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Keyfitz entropy index is a new indicator that measures the sensitivity of life expectancy to a change in mortality rate. Understanding the characteristics of this indicator can significantly help life table studies in survival analysis. In this paper, we take a closer look at some mathematical properties of Keyfitz entropy index. First, using theoretical studies we show that in some cases this index belongs to the interval [0, 1] and in other cases, it is greater than 1. We also provide two inequalities for Keyfitz entropy using Shannon entropy and pth central moments of random variables. Then, we present an empirical value for it. This value can be useful and provides initial information about Keyfitz entropy value to the researcher, especially before estimating the population survival function with common parametric and nonparametric methods. Second, we propose a new nonparametric method for estimating the survival function in life table using information theory which applies existing information from the population, such as average and moments. The survival function estimated by this method provides the maximum value for Keyfitz entropy indicating the maximum sensitivity of life expectancy to changes in age-specific mortality rates. We also demonstrate that the survival function estimated by this method can be a powerful competitor to its counterparts which are estimated by common parametric and nonparametric methods.

In this paper, some new results and inequalities of dynamic failure entropy (DFE) defined by Abbasnejad (2011) will be obtained. Then weighted versions of failure and survival entropies of order β introduced and some properties and inequalities based on these uncertainty measures found. Then weighted DFE will be introduced and some properties investigated. Finally, empirical cumulative entropy is applied to estimate the new information measures.

Estimation of lifetime parameters such as reliability and hazard rate is necessary in many systems. Each method of estimation suffers from its own problems such as complexity of computations, high ...

Estimation of reliability and hazard rate is necessary in many applications. To this aim, different methods of estimation have been employed. Each method suffers from its own problems such as complexity of calculations, high risk and so on. Toward this end, this study employed a new method, E-Bayesian, for estimating the parametric functions of the Generalized Inverted Exponential distribution, which is one of the most noticeable distributions in lifetime studies. Relations are derived under a squared error loss function, type-II censoring and a conjugate prior. E-Bayesian estimations are obtained based on different priors of the hyperparameters to investigate the influence of different prior distributions on these estimations. The asymptotic behaviors of E-Bayesian estimations and relations among them have been investigated. Finally, a comparison among the maximum likelihood, Bayes and E-Bayesian estimations in different sample sizes are made, using a real data and the Monte Carlo simulation. Simulations show that the new presented method is more efficient than previous methods and is also easy to operate. Also, some comparisons among the results of Generalized Inverted Exponential distribution, Exponential distribution and Generalized Exponential distribution are provided.

In this paper, we focused on computing the minimal relative entropy between the original probability and all of the equivalent martin gale measure for the Levy process. For this purpose, the quasiM...

In this study, we propose a novel approach for the valuation of swing options. Swing options are a kind of American options with multiple exercise rights traded in energy markets. Longstaff and Schwartz have suggested a regression-based Monte Carlo method known as the least-squares Monte Carlo (LSMC) method to value American options. In this work, first we introduce the LSMC method for the pricing of swing options. Then, to achieve a desired accuracy for the price estimation, we combine the idea of LSMC with multilevel Monte Carlo (MLMC) method. Finally, to illustrate the proper behavior of this combination, we conduct numerical results based on the Black–Scholes model. Numerical results illustrate the efficiency of the proposed approach.

In this paper, we propose three discrete analogues of the Burr XII-gamma distribution to meet the need of modelling discrete lifetimes in reliability. It is shown that hazard rate functions of discrete analogues are increasing, U-shaped and modified unimodal. Then we investigate estimation of their parameters using three methods (maximum likelihood, moments and empirical). From the results of extensive simulation runs, their accuracies are compared. Finally, we fit three proposed discrete distributions and discrete Burr XII and discrete gamma distributions to a discrete real data set and use AIC to compare them.

In this article, we estimate the parameters of exponential Pareto II distribution by two new methods. The first one is based on the principle of maximum entropy (POME) and the second is by Kullback-Leibler divergence of survival function (KLS). Monte Carlo simulated data are used to evaluate these methods and compare them with the maximum likelihood method. Finally, we fit this distribution to a set of real data by estimation procedures.

Following Sir Anthony and Atkinson who started thinking about the insensitivity of the Gini index to income shares of the lower and the upper income groups, a generalization of the classical Gini index was introduced by Kakwani, Donaldson, Weymark and Yitzhaki which is sensitive to both high and low incomes. In this paper, the maximum entropy method is used to estimate the underlying true income share function based on the limited information of the generalized Gini index about the income shares of a population's percentiles. The income share function is estimated through maximizing both the Shannon entropy and the second-order entropy. In the end, through parametric bootstrap and analyzing a real dataset, the results are compared with the estimator of the share function, which is obtained based on the total information. In contrast to the classic Gini index, the derived share function based on the generalized Gini index provides more accurate approximations for income shares of the lower and the ...

Recently, a new entropy based divergence measure has been introduced which is much like Kullback-Leibler divergence. This entropy measures the distance between an empirical and a prescribed survival function and is a lot easier to compute in continuous distributions than the K-L divergence. In this paper we show that this distance converges to zero with increasing sample size and we apply it to estimate Weibull parameters. Detailed simulations show a higher performance of the new estimation method than the commonly used maximum likelihood and linear regression methods in Weibull scale parameter estimation. Using unbiasing factors provided in this paper for Weibull shape parameter estimation, one can obtain unbiased estimation for Weibull modulus.

In this paper, based upon the M-Entropy measures, two new models for obtaining the ordered weighted averaging (OWA) operators are propoosed. In these models, it is assumed, according to available information, that the OWA weights are in a decreasing or increasing order. Some properties of the models are analyzed, and the method of Lagrange multipliers is used to provide a direct way to find these weights. The models are solved with a specific level of orness comparing the results with some other related models. The results demonstrate the efficiency of the M-Entropy models in generating the OWA operator weights. © 2012 Wiley Periodicals, Inc.

Recently, Liu (2007) defined a new entropy which measures the distance between a prescribed and an empirical survival function. In this article, we use this measure called Differential Cumulative Entropy (DCE) for Weibull parameters estimation. We show that the DCE method provides biased estimations of the Weibull modulus, but utilizing unbiasing factors derived here we enhance the results. A simulation study shows the higher performance of the new method over commonly used maximum likelihood and linear regression methods in Weibull parameters estimation especially in small sample sizes.

Determination of the ordered weighted averaging (OWA) operators is an important issue in the theory of the OWA operator weights. In this paper, the main existing models for determining the OWA operator weights are outlined and the concept of the Bayesian entropy is introduced. Based upon the Bayesian entropy the maximum Bayesian entropy approach for obtaining the OWA operator weights is proposed. In this model it is assumed, according to previous experiences or from theoretical considerations that a decision maker may have reasons to consider a given prior OWA vector. Finally the new model is solved according to the prior OWA vector with specific level of orness comparing the results with other methods. The results demonstrate the efficiency of our model in generating the OWA operator weights. An applied example is also presented to illustrate the applications of the proposed model.

The Shannon interval entropy function as a useful dynamic measure of uncertainty for two sided truncated random variables has been proposed in the literature of reliability. In this paper, we show that interval entropy can uniquely determine the distribution function. Furthermore, we propose a measure of discrepancy between two lifetime distributions at the interval of time in base of Kullback-Leibler discrimination information. We study various properties of this measure, including its connection with residual and past measures of discrepancy and interval entropy, and we obtain its upper and lower bounds.

Measures of uncertainty in past and residual lifetime distributions have been proposed in the information-theoretic literature. Recently, Di Crescenzo and Longobardi (2006) introduced weighted differential entropy and its dynamic versions. These information-theoretic uncertainty measures are shift-dependent. In this paper, we study the weighted differential information measure for two-sided truncated random variables. This new measure is a generalization of recent dynamic weighted entropy measures. We study various properties of this measure, including its connection with weighted residual and past entropies, and we obtain its upper and lower bounds.

In this paper we have designed an acceptance single sampling plan with inspection errors when the fraction of defective items is a fuzzy number. We have shown that the operating characteristics curve of this plan is like a band having high and low bounds, its width depends on the ambiguity of proportion parameter in the lot when the samples size and acceptance numbers are fixed. A comparison of the single sampling plans with and without inspection errors was done to study the effects upon the characteristics. The results of this comparison show that in the sampling plan with inspection errors, there is a lower operating characteristics band in comparison to a sampling plan without inspection errors for good processing quality. We have also shown that the incorrect classification of a good item reduces the fuzzy probability of acceptance and incorrect classification of a defective item results in a higher fuzzy probability of acceptance.

Expected utility maximization problem is one of the most useful tools in mathematical finance, decision analysis and economics. Motivated by statistical model selection, via the principle of expected utility maximization, Friedman and Sandow (J Mach Learn Res 4:257–291, 2003a) considered the model performance question from the point of view of an investor who evaluates models based on the performance of the optimal strategies that the models suggest. They interpreted their performance measures in information theoretic terms and provided new generalizations of Shannon entropy and Kullback–Leibler relative entropy and called them U-entropy and U-relative entropy. In this article, a utility-based criterion for independence of two random variables is defined. Then, Markov’s inequality for probabilities is extended from the U-entropy viewpoint. Moreover, a lower bound for the U-relative entropy is obtained. Finally, a link between conditional U-entropy and conditional Renyi entropy is derived.

In this paper, we derive the β -entropy for Pareto-type and related distributions. Further, the β -entropy for some weighted versions of these distributions, such as order statistics, proportional hazards, proportional reversed hazards, probability weighted moments, upper record and lower record, is obtained.

The aim of this paper is deriving the exact analytical expressions of entropy for the Pareto-types and related distributions. Entropy for $i^{th}$ order statistics of these distributions corresponding to the random sample size $n$ is introduced. We have shown that all the expressions related to finding corresponding entropy for these families and their order statistics distributions are obtained through some particular techniques via integration. Indeed, these techniques for that improper integrals has its own importance also.

In this paper, we review the utility-based generalization of the Shannon entropy and Kullback-Leibler information measure as the U-entropy and the U-relative entropy that was introduced by Friedman et al. (2007). Then, we derive some relations between the U-relative entropy and other information measures based on a parametric family of utility functions.

In this paper, we define the conditional Renyi entropy and show that the so-called chain rule holds for the Renyi entropy. Then, we introduce a relation for the rate of Renyi entropy and use it to derive the rate of the Renyi entropy for an irreducible-aperiodic Markov chain. We also show that the bound for the Renyi entropy rate is simply the Shannon entropy rate.

In this article, results for some well-known families such as Pareto IV, generalized logistic, generalized gamma, double generalized gamma, generalized normal, inverse gamma, and Weibull, and their related families via the links between entropy, variance, Fisher information, and analog of the Fisher information, are derived.

In the field of the wind energy conversion, a precise determination of the probability distribution of wind speed guarantees an efficient use of the wind energy and enhances the position of wind energy against other forms of energy. The present study thus proposes utilizing an accurate numerical-probabilistic algorithm which is the combination of the Newton’s technique and the maximum entropy (ME) method to determine an important distribution in the renewable energy systems, namely the hyper Rayleigh distribution (HRD) which belongs to the family of Weibull distribution. The HRD is mainly used to model the wind speed and the variations of the solar irradiance level with a negligible error. The purpose of this research is to find the unique solution to an optimization problem which occurs when maximizing Shannon’s entropy. To confirm the accuracy and efficiency of our algorithm, we used the long-term data for the average daily wind speed in Toyokawa for 12 yr to examine the Rayleigh distribution (RD). This data set was obtained from the National Climatic Data Center (NCDC) in Japan. It seems that the RD is more closely fitted to the data. In addition, we presented different simulation studies to check the reliability of the proposed algorithm.

In this paper, we discuss different estimators of the records Weibull distributionparameters and also we apply the Kullback-Leibler divergence of survival function methodto estimate record Weibull parameters. Finally, these methods have been compared usingMonte Carlo simulation.

Recently, the concept of generalized entropy has been proposed in the literature of information theory. In the present paper, we introduce and study the notion of generalized entropy in the interval (t1, t2) as uncertainty measure. It is shown that the suggested information measure uniquely determines the distribution function. Also, its properties has been studied. Some results have been obtained and some distributions such as uniform, exponential, Pareto, power series and finite range have been characterized by doubly truncated (interval) generalized entropy. Further, we describe a few orders based on this entropy and show its properties.

The main purpose for this study is to provide a useful algorithm that combines the Maximum Entropy Method (MEM) and a computational method to predict the unique form of a two-dimensional maximum entropy distributions. In this paper, we present the application of MEM to determine the important bivariate distributions which are very effective in industrial and engineering fields especially in Cybernetics and internet systems. The combination of MEM and numerical method as a proposed algorithm is a useful method with the ability to model datasets which includes both missing and presence datas, The new algorithm provides reasonable estimations for reconstructing the target bivariate distributions which has maximum entropy. We examined the effectiveness of our algorithm to make predictions for two famus distributions. The resulting distributions have minimum error with respect to missing information. In fact, maximum entropy distribution is able to fit the known density given the prior knowledge of the target distribution. So, the MEM modeling procedure can be applied in present form for a variety of applications with precsence-only datasets. Possessing the simple and accurate mathematical formulation and using presence-only data, MEM has become a well-suited method for different kinds of distribution modeling.

In the linear regression method for estimating parameters of a Weibull distribution, multiple flaw distributions may be further evidenced by derivation form the linearity of data from a single Weibull distribution. In this paper, a new technique of estimating multiple Weibull parameters is conducted, compared to commonly used regression probability estimator.

Measures of cumulative residual entropy (CRE) and cumulative entropy (CE) about predictability of failure time of a system have been introduced in [8, 4] respectively. In this paper, we propose the weighted forms of CRE and CE. These information-theoretic uncertainty measures are shift-dependent. We study various properties of these measures, including their connections with CRE and CE and weighted mean residual lifetime. We also obtain the estimators of proposed measures by means of order statistics.

Measure of uncertainty about predictability of failure time of a system at the interval of time has been introduced in [13]. In this paper, a measure of discrepancy between two lifetime distributions at the interval of time is proposed on the basis of Kullback-Leibler discrimination information. We study the Kullback-Leibler measure for two-sided truncated random variables. We study various properties of this measure, including its connection with residual and past measures of discrepancy, and we obtain its upper and lower bounds.

In this paper we derive the exact analytical expressions for the information and covariance matrices of the multivariate Burr and related distributions. These distributions arise as tractable parametric models in reliability, actuarial science, economics, finance and telecommunications. We show that all the calculations can be obtained from one main moment multi dimensional integral whose expression is obtained through some particular change of variables., Comment: submitted to Communications in Statistics

This paper derives a new family of Burr-type distributions as new Burr distribution. This particular skewed distribution that can be used quite effectively in analyzing lifetime data. It is observed that the new distribution has modified unimodal hazard function. Various properties of the new Burr distribution, such that moments, quantile functions, hazard function, and Shannon’s entropy are obtained. The exact form of the probability density function and moments of $$i{\mathrm{th}}$$ -order statistics in a sample of size n from new Burr distribution are derived. Estimation of parameters and change-point of hazard function by the maximum likelihood method are discussed. Change-point of hazard function is usually of great interest in medical or industrial applications. The flexibility of the new model is illustrated with an application to a real data set. In addition, a goodness-of-fit test statistic based on the Renyi Kullback–Leibler information is used.