Some Characterization Results on Dynamic Cumulative Residual Tsallis Entropy (original) (raw)

On weighted cumulative residual Tsallis entropy and its dynamic version

Physica A: Statistical Mechanics and its Applications, 2018

Recently, Sati and Gupta [1] have introduced a generalized cumulative residual entropy based on the non-additive Tsallis entropy. The cumulative residual entropy, introduced by Rao et al. [2] is a generalized measure of uncertainty which is applied in reliability and image alignment and non-additive measures of entropy. This entropy finds justifications in many physical, biological and chemical phenomena. In this paper, we derive the weighted form of this measure and call it Weighted Cumulative Residual Tsallis Entropy (WCRTE). Being a "lengthbiased" shift-dependent information measure, WCRTE is related to the differential information in which higher weight is assigned to large values of observed random variables. Based on the dynamic version of this new information measure, we propose ageing classes and it is shown that it can uniquely determine the survival function and Rayleigh distribution. Several properties, including linear transformations, bounds and related results to stochastic orders are obtained for these measures. Also, we identify classes of distributions in which some well-known distributions are maximum dynamic version of WCRTE. The empirical WCRTE is finally proposed to estimate the new information measure.

Some new results on the cumulative residual entropy

Journal of Statistical Planning and Inference, 2010

The residual entropy function is a relevant dynamic measure of uncertainty in reliability and survival studies. Recently, Rao et al. [2004. Cumulative residual entropy: a new measure of information. IEEE Transactions on Information Theory 50, 1220-1228] and Asadi and Zohrevand [2007. On the dynamic cumulative residual entropy. Journal of Statistical Planning and Inference 137, 1931-1941] define the cumulative residual entropy and the dynamic cumulative residual entropy, respectively, as some new measures of uncertainty. They study some properties and applications of these measures showing how the cumulative residual entropy and the dynamic cumulative residual entropy are connected with the mean residual life function.

On the dynamic cumulative residual entropy

Journal of Statistical Planning and Inference, 2007

Recently, Rao et al. [(2004) Cumulative residual entropy: a new measure of information. IEEE Trans. Inform. Theory 50(6), 1220-1228] have proposed a new measure of uncertainty, called cumulative residual entropy (CRE), in a distribution function F and obtained some properties and applications of that. In the present paper, we propose a dynamic form of CRE and obtain some of its properties. We show how CRE (and its dynamic version) is connected with well-known reliability measures such as the mean residual life time.

Entropy and Residual Entropy Functions and Some Characterization Results

Pakistan Journal of Statistics and Operation Research, 2012

In this paper, we have developed conditions under which the entropy function and the residual entropy function characterize the distribution. We have also studied some stochastic comparisons based on the entropy measure and established relations between entropy comparisons and comparisons with respect to other measures in reliability. Conditions for decreasing (increasing) uncertainty in a residual life distribution are obtained. Some relations between the classes of distribution in reliability and the classes of distribution, based on the monotonic properties of uncertainty, in a residual life distribution are obtained.

A Generalized Measure of Cumulative Residual Entropy

2022

In this work, we introduce a generalized measure of cumulative residual entropy and study its properties. We show that several existing measures of entropy such as cumulative residual entropy, weighted cumulative residual entropy and cumulative residual Tsallis entropy, are all special cases of the generalized cumulative residual entropy. We also propose a measure of generalized cumulative entropy, which includes cumulative entropy, weighted cumulative entropy and cumulative Tsallis entropy as special cases. We discuss generating function approach using which we derive different entropy measures. Finally, using the newly introduced entropy measures, we establish some relationships between entropy and extropy measures.

Weighted survival functional entropy and its properties

Open Physics, 2023

The weighted generalized cumulative residual entropy is a recently defined dispersion measure. This article introduces a new uncertainty measure as a generalization of the weighted generalized cumulative residual entropy, called it the weighted fractional generalized cumulative residual entropy of a nonnegative absolutely continuous random variable, which equates to the weighted fractional Shannon entropy. Several stochastic analyses and connections of this new measure to some famous stochastic orders are presented. As an application, we demonstrate this measure in random minima. The new measure can be used to study the coherent and mixed systems, risk measure, and image processing.

On Shift-Dependent Cumulative Entropy Measures

International Journal of Mathematics and Mathematical Sciences, 2016

Measures of cumulative residual entropy (CRE) and cumulative entropy (CE) about predictability of failure time of a system have been introduced in the studies of reliability and life testing. In this paper, cumulative distribution and survival function are used to develop weighted forms of CRE and CE. These new measures are denominated as weighted cumulative residual entropy (WCRE) and weighted cumulative entropy (WCE) and the connections of these new measures with hazard and reversed hazard rates are assessed. These information-theoretic uncertainty measures are shift-dependent and various properties of these measures are studied, including their connections with CRE, CE, mean residual lifetime, and mean inactivity time. The notions of weighted mean residual lifetime (WMRL) and weighted mean inactivity time (WMIT) are defined. The connections of weighted cumulative uncertainties with WMRL and WMIT are used to calculate the cumulative entropies of some well-known distributions. The ...

On Some Characterization Results of Life Time Distributions using Mathai-Haubold Residual Entropy

In the context of information theory, Shannon's entropy 16 plays an important role. In case, one has information about the current age of the component which can be taken into account for measuring its uncertainty, Shannon's entropy is not suitable as such. Consequently, Ebrahimi 3 proposed an alternative approach for characterization of distribution functions in terms of conditional Shannon's measure of uncertainty. In this paper, we propose generalized residual entropy function for characterization of some life time models. Also, upper and lower bound of hazard rate function in terms of generalized residual entropy have been obtained. Based on the proposed measure, we derive the generalized residual entropy function for some continuous lifetime models.

A quantile-based study of cumulative residual Tsallis entropy measures

Physica A: Statistical Mechanics and its Applications, 2018

Highlights (to review)  We have proposed a quantile-based cumulative residual Tsallis entropy (CRTE) and quantile-based CRTE for order statistics. It is shown that unlike the distribution function approach, these quantile-based measures uniquely determine the distribution function through an explicit relationship.  We have illustrated the usefulness of these measures in the quantile set up and in situations where there is no tractable distribution function available.  We have obtained some characterizations for distributions using the quantile versions of CRTE and its order statistics and derived certain bounds.

Notes on Cumulative Entropy as a Risk Measure

Stochastics and Quality Control, 2019

Di Crescenzo and Longobardi [Di Crescenzo and Longobardi, On cumulative entropies, J. Statist. Plann. Inference 139 (2009), no. 12, 4072-4087] proposed the cumulative entropy (CE) as an alternative to the differential entropy. They presented an estimator of CE using empirical approach. In this paper, we consider a risk measure based on CE and compare it with the standard deviation and the Gini mean difference for some distributions. We also make empirical comparisons of these measures using samples from stock market in members of the Organization for Economic Cooperation and Development (OECD) countries.