Stochastic Comparisons of Second-Order Statistics from Dependent and Heterogenous Modified Proportional Hazard Rate Observations

In this manuscript, we study stochastic comparisons of the second-order statistics from dependent or independent observations with modified proportional hazard rates models. First, we establish the usual stochastic order of the second-order statistics from dependent and heterogeneous observations. Second, sufficient conditions are provided in the hazard rate order of the second-order statistics from independent observations. Then, we investigate the hazard rate order of the second-order statistics arising from two sets of independent multiple-outlier modified proportional hazard rates observations. Finally, some numerical examples are given to illustrate the theoretical findings.