

parameters ={{nowrap|μ ∈ R}} — mean (location)
k > 0 shape
\theta > 0 scale |

support =x \in (-\infty, \infty)|

pdf =\propto \exp{\left(\frac{(x-\mu)^2}{4\theta^2}\right)}D_{-2k-1}\left(\frac

x-\mu
{\theta}\right)|

cdf =|

mean =\mu|

median =\mu|

mode = \mu|

variance = \frac{\theta^2}{k-1} for k>1 |

skewness =0|

kurtosis =|

entropy =|

mgf =|

char =|

}}

In probability theory and statistics, the normal-exponential-gamma distribution (sometimes called the NEG distribution) is a three-parameter family of continuous probability distributions. It has a location parameter \mu, scale parameter \theta and a shape parameter k .

Probability density function

The probability density function (pdf) of the normal-exponential-gamma distribution is proportional to

:f(x;\mu, k,\theta) \propto \exp{\left(\frac{(x-\mu)^2}{4\theta^2}\right)}D_{-2k-1}\left(\frac

x-\mu
{\theta}\right),

where D is a parabolic cylinder function.http://www.newton.ac.uk/programmes/SCB/seminars/121416154.html {{dead link|date=October 2014}}

As for the Laplace distribution, the pdf of the NEG distribution can be expressed as a mixture of normal distributions,

:f(x;\mu, k,\theta)=\int_0^\infty\int_0^\infty\ \mathrm{N}(x| \mu, \sigma^2)\mathrm{Exp}(\sigma^2|\psi)\mathrm{Gamma}(\psi|k, 1/\theta^2) \, d\sigma^2 \, d\psi,

where, in this notation, the distribution-names should be interpreted as meaning the density functions of those distributions.

Within this scale mixture, the scale's mixing distribution (an exponential with a gamma-distributed rate) actually is a Lomax distribution.

Applications

The distribution has heavy tails and a sharp peak at \mu and, because of this, it has applications in variable selection.

See also

{{refimprove|date=December 2010}}

References

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{{ProbDistributions|continuous-infinite}}

{{DEFAULTSORT:Normal-Exponential-Gamma Distribution}}

Category:Continuous distributions

Category:Compound probability distributions

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