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{{Short description|Probability distribution}}
{{Draft topics|mathematics}}
{{AfC topic|stem}}

{{Distinguish|text=the [[exponential family]] of probability distributions}}
{{Probability distribution
  | name       = Xgamma distribution
  | type       = continuous
  | parameters = <math>\theta > 0,</math> [[shape parameter|shape]]
  | support    = <math>x \in [0, \infty)</math>
  | pdf        = <math>\frac{\theta^2}{(1+\theta)}\left( 1+\frac{\theta}{2}{x^2}\right)e^{-\theta x}</math>
  | cdf        = <math>1-\frac{(1+\theta+\theta x+\frac{\theta^2 x^2}{2})}{(1+\theta)} e^{-\theta x}</math>
  | mean       = <math>\frac{(\theta+3)}{\theta(1+\theta)}</math>
  | mode       = <math>\frac{1+\sqrt{1-2 \theta}}{\theta},\text{ for } \theta \le 1/2 </math>
  | variance   = <math> \frac{(\theta^2+8 \theta+3)}{\theta^2(1+\theta)^2}</math>
  | skewness   = <math>\frac{2(\theta^3+15 \theta^2+9 \theta+3)}{(\theta^2+8 \theta+3)^{3/2}}</math>
  | mgf        = <math>\frac{\theta^2}{(1+\theta)}\left[\frac{1}{(\theta-t)}+\frac{\theta}{(\theta-t)^3}\right]</math>
  | char       = <math>\frac{\theta^2}{(1+\theta)}\left[\frac{1}{(\theta-it)}+\frac{\theta}{(\theta-it)^3}\right]</math>}}

In [[probability theory]] and [[statistics]], the '''xgamma distribution''' is continuous probability distribution (introduced by Sen et al. in 2016 [https://digitalcommons.wayne.edu/cgi/viewcontent.cgi?article=1916&context=jmasm]). This distribution is obtained as a special finite mixture of [[Exponential distribution|exponentia]]<nowiki/>l and [[Gamma distribution|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,  <math>\frac{\theta}{(1+\theta)}</math> and <math>\frac{1}{(1+\theta)}</math>, respectively, to obtain the density form of the distribution.

==Definitions==

===Probability density function===
The [[probability density function]] (pdf) of an xgamma distribution is<ref>The xgamma Distribution: Statistical Properties and Application. https://digitalcommons.wayne.edu/cgi/viewcontent.cgi?article=1916&context=jmasm</ref>
:<math> f(x;\theta) = \begin{cases}
\frac{\theta^2}{(1+\theta)}\left( 1+\frac{\theta}{2}{x^2}\right)e^{-\theta x} & x \ge 0, \\
0 & x < 0.
\end{cases}</math>
Here ''θ'' > 0 is the parameter of the distribution, often called the ''shape parameter''. The distribution is supported on the interval&nbsp;{{closed-open|0, ∞}}. If a [[random variable]] ''X'' has this distribution, we write&nbsp;{{math|''X'' ~ XG(''θ'')}}.

===Cumulative distribution function===
The [[cumulative distribution function]] is given by
:<math>F(x;\theta) = \begin{cases}
1-\frac{(1+\theta+\theta x+\frac{\theta^2 x^2}{2})}{(1+\theta)} e^{-\theta x} & x \ge 0, \\
0 & x < 0.
\end{cases}</math>
===Characteristic and generating functions ===
The [[characteristic function]] of a random variable following xgamma distribution with parameter ''θ'' is given by<ref>Survival estimation in xgamma distribution under progressively type-II right censored scheme. https://content.iospress.com/articles/model-assisted-statistics-and-applications/mas423</ref>
:<math>\phi_{X}(t)=E[e^{itX}]=\frac{\theta^2}{(1+\theta)}\left[\frac{1}{(\theta-it)}+\frac{\theta}{(\theta-it)^3}\right] ; t \in \R, i=\sqrt{-1}.</math>

The [[moment generating function]] of xgamma distribution is given by
:<math>M_{X}(t)=E[e^{tX}]=\frac{\theta^2}{(1+\theta)}\left[\frac{1}{(\theta-t)}+\frac{\theta}{(\theta-t)^3}\right]; t \in \R. </math>

==Properties==
===Mean, variance, moments, and mode===
The non-central [[Moment (mathematics)|moments]] of ''X'', for <math>r\in\N</math> are given by
:<math>\mu_{r}'=\frac{r![2\theta+(r+1)(r+2)]}{2\theta^{r}(1+\theta)}.</math>
In particular,
The mean or [[expected value]] of a random variable ''X'' following xgamma distribution with parameter ''θ'' is given by
<math display="block">\operatorname{E}[X] = \frac{(\theta+3)}{\theta(1+\theta)}.</math>

The <math>r^{th}</math> <math>(r\in\N)</math> order central moment of xgamma distribution can be obtained from the relation,
<math>
\mu_{r}=E[{(X-\mu)}^{r}]=\sum_{j=0}^{r}\binom rj \mu_{r}{'} (-{\mu})^{r-j},
</math>
where <math> \mu </math> is the mean of the distribution.

The [[variance]] of ''X'' is given by
<math display="block">\operatorname{Var}[X] = \frac{(\theta^2+8 \theta+3)}{\theta^2(1+\theta)^2}.</math>

The mode of xgamma distribution is given by
:<math>
Mode[X]=\begin{cases}
\frac{1+\sqrt{1-2 \theta}}{\theta} & \text{if} &  0<\theta \le 1/2,\\
0 & \text{otherwise}.
\end{cases}</math>

=== 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:
<math>\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}}.</math>

Measure of kurtosis:
<math>\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}}.</math>

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

=== Failure rate or Hazard rate function ===
For xgamma distribution, the hazard rate (or failure rate) function is obtained as
:<math>
h(t;\theta)=\frac{\theta^2(1+\frac{\theta}{2}t^2)}{(1+\theta+\theta t+\frac{\theta^2}{2} t^2)}.
</math>
The hazard rate function in possesses the following properties.
* <math>\lim_{t \to 0}h(t;\theta)=\frac{\theta^2}{(1+\theta)}=\lim_{t \to 0}f(t;\theta).</math>
* <math>h(t;\theta)</math> is an increasing function in <math>t>\sqrt{2/\theta}.</math>
* <math>\theta^2/(1+\theta)<h(t;\theta)<\theta.</math>
=== 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
<math> m(t;\theta)=\frac{1}{\theta}+\frac{(2+\theta t)}{\theta(1+\theta+\theta t+\frac{\theta^2}{2}t^2)}.</math>

This MRL function has the following properties.
* <math>\lim_{t \to 0}m(t;\theta)=E[X]=\frac{(\theta+3)}{\theta(1+\theta)}</math>.
* <math>m(t;\theta)</math> in decreasing in ''t'' and <math>\theta</math> with <math>\frac{1}{\theta}<m(t;\theta)<\frac{(\theta+3)}{\theta(1+\theta)}.</math>
=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 <math>x=(x_{1},x_{2},\ldots,x_{n})</math> be n observations on a random sample <math>X_{1},X_{2},\cdots, X_{n}</math> of size n drawn from xgamma distribution. Then, the likelihood function is given by
<math>
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}}.</math>
The log-likelihood function is obtained as
<math>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}.</math>

To obtain maximum likelihood estimator (MLE) of <math>\theta</math>, <math>\hat{\theta}</math>(say), one can maximize the log-likelihood equation directly with respect to <math>\theta</math> or can solve the non-linear equation, 
<math>\frac{\partial \ln{L(\theta|x)}}{\partial \theta}=0.</math>
It is seen that 
<math>\frac{\partial \ln{L(\theta|x)}}{\partial \theta}=0</math> 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 <math>\theta_{0}=\frac{n}{\sum_{i=1}^{n}x_{i}}.</math> Using this initial solution, we have,
<math>\theta^{(i)}=\theta^{(i-1)}-\frac{l(\theta^{(i-1)}|x)}{l^{'}(\theta^{(i-1)}|x)} </math> for the ith iteration.
one chooses <math>\theta^{(i)}</math> such that <math>\theta^{(i)}\cong\theta^{(i-1)}</math>.

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

==Random variate generation==
To generate random data <math>X_{i}; i=1,2,\ldots,n,</math> from xgamma distribution with parameter <math>\theta</math>, the following algorithm is proposed.

* Generate <math>U_{i} \sim uniform(0,1),i=1,2,\ldots,n.</math>
* Generate <math>V_{i}\sim exp(\theta),i=1,2,\ldots,n.</math>
* Generate <math> W_{i}\sim gamma(3,\theta),i=1,2,\ldots,n.</math>
* If <math>U_{i}\le \theta/(1+\theta)</math>, then set  <math>X_{i}=V_{i}</math>, otherwise, set <math>X_{i}=W_{i}.</math> 

== References ==
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