Normal-exponential-gamma distribution

Normal-Exponential-GammaParameters|  μ ∈ R — mean (location)
[math]\displaystyle{ k \gt 0 }[/math] shape
[math]\displaystyle{ \theta \gt 0 }[/math] scale

|

Support|  [math]\displaystyle{ x \in (-\infty, \infty) }[/math]
PDF|  [math]\displaystyle{ \propto \exp{\left(\frac{(x-\mu)^2}{4\theta^2}\right)}D_{-2k-1}\left(\frac{|x-\mu|}{\theta}\right) }[/math]
Mean|  [math]\displaystyle{ \mu }[/math]
Median|  [math]\displaystyle{ \mu }[/math]
Mode|  [math]\displaystyle{ \mu }[/math]
Variance|  [math]\displaystyle{ \frac{\theta^2}{k-1} }[/math] for [math]\displaystyle{ k\gt 1 }[/math]
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 [math]\displaystyle{ \mu }[/math], scale parameter [math]\displaystyle{ \theta }[/math] and a shape parameter [math]\displaystyle{ k }[/math] .

## Contents

* 1 Probability density function
* 2 Applications
* 3 See also
* 4 References

## Probability density function

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

[math]\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) }[/math],

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,

[math]\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, }[/math]

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[1] at [math]\displaystyle{ \mu }[/math] and, because of this, it has applications in variable selection.

## See also

* Compound probability distribution
* Lomax distribution

## References

1. ↑ 1.0 1.1 http://www.newton.ac.uk/programmes/SCB/seminars/121416154.html[|permanent dead link|dead link}}]

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