Normal-exponential-gamma distribution

Template:Probability distribution 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.

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

$f(x;\mu ,k,\theta )=\underset{0}{\overset{\mathrm{\infty }}{\int }}\underset{0}{\overset{\mathrm{\infty }}{\int }}\mathrm{N}(x|\mu ,{\sigma }^2)\mathrm{E}\mathrm{x}\mathrm{p}({\sigma }^2|\psi )\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(\psi |k,1/{\theta }^2)d{\sigma }^{2}d\psi ,$

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

• Compound probability distribution
• Lomax distribution

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