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Connections of Gini, Fisher, and Shannon by Bayes risk under proportional hazards

Part of: Decision theory Communication, information Survival analysis and censored data Sufficiency and information

Published online by Cambridge University Press:  30 November 2017

Majid Asadi ,
Nader Ebrahimi  and
Ehsan S. Soofi
Majid Asadi*
Affiliation:
University of Isfahan and Institute of Research in Fundamental Sciences (IPM)
Nader Ebrahimi*
Affiliation:
Northern Illinois University
Ehsan S. Soofi*
Affiliation:
University of Wisconsin-Milwaukee
*
* Postal address: Department of Statistics, University of Isfahan, Isfahan, 81744, Iran. Email address: [email protected]
** Postal address: Division of Statistics, Northern Illinois University, DeKalb, IL 60155, USA. Email address: [email protected]
*** Postal address: Lubar School of Business, University of Wisconsin-Milwaukee, P.O. Box 742, Milwaukee, WI 53201, USA. Email address: [email protected]
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Abstract

The proportional hazards (PH) model and its associated distributions provide suitable media for exploring connections between the Gini coefficient, Fisher information, and Shannon entropy. The connecting threads are Bayes risks of the mean excess of a random variable with the PH distribution and Bayes risks of the Fisher information of the equilibrium distribution of the PH model. Under various priors, these Bayes risks are generalized entropy functionals of the survival functions of the baseline and PH models and the expected asymptotic age of the renewal process with the PH renewal time distribution. Bounds for a Bayes risk of the mean excess and the Gini's coefficient are given. The Shannon entropy integral of the equilibrium distribution of the PH model is represented in derivative forms. Several examples illustrate implementation of the results and provide insights for potential applications.

Keywords

Entropy functionalgeneralized entropyequilibrium distributionescort distributionexcess costmean residual lifecumulative hazard

MSC classification

Primary: 62B10: Information-theoretic topics 62C05: General considerations 62N05: Reliability and life testing 94A15: Information theory, general
Secondary: 94A17: Measures of information, entropy
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 4 , December 2017 , pp. 1027 - 1050
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Abe, S. (2000). Axioms and uniqueness theorem for Tsallis entropy. Phys. Lett. A 271, 74–79. Google Scholar
[2] Ardakani, O. M., Ebrahimi, N. and Soofi, E. S. (2018). Ranking forecasts by stochastic error distance, information, and reliability measures. To appear in Internat. Statist. Rev. Google Scholar
[3] Ardakani, O. M., Asadi, M., Ebrahimi, N. and Soofi, E. S. (2017). On the Bayes risk of mean residual with system reliability and empirical applications. Submitted. Google Scholar
[4] Asadi, M. and Zohrevand, Y. (2007). On the dynamic cumulative residual entropy. J. Statist. Planning Infer. 137, 1931–1941. Google Scholar
[5] Asadi, M., Ebrahimi, N., Hamedani, G. G. and Soofi, E. S. (2004). Maximum dynamic entropy models. J. Appl. Prob. 41, 379–390. Google Scholar
[6] Asadi, M., Ebrahimi, N., Soofi, E. S. and Zarezadeh, S. (2014). New maximum entropy methods for modeling lifetime distributions. Naval Res. Logistics 61, 427–434. CrossRefGoogle Scholar
[7] Bairamov, I. and Arnold, B. C. (2008). On the residual lifelengths of the remaining components in an n - k + 1 out of n system. Statist. Prob. Lett. 78, 945–952. CrossRefGoogle Scholar
[8] Bercher, J.-F. (2012). A simple probabilistic construction yielding generalized entropies and divergences, escort distributions and q-Gaussians. Physica A 391, 4460–4469. Google Scholar
[9] Borland, L., Plastino, A. R. and Tsallis, C. (1998). Information gain within nonextensive thermostatistics. J. Math. Phys. 39, 6490–6501. (Erratum: 40 (1999), 2196 Google Scholar
[10] Dabrowska, D. M. and Doksum, K. A. (1988). Estimation and testing in a two-sample generalized odds-rate model. J. Amer. Statist. Assoc. 83, 744–749. Google Scholar
[11] Di Crescenzo, A. and Longobardi, M. (2002). Entropy-based measure of uncertainty in past lifetime distributions. J. Appl. Prob. 39, 434–440. CrossRefGoogle Scholar
[12] Di Crescenzo, A. and Toomaj, A. (2015). Extension of the past lifetime and its connection to the cumulative entropy. J. Appl. Prob. 52, 1156–1174. Google Scholar
[13] Ebrahimi, N., Soofi, E. S. and Soyer, R. (2013). When are observed failures more informative than observed survivals? Naval Res. Logistics 60, 102–110. Google Scholar
[14] Havrda, J. and Charvát, F. (1967). Quantification method of classification processes: Concept of structural a-entropy. Kybernetika 3, 30–35. Google Scholar
[15] Lomnicki, Z. A. (1966). A Note on the Weibull renewal process. Biometrika 53, 375–381. Google Scholar
[16] Maasoumi, E. (1986). The measurement and decomposition of multi-dimensional inequality. Econometrica 54, 991–998. Google Scholar
[17] Navarro, J., del Aguila, Y. and Asadi, M. (2010). Some new results on the cumulative residual entropy. J. Statist. Planning Infer. 140, 310–322. Google Scholar
[18] Oja, H. (1981). On location, scale, skewness and kurtosis of univariate distributions. Scand. J. Statist. 8, 154–168. Google Scholar
[19] Rao, M., Chen, Y., Vemuri, B. C. and Wang, F. (2004). Cumulative residual entropy: a new measure of information. IEEE Trans. Inf. Theory 50, 1220–1228. Google Scholar
[20] Ross, S. M. (1983). Stochastic Processes. John Wiley, New York. Google Scholar
[21] Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York. Google Scholar
[22] Shalit, H. and Yitzhaki, S. (2005). The Mean-Gini efficient portfolio frontier. J. Financial Res. XXVIII, 59–75. Google Scholar
[23] Tsallis, C. (1988). Possible generalization of Boltzmann-Gibbs statistics. J. Statist. Phys. 52, 479–487. Google Scholar
[24] Vakili-Nezhaad, G. R. and Mansoori, G. A. (2004). An application of non-extensive statistical mechanics to nanosystems. J. Comput. Theort. Nanosci. 1, 227–229. Google Scholar
[25] Yannaros, N. (1994). Weibull renewal processes. Ann. Inst. Statist. Math. 46, 641–648. Google Scholar
[26] Yannaros, N. (1988). On Cox processes and gamma renewal processes. J. Appl. Prob. 25, 423–427. Google Scholar
[27] Yitzhaki, S. (1998). More than a dozen alternative ways of spelling Gini. Res. Econom. Ineq. 8, 13–30. Google Scholar
[28] Zografos, K. and Nadarajah, S. (2005). Survival exponential entropies. IEEE Trans. Inf. Theory 51, 1239–1246. Google Scholar