Statistical Papers

, Volume 60, Issue 6, pp 2225–2252 | Cite as

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

  • Amit Ghosh
  • Chanchal KunduEmail author
Regular Article


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.


Cumulative residual (past) inaccuracy Conditionally specified model Conditional proportional (reversed) hazard rate Usual stochastic order 

Mathematics Subject Classification

Primary: 62B10 Secondary: 94A17 62N05 62H05 



The authors warmly thank the Editor-in-Chief of the journal and two anonymous reviewers for their constructive comments and suggestions leading to an improved version of the manuscript. The financial support (Ref. No. SR/FTP/MS-016/2012) rendered by the Department of Science and Technology, Government of India is acknowledged with thanks by C. Kundu for carrying out this research work.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsRajiv Gandhi Institute of Petroleum TechnologyJaisIndia

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