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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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