Normal exponential gamma distribution - Alchetron, the free social encyclopedia


Parameters μ ∈ R — mean (location) k > 0 {\displaystyle k>0\,} shape θ > 0 {\displaystyle \theta >0\,} scale Support x ∈ ( − ∞ , ∞ ) {\displaystyle x\in (-\infty ,\infty )\!} PDF ∝ exp ⁡ ( ( x − μ ) 2 4 θ 2 ) D − 2 k − 1 ( \| x − μ \| θ ) {\displaystyle \propto \exp {\left({\frac {(x-\mu )^{2}}{4\theta ^{2}}}\right)}D_{-2k-1}\left({\frac {\|x-\mu \|}{\theta }}\right)\,\!} Mean μ {\displaystyle \mu } Median μ {\displaystyle \mu } Mode μ {\displaystyle \mu }

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 , scale parameter θ and a shape parameter 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 ) ,

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 ,

where, in this notation, the distribution-names should be interpreted as meaning the density functions of those distributions.

The distribution has heavy tails and a sharp peak at μ and, because of this, it has applications in variable selection.

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