To illustrate the usefulness of Theorem
2.14, let us consider the survival function of the bivariate-Pareto family with scale parameters
\(\sigma_{1}>0\) and
\(\sigma_{2}>0\) and the shape parameter
\(\alpha>1\) given by
$$ \bar{F}_{\sigma_{1},\sigma_{2},\alpha}(x_{1},x_{2})= \biggl[ 1+ \frac {x_{1}}{\sigma_{1}}+\frac{x_{2}}{\sigma_{2}} \biggr] ^{-\alpha}, \quad x_{1}\geq0, x_{2}\geq0. $$
Let
\(m_{\sigma_{1},\sigma_{2},\alpha}(t_{1},t_{2})= [ m_{\sigma _{1},\sigma_{2},\alpha}^{(1)}(t_{1},t_{2}),m_{\sigma_{1},\sigma _{2},\alpha}^{(2)}(t_{1},t_{2}) ] \),
\(t_{1},t_{2}\geq0\) be the bivariate mean residual life function associated to
\(\bar{F}_{\sigma _{1},\sigma_{2},\alpha}\). Carrying out easy integrations we get
$$\begin{aligned} m_{\sigma_{1},\sigma_{2},\alpha}^{(1)}(t_{1},t_{2}) =& \frac{\int_{t_{1}}^{\infty}\bar{F}_{\sigma_{1},\sigma_{2},\alpha }(u,t_{2})\,du}{\bar{F}_{\sigma_{1},\sigma_{2},\alpha}(t_{1},t_{2})} \\ =&\frac{\sigma_{1}}{1-\alpha} \biggl[ 1+\frac{t_{1}}{\sigma_{1}}+\frac {t_{2}}{\sigma_{2}} \biggr],\quad t_{1},t_{2}\geq0, \end{aligned}$$
and
$$\begin{aligned} m_{\sigma_{1},\sigma_{2},\alpha}^{(2)}(t_{1},t_{2}) =& \frac{\int_{t_{2}}^{\infty}\bar{F}_{\sigma_{1},\sigma_{2},\alpha }(t_{1},u)\,du}{\bar{F}_{\sigma_{1},\sigma_{2},\alpha}(t_{1},t_{2})} \\ =&\frac{\sigma_{2}}{1-\alpha} \biggl[ 1+\frac{t_{1}}{\sigma_{1}}+\frac {t_{2}}{\sigma_{2}} \biggr] ,\quad t_{1},t_{2}\geq0. \end{aligned}$$
Let
\(\mathbf{X}_{1}=(X_{1},X_{2})\) and
\(\mathbf{Y}_{2}=(Y_{1},Y_{2})\) be a random pairs with mean residual life functions
\(m_{\sigma_{1},\sigma _{2},\alpha}(t_{1},t_{2})\) and
\(m_{\sigma_{1}^{\prime},\sigma _{2}^{\prime},\alpha^{\prime}}(t_{1},t_{2})\),
\(t_{1},t_{2}\geq0\). In order to characterize the order
\(\preceq_{mrlmr}\) in terms of scale and shape parameters, first observe that
$$ \frac{m_{\sigma_{1}^{\prime},\sigma_{2}^{\prime},\alpha^{\prime }}^{(1)}(t_{1},t_{2})}{m_{\sigma_{1},\sigma_{2},\alpha }^{(1)}(t_{1},t_{2})}\propto\frac{m_{\sigma_{1}^{\prime},\sigma_{2}^{\prime},\alpha ^{\prime}}^{(2)}(t_{1},t_{2})}{m_{\sigma_{1},\sigma_{2},\alpha }^{(2)}(t_{1},t_{2})}. $$
Thus, one has
$$\begin{aligned}& \mathbf{X} \preceq_{mrlmr}\mathbf{Y} \quad\Leftrightarrow \quad \frac{m_{\sigma_{1}^{\prime},\sigma_{2}^{\prime},\alpha^{\prime }}^{(1)}(t_{1},t_{2})}{m_{\sigma_{1},\sigma_{2},\alpha }^{(1)}(t_{1},t_{2})}\mbox{ is increasing in }t_{1},t_{2}\geq0 \\& \quad \Leftrightarrow\quad \phi (t_{1},t_{2})= \frac{\sigma_{2}^{\prime}t_{1}+\sigma_{1}^{\prime }t_{2}+\sigma_{1}^{\prime}\sigma_{2}^{\prime}}{\sigma _{2}t_{1}+\sigma _{1}t_{2}+\sigma_{1}\sigma_{2}}\mbox{ is increasing in }t_{1},t_{2}\geq 0 \\& \quad \Leftrightarrow \quad\frac{\partial \phi (t_{1},t_{2})}{\partial t_{1}}\geq0\quad \mbox{and}\quad \frac{\partial\phi(t_{1},t_{2})}{\partial t_{2}}\geq0\quad\mbox{for all }t_{1},t_{2} \geq0 \\& \quad\Leftrightarrow \quad \left \{ \textstyle\begin{array}{@{}l@{\quad}l} \sigma_{2}^{\prime}(\sigma_{1}\sigma_{2}-\sigma_{1}^{\prime }\sigma _{2})+(\sigma_{1}\sigma_{2}^{\prime}-\sigma_{1}^{\prime}\sigma _{2})t_{2}\geq0&\mbox{for all }t_{2}\geq0, \\ \sigma_{1}(\sigma_{1}^{\prime}\sigma_{2}-\sigma_{1}^{\prime }\sigma _{2}^{\prime})+(\sigma_{1}^{\prime}\sigma_{2}-\sigma_{1}\sigma _{2}^{\prime})t_{1}\geq0&\mbox{for all }t_{1}\geq0\end{array}\displaystyle \right . \\& \quad \Leftrightarrow \quad \left \{ \textstyle\begin{array}{@{}l@{\quad}l} \sigma_{2}^{\prime}(\sigma_{1}\sigma_{2}-\sigma_{1}^{\prime }\sigma _{2})+(\sigma_{1}\sigma_{2}^{\prime}-\sigma_{1}^{\prime}\sigma _{2})t_{2}\geq0&\mbox{for all }t_{2}\geq0, \\ \sigma_{1}(\sigma_{1}^{\prime}\sigma_{2}-\sigma_{1}^{\prime }\sigma _{2}^{\prime})+(\sigma_{1}^{\prime}\sigma_{2}-\sigma_{1}\sigma _{2}^{\prime})t_{1}\geq0&\mbox{for all }t_{1}\geq0\end{array}\displaystyle \right . \\& \quad\Leftrightarrow \quad\frac{\sigma _{1}^{\prime}}{\sigma_{1}}=\frac{\sigma_{2}^{\prime}}{\sigma _{2}}\leq 1. \end{aligned}$$
Now, we use Theorem
2.14 to establish a sufficient condition for the order
\(\preceq_{mwhr}\). Clearly,
$$ 0< \lim_{x_{i}\rightarrow\infty}\frac{\bar{F}_{\sigma_{1}^{\prime },\sigma _{2}^{\prime},\alpha^{\prime}}(x_{1},x_{2})}{\bar{F}_{\sigma _{1},\sigma _{2},\alpha}(x_{1},x_{2})}< \infty,\quad i=1,2 \quad \Leftrightarrow\quad \alpha=\alpha^{\prime}. $$
Thus, from (ii) of Theorem
2.14, one has
$$ \alpha=\alpha^{\prime} \quad\mbox{and}\quad \frac {\sigma _{1}^{\prime}}{\sigma_{1}}= \frac{\sigma_{2}^{\prime}}{\sigma _{2}}\leq 1 \quad\Rightarrow \quad\mathbf{X}\preceq_{mwhr} \mathbf{Y}. $$
We use again Theorem
2.14 to obtain a sufficient condition for the order
\(\preceq_{mwhr}\). Indeed, one can easily derive the next equivalence
$$ E(X_{i})\leq E(Y_{i}),\quad i=1,2 \quad\Leftrightarrow \quad\frac {\sigma _{1}}{1-\alpha}\leq\frac{\sigma_{1}^{\prime}}{1-\alpha^{\prime }}\mbox{ and } \frac{\sigma_{2}}{1-\alpha} \leq\frac{\sigma_{2}^{\prime }}{1-\alpha ^{\prime}}. $$
It follows that
$$\begin{aligned}& \mathbf{X}\preceq_{mrlmr}\mathbf{Y} \quad\mbox{and} \quad E(X_{i})\leq E(Y_{i}),\quad i=1,2 \quad\Leftrightarrow\quad \alpha\leq\alpha^{\prime}, \\& \mathbf{X} \preceq_{mrlmr}\mathbf{Y} \quad\mbox{and} \quad E(X_{i})\leq E(Y_{i}),\quad i=1,2 \quad\Leftrightarrow\quad \alpha\leq\alpha^{\prime}\mbox{ and }\frac{\sigma _{1}^{\prime}}{\sigma_{1}}= \frac{\sigma_{2}^{\prime}}{\sigma _{2}}\leq1. \end{aligned}$$
From (iii) of Theorem
2.14, we get
$$ \alpha\leq\alpha^{\prime} \quad\mbox{and}\quad \frac {\sigma_{1}^{\prime}}{\sigma_{1}}= \frac{\sigma_{2}^{\prime}}{\sigma _{2}}\leq1 \quad\Rightarrow\quad \mathbf{X} \preceq_{mmr}\mathbf{Y}. $$
In addition, we observe from this analysis that the order
\(\preceq_{rmrm}\) is only affected by the scale parameters.