Kaniadakis Gamma distribution

Short description: Continuous probability distribution

κ-Gamma distribution

Probability density function
Parameters	[math]\displaystyle{ 0 \leq \kappa \lt 1 }[/math] [math]\displaystyle{ \alpha \gt 0 }[/math] shape (real) [math]\displaystyle{ \beta\gt 0 }[/math] rate (real) [math]\displaystyle{ 0 \lt \nu \lt 1/\kappa }[/math]
Support	[math]\displaystyle{ x \in [0, +\infty) }[/math]
PDF	[math]\displaystyle{ (1 + \kappa \nu) (2 \kappa)^\nu \frac{\Gamma \big(\frac{1}{2 \kappa} + \frac{\nu}{2} \big)}{\Gamma \big(\frac{1}{2 \kappa} - \frac{\nu}{2} \big)} \frac{\alpha \beta^\nu}{\Gamma \big(\nu\big)} x^{\alpha \nu - 1} \exp_\kappa(-\beta x^\alpha) }[/math]
CDF	[math]\displaystyle{ (1 + \kappa \nu) (2 \kappa)^\nu \frac{\Gamma \big(\frac{1}{2 \kappa} + \frac{\nu}{2} \big)}{\Gamma \big(\frac{1}{2 \kappa} - \frac{\nu}{2} \big)} \frac{\alpha \beta^\nu}{\Gamma \big(\nu\big)} \int_0^x z^{\alpha \nu - 1} \exp_\kappa(-\beta z^\alpha) dz }[/math]
Mode	[math]\displaystyle{ \beta^{ -1 / \alpha } \Bigg( \nu - \frac{ 1 }{ \alpha } \Bigg)^{ \frac{ 1 }{ \alpha } } \Bigg[ 1 - \kappa^2 \bigg( \nu - \frac{ 1 }{ \alpha } \bigg)^2 \Bigg]^{ - \frac{ 1 }{ 2 \alpha } } }[/math]

The Kaniadakis Generalized Gamma distribution (or κ-Generalized Gamma distribution) is a four-parameter family of continuous statistical distributions, supported on a semi-infinite interval [0,∞), which arising from the Kaniadakis statistics. It is one example of a Kaniadakis distribution. The κ-Gamma is a deformation of the Generalized Gamma distribution.

Definitions

Probability density function

The Kaniadakis κ-Gamma distribution has the following probability density function:

[math]\displaystyle{ f_{_{\kappa}}(x) = (1 + \kappa \nu) (2 \kappa)^\nu \frac{\Gamma \big(\frac{1}{2 \kappa} + \frac{\nu}{2} \big)}{\Gamma \big(\frac{1}{2 \kappa} - \frac{\nu}{2} \big)} \frac{\alpha \beta^\nu}{\Gamma \big(\nu\big)} x^{\alpha \nu - 1} \exp_\kappa(-\beta x^\alpha) }[/math]

valid for [math]\displaystyle{ x \geq 0 }[/math], where [math]\displaystyle{ 0 \leq |\kappa| \lt 1 }[/math] is the entropic index associated with the Kaniadakis entropy, [math]\displaystyle{ 0 \lt \nu \lt 1/\kappa }[/math], [math]\displaystyle{ \beta \gt 0 }[/math] is the scale parameter, and [math]\displaystyle{ \alpha \gt 0 }[/math] is the shape parameter.

The ordinary generalized Gamma distribution is recovered as [math]\displaystyle{ \kappa \rightarrow 0 }[/math]: [math]\displaystyle{ f_{_{0}}(x) = \frac{|\alpha| \beta ^\nu }{\Gamma \left( \nu \right)} x^{\alpha \nu - 1} \exp_\kappa(-\beta x^\alpha) }[/math].

Cumulative distribution function

The cumulative distribution function of κ-Gamma distribution assumes the form:

[math]\displaystyle{ F_\kappa(x) = (1 + \kappa \nu) (2 \kappa)^\nu \frac{\Gamma \big(\frac{1}{2 \kappa} + \frac{\nu}{2} \big)}{\Gamma \big(\frac{1}{2 \kappa} - \frac{\nu}{2} \big)} \frac{\alpha \beta^\nu}{\Gamma \big(\nu\big)} \int_0^x z^{\alpha \nu - 1} \exp_\kappa(-\beta z^\alpha) dz }[/math]

valid for [math]\displaystyle{ x \geq 0 }[/math], where [math]\displaystyle{ 0 \leq |\kappa| \lt 1 }[/math]. The cumulative Generalized Gamma distribution is recovered in the classical limit [math]\displaystyle{ \kappa \rightarrow 0 }[/math].

Properties

Moments and mode

The κ-Gamma distribution has moment of order [math]\displaystyle{ m }[/math] given by

[math]\displaystyle{ \operatorname{E}[X^m] = \beta^{-m/ \alpha} \frac{(1 + \kappa \nu) (2 \kappa)^{-m/\alpha}}{1 + \kappa \big( \nu + \frac{m}{\alpha}\big)} \frac{\Gamma \big( \nu + \frac{m}{ \alpha } \big) }{\Gamma(\nu)} \frac{\Gamma\Big(\frac{1}{2\kappa} + \frac{\nu}{2}\Big)}{\Gamma\Big(\frac{1}{2\kappa} - \frac{\nu}{2}\Big)} \frac{\Gamma\Big(\frac{1}{2\kappa} - \frac{\nu}{2} - \frac{m}{2\alpha}\Big)}{\Gamma\Big(\frac{1}{2\kappa} + \frac{\nu}{2} + \frac{m}{2\alpha}\Big)} }[/math]

The moment of order [math]\displaystyle{ m }[/math] of the κ-Gamma distribution is finite for [math]\displaystyle{ 0 \lt \nu + m/\alpha \lt 1/\kappa }[/math].

The mode is given by:

[math]\displaystyle{ x_{\textrm{mode}} = \beta^{-1/\alpha} \Bigg( \nu - \frac{1}{\alpha} \Bigg)^{\frac{1}{\alpha}} \Bigg[ 1 - \kappa^2 \bigg( \nu - \frac{1}{\alpha}\bigg)^2\Bigg]^{-\frac{1}{2\alpha}} }[/math]

Asymptotic behavior

The κ-Gamma distribution behaves asymptotically as follows:

[math]\displaystyle{ \lim_{x \to +\infty} f_\kappa (x) \sim (2\kappa \beta)^{-1/\kappa} (1 + \kappa \nu) (2 \kappa)^\nu \frac{\Gamma \big(\frac{1}{2 \kappa} + \frac{\nu}{2} \big)}{\Gamma \big(\frac{1}{2 \kappa} - \frac{\nu}{2} \big)} \frac{\alpha \beta^\nu}{\Gamma \big(\nu\big)} x^{\alpha \nu - 1 - \alpha /\kappa} }[/math]

[math]\displaystyle{ \lim_{x \to 0^+} f_\kappa (x) = (1 + \kappa \nu) (2 \kappa)^\nu \frac{\Gamma \big(\frac{1}{2 \kappa} + \frac{\nu}{2} \big)}{\Gamma \big(\frac{1}{2 \kappa} - \frac{\nu}{2} \big)} \frac{\alpha \beta^\nu}{\Gamma \big(\nu\big)} x^{\alpha \nu - 1} }[/math]

Related distributions

• The κ-Gamma distributions is a generalization of:
  • κ-Exponential distribution of type I, when [math]\displaystyle{ \alpha = \nu = 1 }[/math];
  • Kaniadakis κ-Erlang distribution, when [math]\displaystyle{ \alpha = 1 }[/math] and [math]\displaystyle{ \nu = n = }[/math] positive integer.
  • κ-Half-Normal distribution, when [math]\displaystyle{ \alpha = 2 }[/math] and [math]\displaystyle{ \nu = 1/2 }[/math];
• A κ-Gamma distribution corresponds to several probability distributions when [math]\displaystyle{ \kappa = 0 }[/math], such as:
  • Gamma distribution, when [math]\displaystyle{ \alpha = 1 }[/math];
  • Exponential distribution, when [math]\displaystyle{ \alpha = \nu = 1 }[/math];
  • Erlang distribution, when [math]\displaystyle{ \alpha = 1 }[/math] and [math]\displaystyle{ \nu = n = }[/math] positive integer;
  • Chi-Squared distribution, when [math]\displaystyle{ \alpha = 1 }[/math] and [math]\displaystyle{ \nu = }[/math] half integer;
  • Nakagami distribution, when [math]\displaystyle{ \alpha = 2 }[/math] and [math]\displaystyle{ \nu \gt 0 }[/math];
  • Rayleigh distribution, when [math]\displaystyle{ \alpha = 2 }[/math] and [math]\displaystyle{ \nu = 1 }[/math];
  • Chi distribution, when [math]\displaystyle{ \alpha = 2 }[/math] and [math]\displaystyle{ \nu = }[/math] half integer;
  • Maxwell distribution, when [math]\displaystyle{ \alpha = 2 }[/math] and [math]\displaystyle{ \nu = 3/2 }[/math];
  • Half-Normal distribution, when [math]\displaystyle{ \alpha = 2 }[/math] and [math]\displaystyle{ \nu = 1/2 }[/math];
  • Weibull distribution, when [math]\displaystyle{ \alpha \gt 0 }[/math] and [math]\displaystyle{ \nu = 1 }[/math];
  • Stretched Exponential distribution, when [math]\displaystyle{ \alpha \gt 0 }[/math] and [math]\displaystyle{ \nu = 1/\alpha }[/math];

See also

• Giorgio Kaniadakis
• Kaniadakis statistics
• Kaniadakis distribution
• Kaniadakis κ-Exponential distribution
• Kaniadakis κ-Gaussian distribution
• Kaniadakis κ-Weibull distribution
• Kaniadakis κ-Logistic distribution
• Kaniadakis κ-Erlang distribution

External links

• Kaniadakis Statistics on arXiv.org