Normal-exponential-gamma distribution

Normal-Exponential-Gamma
Parameters	μ ∈ R — mean (location) ${\displaystyle k>0}$ shape ${\displaystyle \theta >0}$ scale
Support	${\displaystyle x\in (-\infty ,\infty )}$
PDF	${\displaystyle \propto \exp {\left({\frac {(x-\mu )^{2}}{4\theta ^{2}}}\right)}D_{-2k-1}\left({\frac {|x-\mu |}{\theta }}\right)}$
Mean	${\displaystyle \mu }$
Median	${\displaystyle \mu }$
Mode	${\displaystyle \mu }$
Variance	${\displaystyle {\frac {\theta ^{2}}{k-1}}}$ for ${\displaystyle k>1}$
Skewness	0

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 ${\displaystyle \mu }$, scale parameter ${\displaystyle \theta }$ and a shape parameter ${\displaystyle k}$ .

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

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

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,

${\displaystyle f(x;\mu ,k,\theta )=\int _{0}^{\infty }\int _{0}^{\infty }\ \mathrm {N} (x|\mu ,\sigma ^{2})\mathrm {Exp} (\sigma ^{2}|\psi )\mathrm {Gamma} (\psi |k,1/\theta ^{2})\,d\sigma ^{2}\,d\psi ,}$

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.

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