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 μ {\displaystyle \mu } , scale parameter θ {\displaystyle \theta } and a shape parameter k {\displaystyle k} .
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 ) 2 4 θ 2 ) D − 2 k − 1 ( | x − μ | θ ) {\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)} , 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 , σ 2 ) E x p ( σ 2 | ψ ) G a m m a ( ψ | k , 1 / θ 2 ) d σ 2 d ψ , {\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 ,}
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 {\displaystyle \mu } and, because of this, it has applications in variable selection.
See also Compound probability distribution Lomax distribution
References