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 location, $\mu \in \mathbb {R} ^{p}$, 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 the normalized factor C equals to..

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

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

With 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 1,

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

For $p=2$ a typical plot is Figure 2

Figure 2: The pdf of the standardized ϕγ(x) for γ = 2 (Normal), γ = 3 (fat-tailed) with p = 2.

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

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

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

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

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

and

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

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

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

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

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

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

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

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

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

with $C$ as in (2) with $p=1$.

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

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

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

References

[1] Theodore W. Anderson. An Introduction to Multivariate Statistical Analysis. Wiley-Interscience, 3rd edition, July 2003.

[2] Leonard Gross. Logarithmic Sobolev inequalities. Amer. J. Math., 97(4):1061–1083, 1975.

[3] Christos P. Kitsos and Nikolaos K. Tavoularis. Logarithmic Sobolev Inequalities for Information measures. IEEE Trans. Inform. Theory, 55(6):2554–2561, June 2009.

[4] Christos P. Kitsos and Thomas L. Toulias. On the family of the γ-ordered normal distributions. Far East Journal of Theoretical Statistics, 35(2):95–114, January 2011.

[5] Christos P. Kitsos and Ioannis S. Stamatiou. Laplace transformation for the γ-order generalized Normal $N_{\gamma }(\mu ,\sigma ^{2})$. Far East Journal of Theoretical Statistics, 68(1):1–21, 2024.

[6] Christos P. Kitsos, Vassilios G. Vassiliadis, and Thomas L. Toulias. MLE for the γ-order generalized normal distribution. Discussiones Mathematicae - Probability and Statistics, 34(1–2):143–158, 2014.

[7] Thomas L. Toulias and Christos P. Kitsos. On the generalized lognormal distribution. Journal of Probability and Statistics, 2013(1/432642):15 pages, July 2013.

[8] Samuel Karlin and Howard M. Taylor. A First Course in Stochastic Processes. Academic Press, New York, 2nd edition, 1975.

[9] Christos P. Kitsos. Generalizing the Heat Equation. Revstat - Statistical Journal, 23(2):207–221, April 2025.