On weighted Renyi’s entropy for double-truncated distribution

In a recent paper, Nourbakhsh and Yari (2017 Nourbakhsh, M., and G. Yari. 2017. Weighted Renyi’s entropy for lifetime distributions. Communications in Statistics—Theory and Methods 46 (14):7085–98.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) introduce the weighted version of Renyi’s entropy for left/right truncated random variables and studied their properties in context of reliability analysis. In the present communication we extend the notion of weighted Renyi’s entropy for two-sided truncated random variable. In reliability theory and survival analysis, this measure may help to study the quantitative-qualitative information spectrum of a system/component when it fails between two time points. Various aspects of weighted Renyi’s interval entropy have been discussed and some mistakes in the preceding literature have also been corrected. These results generalize and enhance the related existing results that are developed based on weighted Renyi’s entropy for one-sided truncated random variable. Finally, a simulation study is added to provide the estimates of the proposed measure and to demonstrate the performance of the estimates.     

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.     

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.     

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.     

Characterizations based on generalized cumulative residual entropy functions

The Shannon entropy and the cumulative residual entropy (CRE) of a random variable are useful tools in probability theory. Recently, a new concept called generalized cumulative residual entropy (GCRE) of order n was introduced and studied. It is related with the record values of a sequence of i.i.d. random variables and with the relevation transform. In this paper, we show that, under some assumptions, the GCRE function of a fixed order n uniquely determines the distribution function. Some characterizations of particular probability models are obtained from this general result.     

Maximum entropy estimation of income share function from generalized Gini index

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 upper percentiles.     

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 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.     

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.     

Two steps generalized maximum entropy estimation procedure for fitting linear regression when both covariates are subject to error

This paper presents a procedure utilizing the generalized maximum entropy (GME) estimation method in two steps to quantify the uncertainty of the simple linear structural measurement error model parameters exactly. The first step estimates the unknowns from the horizontal line, and then the estimates were used in a second step to estimate the unknowns from the vertical line. The proposed estimation procedure has the ability to minimize the number of unknown parameters in formulating the GME system within each step, and hence reduce variability of the estimates. Analytical and illustrative Monte Carlo simulation comparison experiments with the maximum likelihood estimators and a one-step GME estimation procedure were presented. Simulation experiments demonstrated that the two steps estimation procedure produced parameter estimates that are more accurate and more efficient than the classical estimation methods. An application of the proposed method is illustrated using a data set gathered from the Centre for Integrated Government Services in Delma Island – UAE to predict the association between perceived quality and the customer satisfaction.     

Characterizations of arcsin and related distributions based on a new generalized unimodality

A general characterization for α-unimodal distributions was provided in [Alamatsaz, M.H. 1985: A Note on an Article by Artikis, Acta Mathematica Hungarica 45, 159–162] and its extension to a multivariate case in [Alamatsaz, M.H. 1993: On Characterizations of Exponential and Gamma Distributions, Statistics and Probability Letters 17, 315–319]. Here, by solving the related equations, another generalization for unimodality is presented. As a result of this generalization, a simpler proof of a conjecture, as well as a characterization for generalized arcsin distributions and some generalizations of the author’s earlier works, is obtained. Last, but not the least, it is shown that some elementary methods can be more powerful than some more advanced techniques.     

A new generalized confidence interval for the among-group variance in the heteroscedastic one-way random effects model

Based on the generalized inference idea, a new kind of generalized confidence intervals is derived for the among-group variance component in the heteroscedastic one-way random effects model. We construct structure equations of all variance components in the model based on their minimal sufficient statistics; meanwhile, the fiducial generalized pivotal quantity (FGPQ) can be obtained through solving an implicit equation of the parameter of interest. Then, the confidence interval is derived naturally from the FGPQ. Simulation results demonstrate that the new procedure performs very well in terms of both empirical coverage probability and average interval length.     

On the ratio of generalized variances

This paper examines the ratio of the determinants of two independent central Wishart matrices. The ratio is represented as a product of independent beta type-2 random variables and then the exact density is evaluated for the general case. Different representations and particular cases are also discussed.     

On variable selection in generalized linear and related regression models

This paper is concerned with selection of explanatory variables in generalized linear models (GLM). The class of GLM's is quite large and contains e.g. the ordinary linear regression, the binary logistic regression, the probit model and Poisson regression with linear or log-linear parameter structure. We show that, through an approximation of the log likelihood and a certain data transformation, the variable selection problem in a GLM can be converted into variable selection in an ordinary (unweighted) linear regression model. As a consequence no specific computer software for variable selection in GLM's is needed. Instead, some suitable variable selection program for linear regression can be used. We also present a simulation study which shows that the log likelihood approximation is very good in many practical situations. Finally, we mention briefly possible extensions to regression models outside the class of GLM's.     

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.     

Conditional entropy,entropy density,and strong law of large numbers for generalized controlled tree-indexed Markov chains

In this paper, we first introduces a tree model without degree boundedness restriction namely generalized controlled tree T, which is an extension of some known tree models, such as homogeneous tree model, uniformly bounded degree tree model, controlled tree model, etc. Then some limit properties including strong law of large numbers for generalized controlled tree-indexed non homogeneous Markov chain are obtained. Finally, we establish some entropy density properties, monotonicity of conditional entropy, and entropy properties for generalized controlled tree-indexed Markov chains.     

A new generalized two-sided class of distributions with an emphasis on two-sided generalized normal distribution

The ordinary-G class of distributions is defined to have the cumulative distribution function (cdf) as the value of the cdf of the ordinary distribution F whose range is the unit interval at G, that is, F(G), and it generalizes the ordinary distribution. In this work, we consider the standard two-sided power distribution to define other classes like the beta-G and the Kumaraswamy-G classes. We extend the idea of two-sidedness to other ordinary distributions like normal. After studying the basic properties of the new class in general setting, we consider the two-sided generalized normal distribution with maximum likelihood estimation procedure.     

Estimating powers of the generalized variance under the Pitman closeness criterion

For estimating powers of the generalized variance under a multivariate normal distribution with an unknown mean, the inadmissibility of the closest affine equivariant estimator is shown for the Pitman closeness criterion.     

On the entropy estimators

The paper introduces an estimator of the entropy of a continuous random variable. The estimator is obtained by modifying the estimator proposed by Ebrahimi et al. [Two measures of sample entropy, Statist. Probab. Lett. 20 (1994), pp. 225–234]. The consistency of the estimator is proved and comparisons are made with Vasicek's estimator [A test for normality based on sample entropy, J. R. Stat. Soc. Ser. B 38 (1976), pp. 54–59], van Es estimator [Estimating functionals related to a density by class of statistics based on spacings, Scand. J. Statist. 19 (1992), pp. 61–72], Ebrahimi et al. estimator and Correa estimator [A new estimator of entropy, Comm. Statist. Theory Methods 24 (1995), pp. 2439–2449]. The results indicate that the proposed estimator has smaller mean-squared error than above estimators. A real example is presented and analysed.