Normal-exponential-gamma 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)

,

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.

The distribution has heavy tails and a sharp peak^[1] at $\mu$ and, because of this, it has applications in variable selection.

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