Normal-exponential-gamma distribution

Normal-Exponential-Gamma

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

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.^[1]

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.

Applications

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

References

1. http://www.newton.ac.uk/programmes/SCB/seminars/121416154.html