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


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

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,


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

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.