Draft:Generalized Exponential Distribution

• Comment: Not quite enough independent, significant coverage. WikiOriginal-9 (talk) 00:23, 7 November 2023 (UTC)

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Generalized Exponential Distribution

Probability density function
Parameters	$\alpha \in \mathbb {R} ^{+}$ (shape ) $\lambda \in \mathbb {R} ^{+}$ (rate) (or inverse scale)
Support	$x\in \mathbb {R} ^{+}$
PDF	$\alpha \lambda \left(1-e^{-\lambda x}\right)^{\alpha -1}e^{-\lambda x}$
CDF	$\left(1-e^{-\lambda x}\right)^{\alpha }$
Quantile	$-{\frac {{\text{ln}}(1-p^{1/\alpha })}{\lambda }}$
Mean	$\lambda ^{-1}\left[\psi (\alpha +1)-\psi (1)\right]$
Variance	$\lambda ^{-2}\left[\psi ^{(1)}(1)-\psi ^{(1)}(\alpha +1)\right]$
Skewness	${\frac {\psi ^{(2)}(\alpha +1)-\psi ^{(2)}(1)}{[\psi ^{(1)}(1)-\psi ^{(1)}(\alpha +1)]^{3/2}}}$
MGF	${\frac {\Gamma (\alpha +1)\Gamma (1-{\frac {t}{\lambda }})}{\Gamma (1+\alpha -{\frac {t}{\lambda }})}};0\leq t<\lambda$

In the theory of probability and statistics, the Generalized Exponential (GE) distribution is a two-parameter family of continuous probability distribution. This distribution was introduced by Gupta and Kundu in a 1999 paper and arises as a special case of the three-parameter exponentiated Weibull distribution that was originally introduced by Mudholkar and Srivastava (1994).

Definition

A continuous random variable $X$ is said to follow the Generalized Exponential (GE) distribution with parameters $\alpha$ and $\lambda$ if its CDF is given as

$$F(x;\alpha ,\lambda )=\left(1-e^{-\lambda x}\right)^{\alpha };~x,\alpha ,\lambda >0,$$

where $\alpha$ is the shape parameter and $\lambda$ is the rate parameter. The corresponding PDF is given as

$$f(x;\alpha ,\lambda )=\alpha \lambda \left(1-e^{-\lambda x}\right)^{\alpha -1}e^{-\lambda x};~x,\alpha ,\lambda >0.$$

This distribution is viewed as an expansion of the familiar exponential distribution, featuring an additional shape parameter that enhances its adaptability to model complex datasets more effectively. Both models coincide when the shape parameter $\alpha$ equals one.

Alternative Parametrization

The GE distribution offers an alternative parameterization utilizing the scale parameter $(\theta )$, which is defined as the inverse of the rate parameter $(\theta =1/\lambda )$.

The CDF and PDF of the GE distribution, expressed with respect to the scale parameter, are given as

$$F(x;\alpha ,\theta )=\left(1-e^{-x/\theta }\right)^{\alpha }~{\textrm {and}}~f(x;\alpha ,\theta )={\frac {\alpha }{\theta }}\left(1-e^{-x/\theta }\right)^{\alpha -1}e^{-x/\theta };~x,\alpha ,\theta >0.$$

The properties of the GE distribution with the scale parameter can be derived by substituting $1/\theta$ for $\lambda$ in the corresponding results associated with the GE distribution parameterized by the rate parameter.

Property

Hazard Function

The hazard function of the GE distribution is given by

$h(x;\alpha ,\lambda )={\frac {\alpha \lambda \left(1-e^{-\lambda x}\right)^{\alpha -1}e^{-\lambda x}}{1-\left(1-e^{-\lambda x}\right)^{\alpha }}};~~~x>0.$

The GE distribution has varying hazard rate depending on the value of the shape parameter. The hazard function is decreasing for $\alpha >1$, constant for $\alpha =1$, and increasing for $\alpha <1$.

Moment Generating Function and Moments

The moment generating function (MGF) of the GE distribution is given by

$M_{X}(t)={\frac {\Gamma (\alpha +1)\Gamma (1-{\frac {t}{\lambda }})}{\Gamma (1+\alpha -{\frac {t}{\lambda }})}};~~0\leq t<\lambda ,$

and differentiating the log of the MGF with respect to t repeatedly and then setting t = 0, we can get the expectation, variance, and skewness of GE distribution as

${\textrm {E}}(X)=\lambda ^{-1}\left[\psi (\alpha +1)-\psi (1)\right],$

${\textrm {V}}(X)=\lambda ^{-2}\left[\psi ^{(1)}(1)-\psi ^{(1)}(\alpha +1)\right],$

${\text{Skewness}}(X)=\left[\psi ^{(2)}(\alpha +1)-\psi ^{(2)}(1)\right]{\bigg /}\left[\psi ^{(1)}(1)-\psi ^{(1)}(\alpha +1)\right]^{\frac {3}{2}},$

where $\psi ^{(m)}(z)={\dfrac {\partial ^{m}}{\partial z^{m}}}\psi (z)={\dfrac {\partial ^{m+1}}{\partial z^{m+1}}}\ln \Gamma (z)$ is the polygamma function of order $m$; for $m=0$, it denotes the digamma function.

Information Matrix

If $X_{1},X_{2},\dots ,X_{n}$ ${\overset {\mathrm {iid} }{\sim }}$ $GE(\alpha ,\lambda )$ random variable, then the log-likelihood function with respect to the data ${\boldsymbol {x_{n}}}=(x_{1},x_{2},\dots ,x_{n})$ is given as

$$l(\alpha ,\lambda |{\boldsymbol {X_{n}}})=n\log(\alpha )+n\log(\lambda )+(\alpha -1)\sum _{i=1}^{n}\log \left(1-e^{-\lambda x_{i}}\right)-\lambda \sum _{i=1}^{n}x_{i}~.$$

The Fisher information matrix is given by ${\mathcal {I}}_{n}(\alpha ,\lambda )=(I_{ij})_{i,j=1,2}$, where
$I_{11}={\frac {1}{\alpha ^{2}}},$
$I_{12}=I_{21}=\lambda ^{-1}\left[\psi (\alpha +1)-\psi (1)-{\frac {\alpha }{\alpha -1}}\left(\psi (\alpha )-\psi (1)\right)\right]{\textrm {for}}~\alpha >1,$${\textstyle {\begin{aligned}I_{22}&={\frac {\alpha (\alpha -1)}{\alpha -2}}\lambda ^{-2}\left(\psi '(1)-\psi '(\alpha -1)+\left(\psi (\alpha -1)-\psi (1)\right)^{2}\right)\\&\quad ~~~~~~-\lambda ^{-2}+\alpha \lambda ^{-2}\left(\psi '(1)-\psi '(\alpha )+\left(\psi (\alpha )-\psi (1)\right)^{2}\right)~{\textrm {for}}~\alpha >2.\end{aligned}}}$

Use in Literature

With its shape and rate parameters, GE Distribution felicitates more rigorous skewness attributes than many other distributions. This is why it is considered to be a better choice as a potential flexible model to incorporate high positive skewness in the data. Additionally, depending on the shape parameter, the varying nature of the hazard function provides GE distribution a lot more compatibility in fitting into complex data structures. Numerous researchers have modeled the experimental data using GE distribution across several disciplines. For instance, Iwona et al. (2015) studies the usefulness of GE distribution in Flood frequency analysis (FFA) in fitting flood extremes data. Madi and Raqab (2007) uses GE distribution for the Bayesian prediction of rainfall data. Kannan, Kundu, and Nair (2010) considers the cure rate model based on the GE distribution. Maximov, Rivas-Davalos, and Cota-Felix (2009) uses GE distribution to evaluate the mean life and standard deviation of power system equipment with limited end-of-life failure data. Al-Nasser and Gogah (2017) uses GE distribution for developing reliability test plans with median ranked set sampling theory. Biondi et al. studies a new stochastic model for episode peak and duration in the context of eco-hydro-climatic studies with the assumption that the conditional distribution of the peak value of an episode follows a GE distribution.