Normal-exponential-gamma distribution

Short description: Theory in statistics

Normal-Exponential-GammaParameters|  μ ∈ R — mean (location)
k>0 shape
θ>0 scale

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Support|  x∈(−∞,∞)
PDF|  ∝exp⁡((x−μ)24θ2)D−2k−1(|x−μ|θ)
Mean|  μ
Median|  μ
Mode|  μ
Variance|  θ2k−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.[1] It has a location parameter μ, scale parameter θ and a shape parameter k .

## 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

f(x;μ,k,θ)∝exp⁡((x−μ)24θ2)D−2k−1(|x−μ|θ),

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;μ,k,θ)=∫0∞∫0∞ N(x|μ,σ2)Exp(σ2|ψ)Gamma(ψ|k,1/θ2)dσ2dψ,

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 μ and, because of this, it has applications in variable selection.

## See also

* Compound probability distribution
* Lomax distribution

## References

1. ↑ Griffin, Jim E.; Brown, Philip J. (2005). ﻿Alternative prior distributions for variable selection with very many more variables than observations﻿ (Report). Working Papers. 2005. Coventry: University of Warwick, Centre for Research in Statistical Methodology. https://wrap.warwick.ac.uk/id/eprint/35585/.

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