Extension of the past lifetime and its connection to the cumulative entropy

Abstract

Given two absolutely continuous nonnegative independent random variables, we define the reversed relevation transform as dual to the relevation transform. We first apply such transforms to the lifetimes of the components of parallel and series systems under suitably proportionality assumptions on the hazards rates. Furthermore, we prove that the (reversed) relevation transform is commutative if and only if the proportional (reversed) hazard rate model holds. By repeated application of the reversed relevation transform we construct a decreasing sequence of random variables which leads to new weighted probability densities. We obtain various relations involving ageing notions and stochastic orders. We also exploit the connection of such a sequence to the cumulative entropy and to an operator that is dual to the Dickson-Hipp operator. Iterative formulae for computing the mean and the cumulative entropy of the random variables of the sequence are finally investigated.

Analysis of the Past Lifetime in a Replacement Model through Stochastic Comparisons and Differential Entropy

“…We remark that the differential entropy of X (Y) u,t is dependent on t through the interval entropy of Y. Namely, from (22) one has that H X (Y) u,t is increasing (decreasing) in t > u if and only if H Y (u, t) is increasing (decreasing) in t > u, for fixed u > 0. Although the monotonicity of H Y (u, t) is relevant to establish that it uniquely determines the underlying distribution function (see Proposition 2.1 of [34]), it is not easy to establish conditions leading to its validity.…”

“…It is interesting to illustrate the meaning of Equation (22). The uncertainty about the failure time of an item distributed as the past lifetime X (Y) u,t can be decomposed into three terms: (i) the uncertainty on whether the item has failed before or after time u (according to the distribution of X), (ii) the uncertainty about the failure time in (0, u) given that the item has failed before t, and (iii) the uncertainty about the failure time in interval (u, t), given that the item has failed after u and, thus, the failure time is distributed as Y ∈ (u, t) since the replacement occurred at time u.…”

“…Moreover, the past entropy is employed to encompass the relevant informational properties. We recall various investigation in this area, due to Di Crescenzo and Longobardi [21], Di Crescenzo and Toomaj [22], Kundu and Nanda [23], Kundu et al [24], and Nanda and Paul [25].…”

A Shift-Dependent Measure of Extended Cumulative Entropy and Its Applications in Blind Image Quality Assessment

“…whereΛ(x) = − log F(x). Recently Di Crescenzo and Toomaj [5] discussed some properties of a new weighted distribution based on a cumulative entropy (CE) function. Psarrakos and Navarro [6] generalized the concept of CRE, relating this concept with the mean time between record values and with the concept of relevation transform, and also considered a dynamic version of this new measure (for more details see Calí, Longobardi and Psarrakos, [7]).…”

A past inaccuracy measure based on the reversed relevation transform

“…We remark that the relevation transform was proposed initially by Krakowski (1973), and subsequently it has been applied in several contexts. As a dual concept, the reversed relevation transform was introduced in Di Crescenzo and Toomaj (2015). For some properties and applications of this notion one may refer also to Kayal (2016), and.…”