Bivariate extension of (dynamic) cumulative residual and past inaccuracy measures

Abstract

In a recent paper, Kundu et al. (Metrika 79:335-356, 2016) study the notion of cumulative residual inaccuracy (CRI) and cumulative past inaccuracy (CPI) measures in univariate setup as a generalization of cumulative residual entropy and cumulative past entropy, respectively. Here we address the question of extending the definition of CRI (CPI) to bivariate setup and study their properties. We also prolong these measures to conditionally specified models of two components having possibly different ages or failed at different time instants called conditional CRI (CCRI) and conditional CPI (CCPI), respectively. We provide some bounds on using usual stochastic order and investigate several properties of CCRI (CCPI) including the effect of linear transformation. Moreover, we characterize some bivariate distributions. KeywordsCumulative residual (past) inaccuracy • Conditionally specified model • Conditional proportional (reversed) hazard rate • Usual stochastic order

On generalized conditional cumulative past inaccuracy measure

“…where t 1 , t 2 0. Several aspects of (2.1)-(2.2) have recently been discussed in Ghosh and Kundu [13]. Along a similar line, CPI(α) for X i and Y i , i = 1, 2, called generalized conditional CPI (GCCPI) measure, can be defined as…”

“…For further applications and perspectives of CPI along the same line, one may refer to Ghosh and Kundu [13].…”

“…Here X i represents the conditional distributions of X i subject to the condition that failure of the first component had occurred in (0, t 1 ) and the second failed before t 2 . Recently, Ghosh and Kundu [13] and Kundu and Kundu [26] have considered the bivariate extension of (1.4) and CPE of order α, respectively, for marginal and conditional past lifetimes. Motivated by this, in this article we extend the concept of CPI(α) in bivariate setup with focus on marginal and conditional past lifetimes and study their properties useful in reliability modeling.…”

“…It is worthwhile to mention that the concepts in past time are more appropriate than those truncated from below when the observations are predominantly from left tail. For some recent work on conditionally specified models, we refer to Navarro et al [32], Ahmadi et al [2], Ghosh and Kundu [14], [13], [15], Kayal and Sunoj [18], Kundu and Kundu [25], [26], and the references therein.…”

A Cumulative Residual Inaccuracy Measure for Coherent Systems at Component Level and Under Nonhomogeneous Poisson Processes

“…Kundu et al [22] considered the measures given by Kumar and Taneja [20,21] and obtained several properties when random variables are left, right, and doubly truncated. Quite recently, bivariate extensions of cumulative residual (past) inaccuracy measures have been discussed by Goosh and Kundu [14]. This paper consists of two parts and proceeds as follows.…”