Ordering Results for Order Statistics from Heterogeneous Log-Logistic Distributions

Abstract

SYNOPTIC ABSTRACT The largest and the smallest order statistics respectively represent the lifetime of a parallel and a series system. Various stochastic orders such as the usual stochastic order, hazard rate order and the reversed hazard rate order are used to obtain a stochastically better system. In the present communication, we consider stochastic comparison of the largest and the smallest order statistics arising from heterogeneous log-logistic distributions. First, we treat the case when the components do not receive random shocks. In other case, we assume that the components receive random shocks. The comparisons are studied in terms of the dispersive, usual stochastic, hazard rate and the reversed hazard rate orders. Majorization-based sufficient conditions are obtained to compare the order statistics. In addition, to illustrate the results, several numerical examples are presented.