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A Conditioned Kullback-Leibler Divergence Measure through Compensator Processes and its Relationship to Cumulative Residual Inaccuracy Measure with Applications

Author

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  • Vanderlei da Costa Bueno

    (São Paulo University)

  • Narayanaswamy Balakrishnan

    (McMaster University)

Abstract

Kullback-Leibler divergence measure between two random variables is quite useful in many contexts and has received considerable attention in numerous fields including statistics, physics, probability, and reliability theory. A cumulative Kullback-Leibler divergence measure has been proposed recently as a suitable extension of this measure upon replacing density functions by cumulative distribution functions. In this paper, we study a dynamic version of it by using a point process martingale approach conditioned on an observed past. Interestingly, this concept is identical to cumulative residual inaccuracy measure introduced by (Bueno and Balakrishnan (Probab Eng Sci 36:294-319, 2022). We also extend the concept of relative cumulative residual information generating measure to a conditional one and get Kullback-Leibler divergence measure through it. We further extend the new versions to non-explosive univariate point processes. In particular, we apply the conditioned Kullback-Leibler divergence to compare measures between two non-explosive point processes. Several applications of the established results are presented, including to a general repair process, minimal repair point process, coherent systems, Markov-modulated Poisson processes and Markov chains.

Suggested Citation

  • Vanderlei da Costa Bueno & Narayanaswamy Balakrishnan, 2025. "A Conditioned Kullback-Leibler Divergence Measure through Compensator Processes and its Relationship to Cumulative Residual Inaccuracy Measure with Applications," Methodology and Computing in Applied Probability, Springer, vol. 27(2), pages 1-25, June.
    • Handle: RePEc:spr:metcap:v:27:y:2025:i:2:d:10.1007_s11009-025-10153-x
    DOI: 10.1007/s11009-025-10153-x
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    References listed on IDEAS

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    1. Murali Rao, 2005. "More on a New Concept of Entropy and Information," Journal of Theoretical Probability, Springer, vol. 18(4), pages 967-981, October.
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