Gamma-order Generalized Normal distribution

Gamma-Ordered Generalized Normal Distribution

The multivariate Normal distribution, [1], is extended due to the Logarithmic Sobolev Inequalities (LSI), [2], and can act as a family of distributions based on a “shape” parameter. This shape parameter $\gamma \in \mathbb {R} -[0,1]$ creates along with parameters of position, $\mu \in \mathbb {R} ^{p}$, and dispersion, $\Sigma \in \mathbb {R} ^{p\times p}$, the $N_{\gamma }(\mu ,\Sigma )$ family of distributions with probability density function, [3]

(1) $\varphi _{\gamma }(x)=C\exp \left\{-{\frac {\gamma -1}{\gamma }}\left[Q(x)\right]^{\frac {\gamma }{2(\gamma -1)}}\right\},$

with

(2) $C=C_{p}(\mu ,\Sigma ;\gamma )={\frac {1}{\pi ^{p/2}|\Sigma |^{1/2}}}\cdot {\frac {\Gamma \left({\frac {p}{2}}+1\right)}{\Gamma \left({\frac {p(\gamma -1)}{\gamma }}+1\right)}}\left({\frac {\gamma -1}{\gamma }}\right)^{p(\gamma -1)/\gamma },\quad \gamma _{0}={\frac {\gamma -1}{\gamma }},\quad \gamma _{0}\gamma _{1}=1,$

(3) $Q(x)=\langle x-\mu ,\Sigma ^{-1}(x-\mu )\rangle ,\quad \mu \in \mathbb {R} ^{p},\quad \Sigma \in \mathbb {R} ^{p\times p},$

For $p=2$ a typical plot is Figure 1 Consider the $N_{\gamma }(\mu ,\sigma ^{2}),\ p=1$, see [6], with position (mean) $\mu$, positive scale parameter $\sigma$, extra shape parameter $\gamma \in \mathbb {R} -[0,1]$ and pdf $\varphi _{\gamma }(x;\mu ,\sigma ^{2})$ coming from (1)–(3) and given by, see Figure 2,

(4) $\varphi _{\gamma }(x;\mu ,\sigma ^{2})={\frac {\lambda _{\gamma }}{\sigma {\sqrt {\pi }}}}\exp \left\{-\gamma _{0}\left({\frac {|x-\mu |}{\sigma }}\right)^{\gamma _{1}}\right\},$

Figure 1: 3D plots of φ_γ(x; 0, I) for γ = 2 (left) and γ = 3 (right), with p = 2