and has more comprehensive applications in actuarial as well as in demography. It was initially proposed and studied by Bonferroni (1920), matthematically, it is defined as; $$\zeta(\nu)_{b}=\dfrac{1}{\nu \mu}\int_{0}^{q}xf(x,\alpha,\beta)dx$$ $$\zeta(\nu)_{l}=\dfrac{1}{ \mu}\int_{0}^{q}xf(x,\alpha,\beta)dx$$ respectively. where $q=F^{-1}(\nu)$ and $\mu=E(X)$. Hence using eqn (2.1), the above two equations are reduces as $$\zeta(\nu)_{b}=\dfrac{\sqrt{\alpha}~IG\left( \frac{1+2\beta}{2\beta},\alpha q^{2\beta}\right) }{\nu\Gamma\left( \frac{3\beta+1}{2\beta}\right) }$$ $$\zeta(\nu)_{l}=\dfrac{\sqrt{\alpha}~IG\left( \frac{1+2\beta}{2\beta},\alpha q^{2\beta}\right) }{\Gamma\left( \frac{3\beta+1}{2\beta}\right) }$$

Stochastic Ordering
-------------------

A random variable $X$ is said to be stochastically greater $(Y \le_{st} X)$ than $Y$ if $F_X(x)\le F_Y (x)$ for all $x$. In the similar way, $X$ is said to be stochastically greater $(X \le_{st} Y )$ than $Y$ in the

-   hazard rate order $(X  \le_{hr} Y )$ if $h_X(x) \ge h_Y (x)$ $\forall x$.

-   mean residual life order $(X \le_{mrl} Y )$ if $m_X(x) \ge m_Y (x)$ $\forall x$.

-   likelihood ratio order $(X \le_{lr} Y )$ if $\left[\dfrac{f_{X}(x)}{f_{Y}(x)}\right]$ decreases in x.

From the above relations, we can veryfied that; $$(X \le_{lr} Y )\Rightarrow (X \le_{hr} Y )\Downarrow (X \le_{st} Y )\Rightarrow (X \le_{mrl} Y )$$ The PMaD is ordered with respect to the strongest likelihood ratio ordering as shown in the following theorem.\
**Theorem:** Let $X\sim PMaD(\alpha_1,\beta_1)$ and $Y\sim PMaD(\alpha_2,\beta_2)$. Then $(X \le_{lr} Y )$ 