Draft:Xgamma distribution

In probability theory and statistics, the xgamma distribution is continuous probability distribution introduced by Sen et al. in 2016. This distribution is obtained as a special finite mixture of exponentia l 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.

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

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

Mean, variance, moments, and mode
The non-central moments of X, for $$r\in\N$$ 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 $$r^{th}$$ $$(r\in\N)$$ order central moment of xgamma distribution can be obtained from the relation, $$ \mu_{r}=E[{(X-\mu)}^{r}]=\sum_{j=0}^{r}\binom rj \mu_{r}{'} (-{\mu})^{r-j}, $$ 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 \le 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 $$\frac{1}{\theta}<m(t;\theta)<\frac{(\theta+3)}{\theta(1+\theta)}.$$