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.

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 $\mu$ and, because of this, it has applications in variable selection.

References

^ ^a ^b http://www.newton.ac.uk/programmes/SCB/seminars/121416154.html ^{[dead link]}

Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric negative Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic mixed Poisson negative binomial Panjer parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous
univariate

supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta PERT raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi chi-squared noncentral inverse scaled Dagum Davis Erlang hyper exponential hyperexponential hypoexponential logarithmic F noncentral folded normal Fréchet gamma generalized inverse gamma/Gompertz Gompertz shifted half-logistic half-normal Hotelling's T-squared inverse Gaussian generalized Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal log-t Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto phase-type Poly-Weibull Rayleigh relativistic Breit–Wigner Rice truncated normal type-2 Gumbel Weibull discrete Wilks's lambda
supported on the whole real line	Cauchy exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t Tracy–Widom variance-gamma Voigt
with support whose type varies	generalized chi-squared generalized extreme value generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda