Some properties of the cumulative residual entropy of coherent and mixed systems

Abstract

Recently, Rao et al. (2004) introduced an alternative measure of uncertainty known as the cumulative residual entropy (CRE). It is based on the survival (reliability) function F̅ instead of the probability density function f used in classical Shannon entropy. In reliability based system design, the performance characteristics of the coherent systems are of great importance. Accordingly, in this paper, we study the CRE for coherent and mixed systems when the component lifetimes are identically distributed. Bounds for the CRE of the system lifetime are obtained. We use these results to propose a measure to study if a system is close to series and parallel systems of the same size. Our results suggest that the CRE can be viewed as an alternative entropy (dispersion) measure to classical Shannon entropy.

Generalized Entropies, Variance and Applications

“…On the other hand, the variance measures an average of distances of outcomes of the probability distribution from the mean. Additionally, as mentioned by Psarrakos and Toomaj [ 13 ] (see also Toomaj et al [ 11 ]), the CRE acts like the standard deviation, i.e., it is a dispersion measure in spite of the similarity shape with the Shannon entropy. Although all of three measures, i.e., entropy, standard deviation, and CRE, are measures of dispersion and uncertainty, the lack of a simple relationship between orderings of a distribution by the three measures derives from their quite substantial and subtle differences.…”

“…Some properties of and its dynamic version are widely discussed in Asadi and Zohrevand [ 6 ] and Navarro et al [ 10 ], among others. An application of the CRE in reliability engineering systems is given by Toomaj et al [ 11 ]. Several properties of the GCRE and the DGCRE, such as bounds, stochastic ordering, aging classes properties, relationships with other entropy concepts, and characterization results, were investigated by Navarro and Psarrakos [ 12 ], Psarrakos and Navarro [ 8 ], and Psarrakos and Toomaj [ 13 ].…”

Some results on information properties of coherent systems

“…A useful merging of the measures of information theory such as Shannon entropy and KullbackLeibler discrimination information to compare systems in terms of the uncertainty and information about the predicting of the lifetimes has been recently shown by Toomaj and doostparast . Moreover, Toomaj et al extended the same results to the case of the cumulative residual entropy when lifetimes of components are dependent and identically distributed. This paper deals with entropy ordering properties of coherent systems in a greater detail.…”

Notes on Cumulative Entropy as a Risk Measure

“…Navarro and Psarrakos [7] proved some interesting characterization results based on GCRE. Toomaj, Sunoj and Navarro [14] obtained an interesting connection between CRE and the Gini mean difference as a dispersion measure alternative to standard deviation. A new information measure similar to CRE has been proposed by Di Crescenzo and Longobardi [3] as…”