Some new results on the cumulative residual entropy

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. In this paper, we obtain some new results on these functions. We also define and study the dynamic cumulative past entropy function. Some results are given connecting these measures of a lifetime distribution and that of the associated weighted distribution.     

On generalized dynamic cumulative past entropy measure

Recently, the concept of dynamic cumulative residual entropy and its generalizations has gained much attention among researchers. In this work, a new generalized dynamic cumulative measure in the past lifetime is proposed. Further, some characterization results connecting this new generalized dynamic entropy measure and other reversed measures are obtained.     

Weighted extensions of generalized cumulative residual entropy and their applications

Abstract

In this paper, we consider weighted extensions of generalized cumulative residual entropy and its dynamic(residual) version. Our results include linear transformations, stochastic ordering, bounds, aging class properties and some relationships with other reliability concepts. We also define the conditional weighted generalized cumulative residual entropy and discuss some properties of its. For these concepts, we obtain some characterization results under some assumptions. Finally, we provide an estimator of the new information measure using empirical approach. In addition, we study large sample properties of this estimator.     

Dynamic version of weighted cumulative residual entropy

Some extensions of Shannon entropy to the survival function have been recently proposed. Misagh et al. (2011 Misagh, F., Panahi, Y., Yari, G.H., Shahi, R. (2011, September). Weighted cumulative entropy and its estimation. In: Quality and Reliability (ICQR), 2011, IEEE International conference (pp. 477–480), IEEE.[Crossref] , [Google Scholar]) introduced weighted cumulative residual entropy (WCRE) that was studied more by Mirali et al. (2015 Mirali, M., Baratpour, S., Fakoor, V. (2015). On weighted cumulative residual entropy. Commun. Stat. Theory Methods. doi:10.1080103610926.2015.1053932.[Web of Science ®] , [Google Scholar]). In this article, the dynamic version of WCRE is proposed. Some relationships of this measure with well-known reliability measures and ageing classes are studied and some characterization results for exponential and Rayleigh distributions are provided. Also, a non parametric estimation of dynamic version of WCRE is introduced and its asymptotic behavior is investigated.     

On weighted cumulative residual entropy

In analogy with the weighted Shannon entropy proposed by Belis and Guiasu (1968 Belis, M., Guiasu, S. (1968). A quantitative-qualitative measure of information in cybernetic systems. IEEE Trans. Inf. Th. IT-4:593–594.[Crossref], [Web of Science ®] , [Google Scholar]) and Guiasu (1986 Guiasu, S. (1986). Grouping data by using the weighted entropy. J. Stat. Plann. Inference 15:63–69.[Crossref], [Web of Science ®] , [Google Scholar]), we introduce a new information measure called weighted cumulative residual entropy (WCRE). This is based on the cumulative residual entropy (CRE), which is introduced by Rao et al. (2004 Rao, M., Chen, Y., Vemuri, B.C., Wang, F. (2004). Cumulative residual entropy: a new measure of information. IEEE Trans. Info. Theory 50(6):1220–1228.[Crossref], [Web of Science ®] , [Google Scholar]). This new information measure is “length-biased” shift dependent that assigns larger weights to larger values of random variable. The properties of WCRE and a formula relating WCRE and weighted Shannon entropy are given. Related studies of reliability theory is covered. Our results include inequalities and various bounds to the WCRE. Conditional WCRE and some of its properties are discussed. The empirical WCRE is proposed to estimate this new information measure. Finally, strong consistency and central limit theorem are provided.     

On the generalized cumulative residual entropy weighted distributions

Recently, Feizjavadian and Hashemi (2015 Feizjavadian, S.H., Hashemi, R. (2015). Mean residual weighted versus the length-biased Rayleigh distribution. J. Stat. Comput. Simul. 85:2823–2838.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) introduced and studied the mean residual weighted (MRW) distribution as an alternative to the length-biased distribution, by using the concepts of the mean residual lifetime and the cumulative residual entropy (CRE). In this article, a new sequence of weighted distributions is introduced based on the generalized CRE. This sequence includes the MRW distribution. Properties of this sequence are obtained generalizing and extending previous results on the MRW distribution. Moreover, expressions for some known distributions are given, and finite mixtures between the new sequence of weighted distributions and the length-biased distribution are studied. Numerical examples are given to illustrate the new results.     

Characterization results based on dynamic Tsallis cumulative residual entropy

In this article, the concept of cumulative residual entropy (CRE) given by Rao et al. (2004 Rao, M., Chen, Y., Vemuri, B.C., Wang, F. (2004). Cumulative residual entropy: A new measure of information. IEEE Trans. Inf. Theory 50:1220–1228.[Crossref], [Web of Science ®] , [Google Scholar]) is extended to Tsallis entropy function and dynamic version, both residual and past of it. We study some properties and characterization results for these generalized measures. In addition, we provide some characterization results of the first-order statistic based on the Tsallis survival entropy.     

Bivariate extension of generalized cumulative past entropy

The cumulative past entropy (CPE) of order α, a dual measure of cumulative residual entropy (CRE) of order α, has recently been proposed as a suitable extension of CPE. In this article, we extend the definition of (dynamic) CPE of order α (DCPE(α)) to bivariate setup and obtain some of its properties including bounds. We also look into the problem of extending DCPE(α) for conditionally specified models. Several properties, including monotonicity, and bounds of DCPE(α) are obtained for conditional distributions. Along with some characterization results it is shown that the proposed generalized measure uniquely determines the distribution function. Moreover, we also propose a stochastic order based on this measure and prove interrelation with some existing stochastic orders.     

On the dynamic cumulative residual entropy

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.     

On generalized conditional cumulative residual inaccuracy measure

Abstract

Recently, the notion of cumulative residual Rényi’s entropy has been proposed in the literature as a measure of information that parallels Rényi’s entropy. Motivated by this, here we introduce a generalized measure of it, namely cumulative residual inaccuracy of order α. We study the proposed measure for conditionally specified models of two components having possibly different ages called generalized conditional cumulative residual inaccuracy measure. Several properties of generalized conditional cumulative residual inaccuracy measure including the effect of monotone transformation are investigated. Further, we provide some bounds on using the usual stochastic order and characterize some bivariate distributions using the concept of conditional proportional hazard rate model.     

A quantile-based generalized dynamic cumulative measure of entropy

The cumulative residual entropy (CRE), introduced by Rao et al. (2004), is a new measure of uncertainty and viewed as a dynamic measure of uncertainty. Asadi and Zohrevand (2007) proposed a dynamic form of the CRE, namely dynamic CRE. Recently, Kumar and Taneja (2011) introduced a generalized dynamic CRE based on the Varma entropy introduced by Varma (1966) and called it dynamic CRE of order α and type β. In the present article, we introduce a quantile version of the dynamic CRE of order α and type β and study its properties. For this measure, we obtain some characterization results, aging classes properties, and stochastic comparisons.     

On cumulative entropies

In analogy with the cumulative residual entropy recently proposed by Wang et al. [2003a. A new and robust information theoretic measure and its application to image alignment. In: Information Processing in Medical Imaging. Lecture Notes in Computer Science, vol. 2732, Springer, Heidelberg, pp. 388–400; 2003b. Cumulative residual entropy, a new measure of information and its application to image alignment. In: Proceedings on the Ninth IEEE International Conference on Computer Vision (ICCV’03), vol. 1, IEEE Computer Society Press, Silver Spring, MD, pp. 548–553], we introduce and study the cumulative entropy, which is a new measure of information alternative to the classical differential entropy. We show that the cumulative entropy of a random lifetime X can be expressed as the expectation of its mean inactivity time evaluated at X. Hence, our measure is particularly suitable to describe the information in problems related to ageing properties of reliability theory based on the past and on the inactivity times. Our results include various bounds to the cumulative entropy, its connection to the proportional reversed hazards model, and the study of its dynamic version that is shown to be increasing if the mean inactivity time is increasing. The empirical cumulative entropy is finally proposed to estimate the new information measure.     

Further results on the residual quantile entropy

It was shown that the decreasing mean residual life class implies the decreasing residual quantile entropy class and the decreasing residual quantile entropy class is not closed under formation of mixture. The less quantile entropy order was proved to be closed under the accelerated life models and the generalized order statistics models. Meanwhile, bounds of the entropy and the residual quantile entropy of some aging classes were established.     

Bivariate extension of (dynamic) cumulative past entropy

Recently, the concept of cumulative residual entropy (CRE) has been studied by many researchers in higher dimensions. In this article, we extend the definition of (dynamic) cumulative past entropy (DCPE), a dual measure of (dynamic) CRE, to bivariate setup and obtain some of its properties including bounds. We also look into the problem of extending DCPE for conditionally specified models. Several properties, including monotonicity, and bounds of DCPE are obtained for conditional distributions. It is shown that the proposed measure uniquely determines the distribution function. Moreover, we also propose a stochastic order based on this measure.     

Renyi's entropy for residual lifetime distribution

In the present paper, we introduce and study Renyi's information measure (entropy) for residual lifetime distributions. It is shown that the proposed measure uniquely determines the distribution. We present characterizations for some lifetime models. Further, we define two new classes of life distributions based on this measure. Various properties of these classes are also given.     

Some results on the mean residual waiting time of records

Often, in reliability theory, risk analysis, renewal processes and actuarial studies, mean residual life function or life expectancy plays an important role in studying the conditional tail measure of lifetime data. In this paper, we introduce the notion of the mean residual waiting time of records and present some monotonic and aging properties. Sharp bounds for the mean residual waiting time of records are also investigated.     

Estimating utility functions using generalized maximum entropy

This paper estimates von Neumann and Morgenstern utility functions using the generalized maximum entropy (GME), applied to data obtained by utility elicitation methods. Given the statistical advantages of this approach, we provide a comparison of the performance of the GME estimator with ordinary least square (OLS) in a real data small sample setup. The results confirm the ones obtained for small samples through Monte Carlo simulations. The difference between the two estimators is small and it decreases as the width of the parameter support vector increases. Moreover, the GME estimator is more precise than the OLS one. Overall, the results suggest that GME is an interesting alternative to OLS in the estimation of utility functions when data are generated by utility elicitation methods.     

Conditional mean residual life functions

In this paper a conditional mean residual life in the context of reliability theory is introduced. The properties of the conditional mean residual life are studied. Various characterizations by the conditional mean residual life are proposed.     

Mean residual weighted versus the length-biased Rayleigh distribution

In some statistical applications, data may not be considered as a random sample of the whole population and some subjects have less probability of belonging to the sample. Consequently, statistical inferences for such data sets, usually yields biased estimation. In such situations, the length-biased version of the original random variable as a special weighted distribution often produces better inferences. An alternative weighted distribution based on the mean residual life is suggested to treat the biasedness. The Rayleigh distribution is applied in many real applications, hence the proposed method of weighting is performed to produce a new lifetime distribution based on the Rayleigh model. In addition, statistical properties of the proposed distribution is investigated. A simulation study and a real data set are prepared to illustrate that the mean residual weighted Rayleigh distribution gives a better fit than the original and also the length-biased Rayleigh distribution.     

On weighted generalized entropy

In the literature of information theory, the concept of generalized entropy has been proposed and the length-based shift dependent information measure has been studied. In this paper, the concept of weighted generalized entropy has been introduced. The properties of weighted generalized residual entropy and weighted generalized past entropy are also discussed.