Cumulative Past Entropy Research Articles

A note on interval cumulative past Renyi entropy

In analogy with cumulative residual entropy, Di Crescenzo and Longobardi (2009) proposed cumulative past entropy which served as the basis for many following works. To this end, we define cumulative past Renyi entropy for double truncated random variable which we call as interval cumulative past Renyi entropy (ICPRE). Some properties, such as effect of monotone transformation and bounds of ICPRE, are studied. Furthermore, it is shown that the proposed measure uniquely determines the distribution function and characterizes certain lifetime distributions. Finally, non-parametric estimators of ICPRE are obtained and the best proposed estimator is investigated with the help of simulated data set and real-life data set.

Information analysis for past life based on geometric vitality function of k-records

The past geometric vitality function is a helpful tool in reliability and survival studies for describing various properties of a system when it breaks down in In this article, we introduce the concept of past GVF in k-records and give the expression of the proposed measure in terms of expectation of truncated gamma distributed random variable arising from any continuous distribution. In the context of information, the study describes the importance of constructing k-records from a random sample. Further, some well-known properties were analyzed using the proposed measure. The study establishes a simple estimator for the past GVF of k-records and presents a numerical illustration based on a real-life data set. In addition to this, we present the numerical comparison of performance of past GVF, cumulative past entropy and dynamic cumulative past entropy in the context of k-records with the help of a real data set. Finally, based on a real-life data set we discuss the aging pattern of the system based on past GVF of k-record values.

Weighted fractional generalized cumulative past entropy and its properties

Weighted fractional generalized cumulative past entropy and its properties

Extended fractional cumulative past and paired [formula omitted]-entropy measures

Very recently, extended fractional cumulative residual entropy (EFCRE) has been proposed by Foroghi et al., (2022). In this paper, we introduce extended fractional cumulative past entropy (EFCPE), which is a dual of the EFCRE. The newly proposed measure depends on the logarithm of fractional order and the cumulative distribution function (CDF). Various properties of the EFCPE have been explored. This measure has been extended to the bivariate setup. Furthermore, the conditional EFCPE is studied and some of its properties are provided. The EFCPE for inactivity time has been proposed. In addition, the extended fractional cumulative paired ϕ-entropy has been introduced and studied. The proposed EFCPE has been estimated using empirical CDF. Furthermore, the EFCPE is studied for coherent systems. A validation of the proposed measure is provided using logistic map. Finally, an application is reported.

English

In this paper, we obtain lower bounds for the variance of a function of random variables in terms of measures of reliability and entropy. Also based on the obtained characterization via the lower bounds for the variance of a function of random variable X, we find a characterization of the weighted function corresponding to density function f(x), in terms of Chernoff-type inequalities. Subsequently, we obtain monotonic relationships between variance residual life and dynamic cumulative residual entropy and between variance past lifetime and dynamic cumulative past entropy. Moreover, we find lower bounds for the variance of functions of weighted random variables with specific weight functions applicable in reliability under suitable conditions.

Characterization Results Based on Generalized Dynamic Information Measure

In this paper, we consider a generalized past entropy of order [Formula: see text] and type [Formula: see text] and have characterized some distributions of random lifetimes in continuous and discrete situations. Further, we propose a weighted version of generalized past entropy for continuous and discrete cases and have proved the characterization results. Also, we study a generalized weighted residual entropy for the discrete case. At the end, the concept of generalized cumulative past entropy are also discussed.

PROPERTIES FOR GENERALIZED CUMULATIVE PAST MEASURES OF INFORMATION

AbstractThe Shannon entropy based on the probability density function is a key information measure with applications in different areas. Some alternative information measures have been proposed in the literature. Two relevant ones are the cumulative residual entropy (based on the survival function) and the cumulative past entropy (based on the distribution function). Recently, some extensions of these measures have been proposed. Here, we obtain some properties for the generalized cumulative past entropy. In particular, we prove that it determines the underlying distribution. We also study this measure in coherent systems and a closely related generalized past cumulative Kerridge inaccuracy measure.

English

The notion of cumulative past inaccuracy (CPI) measure has recently been proposed in the literature as a generalization of cumulative past entropy (CPE) in univariate as well as bivariate setup. In this paper, we introduce the notion of CPI of order α and study the proposed measure for conditionally specified models of two components failed at different time instants, called generalized conditional CPI (GCCPI). Several properties, including the effect of monotone transformation and bounds of GCCPI are discussed. Furthermore, we characterize some bivariate distributions under the assumption of conditional proportional reversed hazard rate model. Finally, the role of GCCPI in reliability modeling has also been investigated for a real-life problem.

Bivariate extension of generalized cumulative past entropy

ABSTRACTThe 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.

Bivariate extension of (dynamic) cumulative residual and past inaccuracy measures

In a recent paper, Kundu et al. (Metrika 79:335–356, 2016) study the notion of cumulative residual inaccuracy (CRI) and cumulative past inaccuracy (CPI) measures in univariate setup as a generalization of cumulative residual entropy and cumulative past entropy, respectively. Here we address the question of extending the definition of CRI (CPI) to bivariate setup and study their properties. We also prolong these measures to conditionally specified models of two components having possibly different ages or failed at different time instants called conditional CRI (CCRI) and conditional CPI (CCPI), respectively. We provide some bounds on using usual stochastic order and investigate several properties of CCRI (CCPI) including the effect of linear transformation. Moreover, we characterize some bivariate distributions.

Bivariate extension of (dynamic) cumulative past entropy

ABSTRACTRecently, 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.

On generalized dynamic cumulative past entropy measure

ABSTRACTRecently, 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.

Further results on discrete mean past lifetime

The present paper aims at studying the mean past lifetime of a discrete random variable. The notion of discrete mean past lifetime is studied in relation to the concepts of reversed hazard rate, reversed lack of memory property, and cumulative past entropy. New classes of distributions characterized by particular forms of discrete mean past life are also investigated. Implications of an increasing mean past lifetime on other reliability notions are studied and finally some bivariate generalizations are discussed.

On study of dynamic survival and cumulative past entropies

Recently, a new class of measure of uncertainty, called “dynamic survival entropy”, has been defined and studied in the literature. Based on this entropy, DSE(α) ordering, IDSE(α), and DDSE(α) classes of life distributions are defined and some results are studied. In this paper, our main aim is to prove some more results of the ordering and the aging classes of life distributions mentioned above. Some important distributions such as exponential, Pareto, Pareto II, and finite range distributions are also characterized. Here we have defined cumulative past entropy and proved some interesting results.

Doubly truncated (interval) cumulative residual and past entropy

Two measures of uncertainty, the cumulative residual entropy (CRE) and doubly truncated Shannon entropy have been introduced respectively by Rao et al. (2004) and Sunoj et al. (2009). In this paper we proposed the doubly truncated (interval) cumulative residual entropy (ICRE), which is an extension of CRE and the dynamic CRE defined by Asadi and Zohrevand (2007). We study some properties and characterization of this measure, including its connections with doubly truncated Shannon entropy and mean residual life. Also, the dual measure, doubly truncated (interval) cumulative past entropy (ICPE) is defined and some of its properties are studied. Furthermore, their monotonicity and associated aging classes of distributions are discussed and the upper (lower) bound for them are obtained. Finally for exponential distribution all entropies are presented.

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.