On the Kullback Leibler information for mixed systems

Abstract

Entropy and Kullback–Leibler (KL) information for engineering systems have been studied in both statistical and reliability contexts. In this paper, we prove that the KL information between distributions of mixed system lifetimes and the corresponding component lifetimes and also the associated order statistics are distribution free and depends only on the signature of the system provided that lifetimes of components are independent and identically distributed (iid). The obtained results are used to find the closest and the farthest distribution of order statistics from the distribution of the system’s lifetime which is useful to approximate stochastic behaviour of mixed systems when the number of components is large. Finally, we provide bounds and also use the results to obtain a more preferable system among all systems. Some illustrative examples are also given.

Keywords:

• entropy
• KL information
• mixed system
• system signature

Acknowledgements

The authors are grateful to the Editor-in-Chief, the Associate Editor, and the anonymous referees for their useful suggestions and comments on an earlier version of this paper, which resulted in this improved version of the manuscript.

Additional information

Notes on contributors

A. Toomaj

Abdolsaeed Toomaj received his Ph.D. degree in Statistical Inference in 2013 under the supervision of Dr. Mahdi Doostparast from Ferdowsi University of Mashhad, Iran. His research interests include reliability theory, stochastic orders, survival analysis and information theory. He also published several articles in scientific journals such as Journal of Statistical Theory and Applications, Naval Research Logistics and Applied Stochastic Models in Business and Industry.

M. Doostparast

Mahdi Doostparast received his Ph.D. degree in Statistical Inference in 2007 from Ferdowsi University of Mashhad, Iran. He is currently Associate Professor at the Faculty of Mathematical Sciences (Department of Statistics), Ferdowsi University of Mashhad, Iran. His research interests include prediction and estimation with ordered and censored data, testing statistical hypotheses, mathematical finance and statistical reliability analysis.