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

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.

Stochastic Properties of Fractional Generalized Cumulative Residual Entropy and Its Extensions

“…In the decision theoretic framework, the MRL function is the optimal prediction of under the conditional quadratic loss function as the mean of the PDF In other words, we have for all The function is a local risk measure, given the value the threshold t takes. Its global risk of the MRL function of X is the Bayes risk where denotes the average based on the prior PDF for the threshold t (see Ardakani et al [ 15 ] and Asadi et al [ 16 ] for more details). The following theorem provides expressions for under different priors.…”

Variants of Mixtures: Information Properties and Applications

“…Based on the variance representation in (4.5), Asadi et al (2017) calledĨ(Θ) the Bayes risk of Fisher information and in Asadi et al (2019a) it is called Bayes-Fisher information, for short.…”

“…This measure is used in Asadi et al (2017), Ardakani et al (2018), andArdakani et al (2020), where it is referred to as the Bayes risk of m(τ), which is a misnomer. The Bayes risk of m(τ) is given by the E τ [Var(X − τ|X > τ)].…”

On reconsidering entropies and divergences and their cumulative counterparts: Csiszár's, DPD's and Fisher's type cumulative and survival measures

“…These papers were the basis for an increasing interest for the definition of informational measures in the context of the paper by Rao et al [73]. In this direction, entropy and divergence type measures are introduced and studied, in the framework of the cumulative distribution function or in terms of the respective survival function, in the papers by Rao [72], Zografos and Nadarajah [98], Di Crescenzo and Longobardi [31], Baratpour and Rad [12], Park et al [66] and the subsequent papers by Di Crescenzo and Longobardi [32], Klein et al [46], Asadi et al [6,7], Park et al [67], Klein and Doll [45], among many others. In these and other treatments, entropy type measures and Kullback-Leibler type divergences have been mainly received the attention of the authors.…”

“…In these and other treatments, entropy type measures and Kullback-Leibler type divergences have been mainly received the attention of the authors. Entropy and divergence type measures are also considered in the papers by Klein et al [46], Asadi et al [6], Klein and Doll [45] by combining the cumulative distribution function and the survival function. However, to the best of our knowledge, it seems that there has not yet appeared in the existing literature a definition of the broad class of Csiszár's type -divergences or a definition of the density power divergence in the framework which was initiated in the paper by Rao et al [73].…”