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

TL;DR

This paper investigates stochastic comparisons of second-order statistics from dependent and independent observations under modified proportional hazard rate models, providing theoretical conditions and numerical illustrations.

Contribution

It establishes stochastic order relations for second-order statistics in dependent and heterogeneous settings and offers new sufficient conditions for independent cases.

Findings

Stochastic order of second-order statistics from dependent observations established.

Sufficient conditions for hazard rate order in independent observations derived.

Numerical examples confirm theoretical results.

Abstract

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.