Gamma-order Generalized Normal distribution

{{Orphan|date=June 2025}}

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

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

with the normalized factor C equals to..

(2) $C\; =\; C\_p(\backslash mu,\; \backslash Sigma;\; \backslash gamma)\; =\; \backslash frac\{1\}\{\backslash pi^\{p/2\}\; |\backslash Sigma|^\{1/2\}\}\; \backslash cdot\; \backslash frac\{\backslash Gamma\backslash left(\backslash frac\{p\}\{2\}\; +\; 1\backslash right)\}\{\backslash Gamma\backslash left(\; \backslash frac\{p(\backslash gamma\; -\; 1)\}\{\backslash gamma\}\; +\; 1\; \backslash right)\}\; \backslash left(\; \backslash frac\{\backslash gamma\; -\; 1\}\{\backslash gamma\}\; \backslash right)^\{p(\backslash gamma\; -\; 1)/\backslash gamma\},\; \backslash quad\; \backslash gamma\_0\; =\; \backslash frac\{\backslash gamma\; -\; 1\}\{\backslash gamma\},\; \backslash quad\; \backslash gamma\_0\; \backslash gamma\_1\; =\; 1,$

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

Consider the $N\_\backslash gamma(\backslash mu,\; \backslash sigma^2).\backslash$

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

(4) $\backslash varphi\_\backslash gamma(x;\; \backslash mu,\; \backslash sigma^2)\; =\; \backslash frac\{\backslash lambda\_\backslash gamma\}\{\backslash sigma\; \backslash sqrt\{\backslash pi\}\}\; \backslash exp\; \backslash left\backslash \{\; -\backslash gamma\_0\; \backslash left(\; \backslash frac$

x - \mu

{\sigma} \right)^{\gamma_1} \right\},

File:GammaPDF1D_Standardized.jpg

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) $\backslash lambda\_\backslash gamma\; =\; \backslash frac\{\backslash Gamma\backslash left(\backslash frac\{1\}\{2\}\; +\; 1\backslash right)\}\{\backslash Gamma\backslash left(\backslash frac\{\backslash gamma\; -\; 1\}\{\backslash gamma\}\; +\; 1\backslash right)\}\; \backslash left(\; \backslash frac\{\backslash gamma\; -\; 1\}\{\backslash gamma\}\; \backslash right)^\{\backslash frac\{\backslash gamma\; -\; 1\}\{\backslash gamma\}\}\; =\; \backslash frac\{\backslash Gamma\backslash left(\backslash frac\{1\}\{2\}\; +\; 1\backslash right)\}\{\backslash Gamma(\backslash gamma\_0\; +\; 1)\}\; \backslash cdot\; \backslash gamma\_0^\{\backslash gamma\_0\}.$

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

File:GammaPDF2D_3DPlot.jpg

Let $Y\; \backslash sim\; N\_\backslash gamma(\backslash mu,\; \backslash sigma^2)$ then the central moments are evaluated as

(6) $\backslash beta\_n\; =\; \backslash mathbb\{E\}(Y^n)\; =\; \backslash sum\_\{\backslash text\{even\; \}\; r\; =\; 0\}^n\; \backslash binom\{n\}\{r\}\; \backslash mu^r\; \backslash sigma^\{n\; -\; r\}\; \backslash cdot\; \backslash gamma\_0^\{-n\}\; \backslash cdot\; \backslash gamma\_0^\{n\; -\; r\}\; \backslash cdot\; \backslash frac\{\backslash Gamma((n\; -\; r\; +\; 1)\backslash gamma\_0)\}\{\backslash Gamma(\backslash gamma\_0)^\{n\; -\; r\}\}.$

When $\backslash gamma\; =\; 2$, then

$\backslash beta\_1\; =\; \backslash mu,\backslash quad\; \backslash beta\_2\; =\; \backslash mu^2\; +\; \backslash sigma^2,\backslash quad\; \backslash beta\_3\; =\; \backslash mu^3\; +\; 3\backslash mu\backslash sigma^2,\backslash quad\; \backslash beta\_4\; =\; \backslash mu^4\; +\; 6\backslash mu^2\backslash sigma^2\; +\; 3\backslash sigma^4.$

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

(7) $\backslash operatorname\{Var\}(Y)\; =\; \backslash gamma\_0^\{-2\}\; \backslash cdot\; \backslash frac\{\backslash Gamma(3\backslash gamma\_0)\}\{\backslash Gamma(\backslash gamma\_0)\}\; \backslash cdot\; \backslash sigma^2$

and

(8) $\backslash operatorname\{Kurt\}(Y)\; =\; \backslash frac\{\backslash Gamma(\backslash gamma\_0)\backslash Gamma(5\backslash gamma\_0)\}\{\backslash Gamma^2(3\backslash gamma\_0)\}\; -\; 3.$

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

(9) $\backslash mathcal\{L\}\_\{\backslash varphi\_\backslash gamma\}(\backslash xi)\; =\; \backslash frac\{e^\{\backslash xi\; \backslash mu\}\}\{\backslash Gamma(\backslash gamma\_0)\}\; \backslash sum\_\{j=0\}^\{\backslash infty\}\; \backslash frac\{1\}\{(2j)!\}\; \backslash left(\; \backslash xi\; \backslash sigma\; (\backslash gamma\_1)^\{\backslash gamma\_0\}\; \backslash right)^\{2j\}\; \backslash Gamma\backslash left(\; (2j\; +\; 1)\backslash gamma\_0\; \backslash right).$

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

(10) $\backslash mathcal\{L\}\_\{\backslash varphi\_2\}(\backslash xi)\; =\; \backslash exp\backslash left\backslash \{\; \backslash xi\; \backslash mu\; +\; \backslash frac\{\backslash xi^2\; \backslash sigma^2\}\{2\}\; \backslash right\backslash \}.$

Consider $X\; \backslash sim\; N\_\backslash gamma(\backslash mu,\; \backslash sigma^2\; I)$. The truncated γ-order Normal to the right at $x\; =\; \backslash tau$, 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 \le \tau

\end{cases}

and the truncated γ-order Normal to the left at $x\; =\; \backslash tau\_0$ 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 \ge \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, $\backslash ln\; X\; \backslash sim\; N\_\backslash gamma(\backslash mu,\; \backslash sigma^2)$.

Then $X$ is said to follow the γ-order Lognormal distribution, denoted by $LN\_\backslash gamma(\backslash mu,\; \backslash sigma)$, 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].