Gamma-order Generalized Normal distribution

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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 ${\displaystyle \gamma \in \mathbb {R} -[0,1]}$ creates along with parameters of location, ${\displaystyle \mu \in \mathbb {R} ^{p}}$, and dispersion, ${\displaystyle \Sigma \in \mathbb {R} ^{p\times p}}$, the ${\displaystyle N_{\gamma }(\mu ,\Sigma )}$ family of distributions with probability density function, [3]

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

with the normalized factor C equals to..

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

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

Consider the ${\displaystyle N_{\gamma }(\mu ,\sigma ^{2}).\ }$

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

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

Figure 1: The pdf of the standardized φ₍γ₎(x) for γ = 2 (Normal), γ = −0.1 (near to Dirac), γ = 1.05 (near to Uniform) and γ = 30 (near to Laplace), with p = 1.

with

(5) ${\displaystyle \lambda _{\gamma }={\frac {\Gamma \left({\frac {1}{2}}+1\right)}{\Gamma \left({\frac {\gamma -1}{\gamma }}+1\right)}}\left({\frac {\gamma -1}{\gamma }}\right)^{\frac {\gamma -1}{\gamma }}={\frac {\Gamma \left({\frac {1}{2}}+1\right)}{\Gamma (\gamma _{0}+1)}}\cdot \gamma _{0}^{\gamma _{0}}.}$

For ${\displaystyle p=2}$ a typical plot is Figure 2

Let ${\displaystyle Y\sim N_{\gamma }(\mu ,\sigma ^{2})}$ then the central moments are evaluated as

(6) ${\displaystyle \beta _{n}=\mathbb {E} (Y^{n})=\sum _{{\text{even }}r=0}^{n}{\binom {n}{r}}\mu ^{r}\sigma ^{n-r}\cdot \gamma _{0}^{-n}\cdot \gamma _{0}^{n-r}\cdot {\frac {\Gamma ((n-r+1)\gamma _{0})}{\Gamma (\gamma _{0})^{n-r}}}.}$

When ${\displaystyle \gamma =2}$, then ${\displaystyle \beta _{1}=\mu ,\quad \beta _{2}=\mu ^{2}+\sigma ^{2},\quad \beta _{3}=\mu ^{3}+3\mu \sigma ^{2},\quad \beta _{4}=\mu ^{4}+6\mu ^{2}\sigma ^{2}+3\sigma ^{4}.}$

Moreover, the variance and the kurtosis have been evaluated, [4] as

(7) ${\displaystyle \operatorname {Var} (Y)=\gamma _{0}^{-2}\cdot {\frac {\Gamma (3\gamma _{0})}{\Gamma (\gamma _{0})}}\cdot \sigma ^{2}}$

and

(8) ${\displaystyle \operatorname {Kurt} (Y)={\frac {\Gamma (\gamma _{0})\Gamma (5\gamma _{0})}{\Gamma ^{2}(3\gamma _{0})}}-3.}$

The Laplace transform of ${\displaystyle \varphi _{\gamma }(.)}$ can be obtained, [5],

(9) ${\displaystyle {\mathcal {L}}_{\varphi _{\gamma }}(\xi )={\frac {e^{\xi \mu }}{\Gamma (\gamma _{0})}}\sum _{j=0}^{\infty }{\frac {1}{(2j)!}}\left(\xi \sigma (\gamma _{1})^{\gamma _{0}}\right)^{2j}\Gamma \left((2j+1)\gamma _{0}\right).}$

When ${\displaystyle \gamma =2}$, (9) is reduced to the well-known form of the Laplace transform of the Normal distribution ${\displaystyle N(\mu ,\sigma ^{2})}$, that is

(10) ${\displaystyle {\mathcal {L}}_{\varphi _{2}}(\xi )=\exp \left\{\xi \mu +{\frac {\xi ^{2}\sigma ^{2}}{2}}\right\}.}$

Consider ${\displaystyle X\sim N_{\gamma }(\mu ,\sigma ^{2}I)}$. The truncated γ-order Normal to the right at ${\displaystyle x=\tau }$, see [6], is defined as

${\displaystyle f_{\gamma ,\tau ^{+}}(x)={\begin{cases}0&{\text{if }}x>\tau \\{\dfrac {C}{\Phi _{\gamma }\left({\frac {\tau -\mu }{\sigma }}\right)}}\exp \left\{-{\frac {\gamma -1}{\gamma }}\left|{\frac {x-\mu }{\sigma }}\right|^{\frac {\gamma }{\gamma -1}}\right\}&{\text{if }}x\leq \tau \end{cases}}}$

and the truncated γ-order Normal to the left at ${\displaystyle x=\tau _{0}}$ is

${\displaystyle f_{\gamma ,\tau _{0}^{-}}(x)={\begin{cases}0&{\text{if }}x<\tau _{0}\\{\dfrac {C}{1-\Phi _{\gamma }\left({\frac {\tau _{0}-\mu }{\sigma }}\right)}}\exp \left\{-{\frac {\gamma -1}{\gamma }}\left|{\frac {x-\mu }{\sigma }}\right|^{\frac {\gamma }{\gamma -1}}\right\}&{\text{if }}x\geq \tau _{0}\end{cases}}}$

with ${\displaystyle C}$ as in (2) with ${\displaystyle p=1}$.

Consider the logarithm of a rv ${\displaystyle X}$ that follows the γ-order Normal, that is, ${\displaystyle \ln X\sim N_{\gamma }(\mu ,\sigma ^{2})}$. Then ${\displaystyle X}$ is said to follow the γ-order Lognormal distribution, denoted by ${\displaystyle LN_{\gamma }(\mu ,\sigma )}$, with pdf, [7]

(11) ${\displaystyle f_{LN_{\gamma }}(x;\mu ,\sigma )={\frac {1}{2x\sigma \Gamma \left({\frac {\gamma -1}{\gamma }}\right)\left({\frac {\gamma -1}{\gamma }}\right)^{\frac {1}{\gamma }}}}\exp \left\{-{\frac {\gamma -1}{\gamma }}\left({\frac {|\ln x-\mu |}{\sigma }}\right)^{\frac {\gamma }{\gamma -1}}\right\}.}$

For an application of the γ-order generalized Normal distribution to the generalization of the Heat Equation, [8], see [9].