Draft:Xgamma distribution

Not to be confused with the exponential family of probability distributions.

Xgamma distribution

Parameters	$\theta >0,$ shape
Support	$x\in [0,\infty )$
PDF	${\frac {\theta ^{2}}{(1+\theta )}}\left(1+{\frac {\theta }{2}}{x^{2}}\right)e^{-\theta x}$
CDF	$1-{\frac {(1+\theta +\theta x+{\frac {\theta ^{2}x^{2}}{2}})}{(1+\theta )}}e^{-\theta x}$
Mean	${\frac {(\theta +3)}{\theta (1+\theta )}}$
Mode	${\frac {1+{\sqrt {1-2\theta }}}{\theta }},{\text{ for }}\theta \leq 1/2$
Variance	${\frac {(\theta ^{2}+8\theta +3)}{\theta ^{2}(1+\theta )^{2}}}$
Skewness	${\frac {2(\theta ^{3}+15\theta ^{2}+9\theta +3)}{(\theta ^{2}+8\theta +3)^{3/2}}}$
MGF	${\frac {\theta ^{2}}{(1+\theta )}}\left[{\frac {1}{(\theta -t)}}+{\frac {\theta }{(\theta -t)^{3}}}\right]$
CF	${\frac {\theta ^{2}}{(1+\theta )}}\left[{\frac {1}{(\theta -it)}}+{\frac {\theta }{(\theta -it)^{3}}}\right]$

In probability theory and statistics, the xgamma distribution is continuous probability distribution (introduced by Sen et al. in 2016 [1]). This distribution is obtained as a special finite mixture of exponential and gamma distributions. This distribution is successfully used in modelling time-to-event or lifetime data sets coming from diverse fields.

Exponential distribution with parameter θ and gamma distribution with scale parameter θ and shape parameter 3 are mixed mixing proportions, ${\frac {\theta }{(1+\theta )}}$ and ${\frac {1}{(1+\theta )}}$, respectively, to obtain the density form of the distribution.

Definitions

Probability density function

The probability density function (pdf) of an xgamma distribution is

$f(x;\theta )={\begin{cases}{\frac {\theta ^{2}}{(1+\theta )}}\left(1+{\frac {\theta }{2}}{x^{2}}\right)e^{-\theta x}&x\geq 0,\\0&x<0.\end{cases}}$

Here θ > 0 is the parameter of the distribution, often called the shape parameter. The distribution is supported on the interval [0, ∞). If a random variable X has this distribution, we write X ~ XG(θ).

Cumulative distribution function

The cumulative distribution function is given by

$F(x;\theta )={\begin{cases}1-{\frac {(1+\theta +\theta x+{\frac {\theta ^{2}x^{2}}{2}})}{(1+\theta )}}e^{-\theta x}&x\geq 0,\\0&x<0.\end{cases}}$

Characteristic and generating functions

The characteristic function of a random variable following xgamma distribution with parameter θ is given by

$\phi _{X}(t)=E[e^{itX}]={\frac {\theta ^{2}}{(1+\theta )}}\left[{\frac {1}{(\theta -it)}}+{\frac {\theta }{(\theta -it)^{3}}}\right];t\in \mathbb {R} ,i={\sqrt {-1}}.$

The moment generating function of xgamma distribution is given by

$M_{X}(t)=E[e^{tX}]={\frac {\theta ^{2}}{(1+\theta )}}\left[{\frac {1}{(\theta -t)}}+{\frac {\theta }{(\theta -t)^{3}}}\right];t\in \mathbb {R} .$

Properties

Mean, variance, moments, and mode

The non-central moments of X, for $r\in \mathbb {N}$ are given by

$\mu _{r}'={\frac {r![2\theta +(r+1)(r+2)]}{2\theta ^{r}(1+\theta )}}.$

In particular, The mean or expected value of a random variable X following xgamma distribution with parameter θ is given by $$\operatorname {E} [X]={\frac {(\theta +3)}{\theta (1+\theta )}}.$$

The $r^{th}$ $(r\in \mathbb {N} )$ order central moment of xgamma distribution can be obtained from the relation, $\mu _{r}=E[{(X-\mu )}^{r}]=\sum _{j=0}^{r}{\binom {r}{j}}\mu _{r}{'}(-{\mu })^{r-j},$ where $\mu$ is the mean of the distribution.

The variance of X is given by $$\operatorname {Var} [X]={\frac {(\theta ^{2}+8\theta +3)}{\theta ^{2}(1+\theta )^{2}}}.$$

The mode of xgamma distribution is given by

$Mode[X]={\begin{cases}{\frac {1+{\sqrt {1-2\theta }}}{\theta }}&{\text{if}}&0<\theta \leq 1/2,\\0&{\text{otherwise}}.\end{cases}}$

Skewness and kurtosis

The coefficients of skewness and kurtosis of xgamma distribution with parameter θ show that the distribution is positively skewed.

Measure of skewness: ${\sqrt {\beta _{1}}}={\sqrt {\frac {\mu _{3}^{2}}{\mu _{2}^{3}}}}={\frac {2(\theta ^{3}+15\theta ^{2}+9\theta +3)}{(\theta ^{2}+8\theta +3)^{3/2}}}.$

Measure of kurtosis: $\beta _{2}={\frac {\mu _{4}}{\mu _{2}^{2}}}={\frac {3(5\theta ^{4}+88\theta ^{3}+310\theta ^{2}+288\theta +177)}{(\theta ^{2}+8\theta +3)^{2}}}.$

Survival properties

Among survival properties, failure rate or hazard rate function, mean residual life function and stochastic order relations are well established for xgamma distribution with parameter θ.

The survival function at time point t(> 0) is given by

$S(t;\theta )=\Pr(X>t)={\frac {(1+\theta +\theta t+{\frac {\theta ^{2}t^{2}}{2}})}{(1+\theta )}}e^{-\theta t}.$

Failure rate or Hazard rate function

For xgamma distribution, the hazard rate (or failure rate) function is obtained as

$h(t;\theta )={\frac {\theta ^{2}(1+{\frac {\theta }{2}}t^{2})}{(1+\theta +\theta t+{\frac {\theta ^{2}}{2}}t^{2})}}.$

The hazard rate function in possesses the following properties.

• $\lim _{t\to 0}h(t;\theta )={\frac {\theta ^{2}}{(1+\theta )}}=\lim _{t\to 0}f(t;\theta ).$
• $h(t;\theta )$ is an increasing function in $t>{\sqrt {2/\theta }}.$
• $\theta ^{2}/(1+\theta )<h(t;\theta )<\theta .$

Mean residual life (MRL) function

Mean residual life (MRL) function related to a life time probability distribution is an important characteristic useful in survival analysis and reliability engineering. The MRL function for xgamma distribution is given by $m(t;\theta )={\frac {1}{\theta }}+{\frac {(2+\theta t)}{\theta (1+\theta +\theta t+{\frac {\theta ^{2}}{2}}t^{2})}}.$

This MRL function has the following properties.

• $\lim _{t\to 0}m(t;\theta )=E[X]={\frac {(\theta +3)}{\theta (1+\theta )}}$.
• $m(t;\theta )$ in decreasing in t and $\theta$ with ${\frac {1}{\theta }}<m(t;\theta )<{\frac {(\theta +3)}{\theta (1+\theta )}}.$

Statistical inference

Below are provided two classical methods, namely maximum of estimation for the unknown parameter of xgamma distribution under complete sample set up.

Parameter estimation

Maximum likelihood estimation

Let $x=(x_{1},x_{2},\ldots ,x_{n})$ be n observations on a random sample $X_{1},X_{2},\cdots ,X_{n}$ of size n drawn from xgamma distribution. Then, the likelihood function is given by $L(\theta |x)=\prod _{i=1}^{n}{\frac {\theta ^{2}}{(1+\theta )}}\left(1+{\frac {\theta }{2}}x_{i}^{2}\right)e^{-\theta x_{i}}.$ The log-likelihood function is obtained as $l(\theta )=\ln {L(\theta |x)}=2n\ln {\theta }-n\ln(1+\theta )+\sum _{i=1}^{n}\ln {\left(1+{\frac {\theta }{2}}x_{i}^{2}\right)}-\theta \sum _{i=1}^{n}x_{i}.$

To obtain maximum likelihood estimator (MLE) of $\theta$, ${\hat {\theta }}$(say), one can maximize the log-likelihood equation directly with respect to $\theta$ or can solve the non-linear equation, ${\frac {\partial \ln {L(\theta |x)}}{\partial \theta }}=0.$ It is seen that ${\frac {\partial \ln {L(\theta |x)}}{\partial \theta }}=0$ cannot be solved analytically and hence numerical iteration technique, such as, Newton-Raphson algorithm is applied to solve. The initial solution for such an iteration can be taken as $\theta _{0}={\frac {n}{\sum _{i=1}^{n}x_{i}}}.$ Using this initial solution, we have, $\theta ^{(i)}=\theta ^{(i-1)}-{\frac {l(\theta ^{(i-1)}|x)}{l^{'}(\theta ^{(i-1)}|x)}}$ for the ith iteration. one chooses $\theta ^{(i)}$ such that $\theta ^{(i)}\cong \theta ^{(i-1)}$.

Method of moments estimation

Given a random sample $X_{1},X_{2},\ldots ,X_{n}$ of size n from the xgamma distribution, the moment estimator for the parameter $\theta$ of xgamma distribution is obtained as follows. Equate sample mean, ${\bar {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}$ with first order moment about origin to get ${\bar {X}}={\frac {(\theta +3)}{\theta (1+\theta )}},$ which provides a quadratic equation in $\theta$ as ${\bar {X}}\theta ^{2}+({\bar {X}}-1)\theta -3=0.$ Solving it, one gets the moment estimator, ${\hat {\theta _{M}}}$ (say), of $\theta$ as ${\hat {\theta }}_{M}={\frac {-({\bar {X}}-1)+{\sqrt {({\bar {X}}-1)^{2}+12{\bar {X}}}}}{2{\bar {X}}}}\;{\text{for}}\;{\bar {X}}>0.$

Random variate generation

To generate random data $X_{i};i=1,2,\ldots ,n,$ from xgamma distribution with parameter $\theta$, the following algorithm is proposed.

• Generate $U_{i}\sim uniform(0,1),i=1,2,\ldots ,n.$
• Generate $V_{i}\sim exp(\theta ),i=1,2,\ldots ,n.$
• Generate $W_{i}\sim gamma(3,\theta ),i=1,2,\ldots ,n.$
• If $U_{i}\leq \theta /(1+\theta )$, then set $X_{i}=V_{i}$, otherwise, set $X_{i}=W_{i}.$