Extended fractional cumulative past and paired phi-entropy measures

TL;DR

This paper introduces the extended fractional cumulative past entropy (EFCPE), a new measure based on the CDF, exploring its properties, extensions to bivariate and conditional cases, and applications including system reliability and logistic map validation.

Contribution

It proposes the EFCPE as a dual to EFCRE, extends it to bivariate and conditional cases, and introduces the paired phi-entropy, providing a comprehensive analysis of these new entropy measures.

Findings

EFCPE depends on the logarithm of fractional order and CDF.

EFCPE is extended to bivariate and conditional cases.

Empirical estimation and applications to systems and logistic map are demonstrated.

Abstract

Very recently, extended fractional cumulative residual entropy (EFCRE) has been proposed by Foroghi et al. (2022). In this paper, we introduce extended fractional cumulative past entropy (EFCPE), which is a dual of the EFCRE. The newly proposed measure depends on the logarithm of fractional order and the cumulative distribution function (CDF). Various properties of the EFCPE have been explored. This measure has been extended to the bivariate setup. Furthermore, the conditional EFCPE is studied and some of its properties are provided. The EFCPE for inactivity time has been proposed. In addition, the extended fractional cumulative paired phi-entropy has been introduced and studied. The proposed EFCPE has been estimated using empirical CDF. Furthermore, the EFCPE is studied for coherent systems. A validation of the proposed measure is provided using logistic map. Finally, an application is reported.