  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.
