Normal-exponential-gamma distribution
| Parameters |
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] | ||
| [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] .
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
References
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