Abstract
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.
Highlights
The concept of entropy was originally introduced in Shannon [1] in the context of communication theory
Reliability and survival analysis is a branch of statistics which deals with death in biological organisms and failure in mechanical systems
There are several uncertainty measures that play a central role in understanding and describing reliability
Summary
The concept of entropy was originally introduced in Shannon [1] in the context of communication theory. Entropy (1) is not scale invariant because H(cX) = log |c| + H(X), but it is translation invariant, so that H(c + X) = H(X) for some constant c The latter property can be interpreted as the shift independence of Shannon information. The notion of cumulative residual entropy (CRE) as an alternative measure of uncertainty was introduced in Wang et al [2]. This measure is based on survival function and is defined as follows:. The dynamic cumulative residual entropy (DCRE) of lifetime X at time t ≥ 0 is defined by. The cumulative entropy of a nonnegative random lifetime X is defined as CE (X) = − ∫ F (x) log F (x) dx = E (μF (X)) , (6). Throughout the remaining of this paper, all random variables are assumed as absolutely continuous
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