Normal-Exponential-GammaParameters |
μ ∈ R — mean (location)
shape
scale |
---|
Support |
![{\displaystyle x\in (-\infty ,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7aea9be5e96822459afc5c7d9f911a586290dc5) |
---|
PDF |
![{\displaystyle \propto \exp {\left({\frac {(x-\mu )^{2}}{4\theta ^{2}}}\right)}D_{-2k-1}\left({\frac {|x-\mu |}{\theta }}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d740e268842ba5fe4b82f38c0e0fb97a23b25ddd) |
---|
Mean |
![{\displaystyle \mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161) |
---|
Median |
![{\displaystyle \mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161) |
---|
Mode |
![{\displaystyle \mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161) |
---|
Variance |
for ![{\displaystyle k>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cda43bd4034dc2d04cd562005d0af81d3d2dbc6) |
---|
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
, scale parameter
and a shape parameter
.
Probability density function
The probability density function (pdf) of the normal-exponential-gamma distribution is proportional to
,
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,
![{\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 ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0cbdab2447cadba8216969e658805b18e63511ac)
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
and, because of this, it has applications in variable selection.
See also
References
|
---|
Discrete univariate | with finite support | |
---|
with infinite support | |
---|
|
---|
Continuous univariate | supported on a bounded interval | |
---|
supported on a semi-infinite interval | |
---|
supported on the whole real line | |
---|
with support whose type varies | |
---|
|
---|
Mixed univariate | |
---|
Multivariate (joint) | |
---|
Directional | |
---|
Degenerate and singular | |
---|
Families | |
---|
|