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Dynamic generalized information measures

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  • Asadi, Majid
  • Ebrahimi, Nader
  • Soofi, Ehsan S.

Abstract

In many reliability and survival analysis problems the current age of an item under study must be taken into account by information measures of the lifetime distribution. Kullback-Leibler information and Shannon entropy have been considered in this context, which led to information measures that depend on time, and thus are dynamic. This paper develops dynamic information divergence and entropy of order [alpha], also known as Rényi information and entropy, which for [alpha]=1 give the Kullback-Leibler information and Shannon entropy, respectively. We give characterizations of the proportional hazards model, the exponential distribution, and Generalized Pareto distributions in terms of dynamic Rényi information and entropy. It is also shown that dynamic Rényi entropy uniquely determines distributions that have monotone densities. A result that relates dynamic Rényi entropy and hazard rate orderings is given. This result leads to a Maximum Dynamic Entropy of order [alpha] formulation and characterizations of some well-known lifetime models. A dynamic entropy hazard rate inequality is developed as an analog of the well-known entropy moment inequality.

Suggested Citation

  • Asadi, Majid & Ebrahimi, Nader & Soofi, Ehsan S., 2005. "Dynamic generalized information measures," Statistics & Probability Letters, Elsevier, vol. 71(1), pages 85-98, January.
    • Handle: RePEc:eee:stapro:v:71:y:2005:i:1:p:85-98
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    References listed on IDEAS

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    1. Golan, Amos & Perloff, Jeffrey M., 2002. "Comparison of maximum entropy and higher-order entropy estimators," Journal of Econometrics, Elsevier, vol. 107(1-2), pages 195-211, March.
    2. Di Crescenzo, Antonio & Longobardi, Maria, 2004. "A measure of discrimination between past lifetime distributions," Statistics & Probability Letters, Elsevier, vol. 67(2), pages 173-182, April.
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    5. Nader Ebrahimi & S. Kirmani, 1996. "A measure of discrimination between two residual life-time distributions and its applications," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 48(2), pages 257-265, June.
    6. Asadi, Majid & Ebrahimi, Nader, 2000. "Residual entropy and its characterizations in terms of hazard function and mean residual life function," Statistics & Probability Letters, Elsevier, vol. 49(3), pages 263-269, September.
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    1. K. Nair & P. Sankaran & S. Smitha, 2011. "Chernoff distance for truncated distributions," Statistical Papers, Springer, vol. 52(4), pages 893-909, November.
    2. Chanchal Kundu, 2015. "Generalized measures of information for truncated random variables," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(4), pages 415-435, May.
    3. Abbasnejad, M. & Arghami, N.R. & Morgenthaler, S. & Mohtashami Borzadaran, G.R., 2010. "On the dynamic survival entropy," Statistics & Probability Letters, Elsevier, vol. 80(23-24), pages 1962-1971, December.
    4. Farrukh Jamal & Muhammad Ijaz & Ghulam Mustafa & Sadaf Khan & Shakaiba Shafiq, 2024. "On the theory of order statistics of the flexible Lomax distribution," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 34(2), pages 33-45.
    5. Asadi, Majid & Ebrahimi, Nader & Soofi, Ehsan S., 2018. "Optimal hazard models based on partial information," European Journal of Operational Research, Elsevier, vol. 270(2), pages 723-733.
    6. Vikas Kumar & Nirdesh Singh, 2023. "Some Results on Quantile Version of R é $\acute {e}$ nyi Entropy of Order Statistics," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 248-273, February.
    7. Kayal, Suchandan, 2015. "On generalized dynamic survival and failure entropies of order (α,β)," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 123-132.
    8. R. Maya & E. Abdul-Sathar & G. Rajesh & K. Muraleedharan Nair, 2014. "Estimation of the Renyi’s residual entropy of order $$\alpha $$ with dependent data," Statistical Papers, Springer, vol. 55(3), pages 585-602, August.
    9. Nanda, Asok K. & Sankaran, P.G. & Sunoj, S.M., 2014. "Rényi’s residual entropy: A quantile approach," Statistics & Probability Letters, Elsevier, vol. 85(C), pages 114-121.

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