and hence $(X \le_{hr} Y )$, $(X \le_{mrl} Y )$ and $(X \le_{st} Y )$ for all values of $\alpha_i,\beta_i$; $i=1, 2$.\
**Proof:** The likelihood ratio is $\left[\dfrac{f_{X}(x)}{f_{Y}(x)}\right]$ i.e. $$\begin{split}
\Phi=\dfrac{f_{X}(x)}{f_{Y}(x)}&=\left( \dfrac{\alpha_1}{\alpha_2}\right)^{\frac{3}{2}} \left( \dfrac{\beta_1}{\beta_2}\right) x^{3(\beta_1-\beta_2)} e^{-(\alpha_1 x^{2\beta_1}+\alpha_2 x^{2\beta_2})}
\end{split}$$ Therefore, $$\Phi^{'}=\log \left( \dfrac{f_{X}(x)}{f_{Y}(x)}\right) =\dfrac{1}{x}\left[ 3(\beta_1-\beta_2)-(\alpha_1 x^{2\beta_1}+\alpha_2 x^{2\beta_2})\right]$$ If $\beta_1=\beta_2=\beta (say)$, then $\Phi^{'}<0$, which shows that $(X \le_{lr} Y )$. The remaining ordering behaviour can be proved in same way. Also, if $\alpha_1=\alpha_2=\alpha (say)$ and $\beta_1<\beta_2$ then again $\Phi^{'}<0$, which shows that $(X \le_{lr} Y )$. The remaining ordering can be proved in same way.

Residual Lifetime
=================

In survival analysis, the term residual lifetime often used to describe the remaining lifetime associated with any particular system. Here, we derived the expression of residual life and reversed residual life for PMaD. The residual lifetime function is defined by $ R_{t}= P[x -t|x > t], t \ge0$ and the reversed residual life is described as $\bar{R }_{t} = P[t -x|x \le t]$ which denotes the time elapsed from the failure of a component given that its life less or equal to $t$.

-   **Residual life time function**\
    The survival function of the residual lifetime is given by $$S_{R_t}(x)=\dfrac{S(t+x)}{S(t)}= \dfrac{\sqrt{\pi}-2\gamma\left[ \frac{3}{2},\alpha (x+t)^{2\beta}\right] }{\sqrt{\pi}-2\gamma\left( 