Residual and Past Entropy for Concomitants of Ordered Random Variables of Morgenstern Family

Abstract

For a system, which is observed at timet, the residual and past entropies measure the uncertainty about the remaining and the past life of the distribution, respectively. In this paper, we have presented the residual and past entropy of Morgenstern family based on the concomitants of the different types of generalized order statistics (gos) and give the linear transformation of such model. Characterization results for these dynamic entropies for concomitants of ordered random variables have been considered.

On concomitants of ordered random variables under general forms of Morgenstern family

“…Remark 2.1. By substituting p = 1 in Huang and Kotz extension, and p 1 = p 2 = q 1 = q 2 = 1 in Bairamov extension, Equations (18) to (43) reduces to the ordinary Morgenstern family, and we obtain the same results that mentioned in Mohie El-Din et al [16].…”

“…Based on Huang and Kotz extension with pd f given by(2) and cd f given by (1), utilizing (3),(16) and(12), the pd f and cd f of the concomitant of rth case-I of d os, Y d[r;n,m,k] , are given by, 1 ≤ r ≤ n, respectively: d1[r;n,m,k] (y) = f Y (y) 1 + αT * 2 (r; n, m, k)…”

A measure of inaccuracy in concomitants of ordered random variables under Farlie-Gumbel-Morgenstern family

“…As q is the mode of the distribution, hence ln f Y (y) ≤ ln Q. Using this fact in (20) we get our result.…”

Some Information Measures Properties of the GOS-Concomitants from the FGM Family

“…Residual and past entropies for concomitants of GOS from the FGM family were determined by Mohie EL-Din et al in [ 29 ]. They considered also concomitants of other types of GOS, but the form of the entropies is similar.…”

“…They considered also concomitants of other types of GOS, but the form of the entropies is similar. The residual entropy of the r-th concomitant of GOS from the FGM family is [ 29 ]: where and …”