Ordering Results for Order Statistics from Heterogeneous Log-Logistic Distributions

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.

Key Words and Phrases:

• Majorization
• parallel and series systems
• random shocks
• stochastic orders

2010 Mathematics Subject Classification:

• 60E15
• 60E10

Acknowledgements

The authors would like to thank the Editor, an associate Editor and an anonymous reviewer for their useful comments, which have improved the earlier version of the manuscript.

Disclosure Statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Suchandan Kayal gratefully acknowledges the partial financial support for this research work under a Grant MTR/2018/000350, SERB, India. The financial support (vide D.O.No. F. 14-34/2011 (CPP-II) dated 11.01.2013, F.No. 16-9(June 2019)/2019(NET/CSIR), UGC-Ref.No.:1238/(CSIR-UGC NET JUNE 2019)) from the University Grants Commission (UGC), Government of India, is sincerely acknowledged with thanks by Raju Bhakta.