Ordering results of extreme order statistics from multiple-outlier scale models with dependence

Many authors have studied ordering results between extreme order statistics from multiple-outlier models when the observations are independent. However, the independence assumption is not very attractive in many situations due to the complexity of the problems. This paper focuses on stochastic comparisons of extreme order statistics stemming from multiple-outlier scale models with dependence. Archimedean copula is used to model dependence structure among nonnegative random variables. Sufficient conditions are obtained to compare the largest order statistics in the sense of the usual stochastic, reversed hazard rate, likelihood ratio, dispersive, star, and Lorenz orders. The smallest order statistics are also compared with respect to the usual stochastic, hazard rate, star, and Lorenz orders. Here, the sufficient conditions are based on the weak-super majorization, weak-sub majorization, and p-larger orders between the model parameters.  To illustrate the theoretical establishments, some examples are provided. Furthermore, some counterexamples are provided to establish that ignorance of sufficient conditions may not lead to the established ordering results between the order statistics.

Keywords:

Archimedean copula , Dispersive order , Hazard rate order , Majorization , Multiple-outlier model , Reversed hazard rate order