Stochastic comparisons of second largest order statistics with dependent heterogeneous random variables

In the context of actuarial science, the second largest claim amount is crucial to insurance analysis since they provide useful information for determining annual premium. In this article, we provide sufficient conditions of the second largest claim amounts arising from two sets of dependent and heterogeneous individual risk models according to various stochastic orders. It is first shown that the reversed hazard order between the occurrence probabilities vectors implies the reversed hazard order of the second largest claim amounts under certain conditions. Second, sufficient conditions are established for the usual stochastic ordering of the second largest claim amounts arising from heterogeneous dependent individual risk models under copula dependence. Finally, several examples illustrating the theoretical results are presented as well.