Farsam Misagh

Measures of cumulative residual entropy (CRE) and cumulative entropy (CE) about predictability of failure time of a system have been introduced in the studies of reliability and life testing. In this paper, cumulative distribution and survival function are used to develop weighted forms of CRE and CE. These new measures are denominated as weighted cumulative residual entropy (WCRE) and weighted cumulative entropy (WCE) and the connections of these new measures with hazard and reversed hazard rates are assessed. These information-theoretic uncertainty measures are shift-dependent and various properties of these measures are studied, including their connections with CRE, CE, mean residual lifetime, and mean inactivity time. The notions of weighted mean residual lifetime (WMRL) and weighted mean inactivity time (WMIT) are defined. The connections of weighted cumulative uncertainties with WMRL and WMIT are used to calculate the cumulative entropies of some well-known distributions. The joint versions of WCE and WCRE are defined which have the additive properties similar to those of Shannon entropy for two independent random lifetimes. The upper boundaries of newly introduced measures and the effect of linear transformations on them are considered. Finally, empirical WCRE and WCE are proposed by virtue of sample mean, sample variance, and order statistics to estimate the new measures of uncertainty. The consistency of these estimators is studied under specific choices of distributions.

The Shannon interval entropy function as a useful dynamic measure of uncertainty for two sided truncated random variables has been proposed in the literature of reliability. In this paper, we show that interval entropy can uniquely determine the distribution function. Furthermore, we propose a measure of discrepancy between two lifetime distributions at the interval of time in base of Kullback-Leibler discrimination information. We study various properties of this measure, including its connection with residual and past measures of discrepancy and interval entropy, and we obtain its upper and lower bounds.

Measure of uncertainty about predictability of failure time of a system at the interval of time has been introduced in [13]. In this paper, a measure of discrepancy between two lifetime distributions at the interval of time is proposed on the basis of Kullback-Leibler discrimination information. We study the Kullback-Leibler measure for two-sided truncated random variables. We study various properties of this measure, including its connection with residual and past measures of discrepancy, and we obtain its upper and lower bounds.

Measures of cumulative residual entropy (CRE) and cumulative entropy (CE) about predictability of failure time of a system have been introduced in [8, 4] respectively. In this paper, we propose the weighted forms of CRE and CE. These information-theoretic uncertainty measures are shift-dependent. We study various properties of these measures, including their connections with CRE and CE and weighted mean residual lifetime. We also obtain the estimators of proposed measures by means of order statistics.