Mathematical Definitions

Introduction

This page provides the mathematical definitions of the expanded gamma distribution, detailing its probability density function, cumulative distribution function, and key statistical measures. These definitions serve as a foundation for understanding the distribution’s properties and for applying it to real-world data analysis scenarios.

Parameters

The expanded gamma distribution is characterized by three parameters:

  • \(\alpha\) (alpha): The shape parameter, which must be positive (:math:alpha > 0).

  • \(\beta\) (beta): The scale parameter, which must be non-zero (:math:beta neq 0). It determines the direction of skewness; positive values lead to right-skewness, while negative values result in left-skewness.

  • \(\delta\) (delta): The location parameter, which can be any real number and shifts the distribution along the x-axis.

The parameters \(\alpha\) and \(\beta\) primarily control the shape and scale of the distribution, respectively. The parameter \(\delta\) shifts the distribution along the x-axis.

Note that while the distribution is strictly either left- or right-skewed, as the parameter \(\alpha\) approaches infinity, the shape of the distribution converges to that of a normal distribution. Thus, for practical purposes, an expanded gamma distribution defined with a high \(\alpha\) value, such as \(1e9\), can approximate symmetrical scenarios quite well.

Distribution Functions

The probability density function (PDF) and cumulative distribution function (CDF) are defined as follows:

Probability Density Function (PDF):

\[\begin{split}f(x; \alpha, \beta, \delta) = \left\{ \begin{array}{ll} \dfrac{|x - \delta|^{\alpha - 1} e^{-(x - \delta) / \beta}}{|\beta|^\alpha \Gamma(\alpha)}, & \text{for } \dfrac{x - \delta}{\beta} > 0 \\ 0, & \text{otherwise} \end{array} \right.\end{split}\]

where \(\Gamma(\alpha)\) is the gamma function \(\displaystyle\int_{0}^{\infty}t^{\alpha-1}e^{-t}\,dt\)

Cumulative Density Function (CDF):

\[\begin{split}F(x; \alpha, \beta, \delta) = \left\{ \begin{array}{ll} \dfrac{\gamma(\alpha, \dfrac{x - \delta}{\beta})}{\Gamma(\alpha)}, & \text{for } \beta > 0, \dfrac{x - \delta}{\beta} > 0 \\ 1 - \dfrac{\gamma(\alpha, \dfrac{x - \delta}{\beta})}{\Gamma(\alpha)}, & \text{for } \beta < 0, \dfrac{x - \delta}{\beta} > 0 \end{array} \right.\end{split}\]

where \(\gamma \left(\alpha, \frac{x-\delta}{\beta}\right)\) is the lower incomplete gamma function \(\displaystyle\int_{0}^{\frac{x-\delta}{\beta}}t^{\alpha-1}e^{-t}\,dt\\.\)

Please note that the cumulative distribution function (CDF) of the gamma distribution does not have a closed-form solution. Consequently, the computation of the inverse CDF, or percent-point function, cannot be performed directly and requires the use of numerical techniques or approximation methods for its calculation.

Statistical measures

Expected value:

\[\begin{align} E[X] &= \alpha \beta + \delta \end{align}\]

Variance:

\[\begin{align} \text{Var}(X) &= \alpha \beta^2 \end{align}\]

Skewness:

\[\begin{align} \text{Skew}(X) &= \dfrac{\beta}{|\beta|}\times\dfrac{2}{\sqrt{\alpha}} \end{align}\]

Kurtosis:

\[\begin{align} \text{Kurt}(X) &= \dfrac{6}{\alpha} \end{align}\]

Mode:

\[\begin{align} \text{Mode}(X) &= (\alpha - 1) \beta + \delta \end{align}\]