Some Results on Dynamic Generalized Survival Entropy

Recently, Zografos and Nadarajah (2005 Zografos, K., Nadarajah, S. (2005). Survival exponential entropies. IEEE Trans. Inform. Theor. 51:1239–1246.[Crossref], [Web of Science ®] , [Google Scholar]) proposed two measures of uncertainty based on the survival function, called the survival exponential entropy and the generalized survival exponential entropy. In this article, we explore properties of the generalized survival entropy and the dynamic version of it. We study conditions under which the generalized survival entropy of first order statistic can uniquely determines the parent distribution. The exponential, Pareto, and finite range distributions, which are commonly used in reliability, have been characterized using this generalized measure. Another measure of entropy is also introduced in analogy with cumulative entropy which has been proposed by Di Crescenzo and Longobardi (2009) and some properties of it are given.     

Weighted Renyi's entropy for lifetime distributions

Recently, Di Crescenzo and Longobardi (2006 Di Crescenzo, A., Longobardi, M. (2006). On weighted residual and past entropies. Sci. Math. Jpn. 64:255–266. [Google Scholar]) have studied “length-biased” shift-dependent information measure and its dynamic versions. On the other hand, Renyi's entropy plays a vital role in the literature of information theory that is a generalization of Shannon's entropy. In this article, the concepts of weighted Renyi's entropy, weighted residual Renyi's entropy, and weighted past Renyi's entropy are introduced and their properties are discussed.     

Nonparametric estimation of intensities of nonhomogeneous Poisson processes

The problem of nonparametric estimation of the intensity of a nonhomogeneous Poisson process is considered. A kernel estimator of the intensity is introduced with data driven bandwidth. The bandwidth is obtained from an L₂ cross validation procedure. Results on almost sure convergence of the estimator are obtained, provided the number of observed realizations n tends to infinity. The limiting distribution of the estimator is presented for n→∞.     

Admissibility of estimators in the non-regular family under entropy loss function

In this paper we have considered the problem of finding admissible estimates for a fairly general class of parametric functions in the so called “non-regular” type of densities. The admissibility of generalized Bayes and Pitman estimates of functions of parameters have been established under entropy loss function.     

An efficient integrated nonparametric entropy estimator of serial dependence

ABSTRACT

We propose an efficient numerical integration-based nonparametric entropy estimator for serial dependence and show that the new entropy estimator has a smaller asymptotic variance than Hong and White’s (2005 Hong, Y., White, H. (2005). Asymptotic distribution theory for nonparametric entropy measures of serial dependence. Econometrica 73:837–901.[Crossref], [Web of Science ®] , [Google Scholar]) sample average-based estimator. This delivers an asymptotically more efficient test for serial dependence. In particular, the uniform kernel gives the smallest asymptotic variance for the numerical integration-based entropy estimator over a class of positive kernel functions. Moreover, the naive bootstrap can be used to obtain accurate inferences for our test, whereas it is not applicable to Hong and White’s (2005 Hong, Y., White, H. (2005). Asymptotic distribution theory for nonparametric entropy measures of serial dependence. Econometrica 73:837–901.[Crossref], [Web of Science ®] , [Google Scholar]) sample averaging approach. A simulation study confirms the merits of our approach.     

Characterizations based on Rényi entropy of order statistics and record values

Two different distributions may have equal Rényi entropy; thus a distribution cannot be identified by its Rényi entropy. In this paper, we explore properties of the Rényi entropy of order statistics. Several characterizations are established based on the Rényi entropy of order statistics and record values. These include characterizations of a distribution on the basis of the differences between Rényi entropies of sequences of order statistics and the parent distribution.     

Estimation for a scale parameter with known coefficient of variation

A loss function proposed by Wasan (1970) is well-fitted for a measure of inaccuracy for an estimator of a scale parameter of a distribution defined onR ⁺=(0, ∞). We refer to this loss function as the K-loss function. A relationship between the K-loss and squared error loss functions is discussed. And an optimal estimator for a scale parameter with known coefficient of variation under the K-loss function is presented.     

Maximum expected entropy transformed Latin hypercube designs

Existing projection designs (e.g. maximum projection designs) attempt to achieve good space-filling properties in all projections. However, when using a Gaussian process (GP), model-based design criteria such as the entropy criterion is more appropriate. We employ the entropy criterion averaged over a set of projections, called expected entropy criterion (EEC), to generate projection designs. We show that maximum EEC designs are invariant to monotonic transformations of the response, i.e. they are optimal for a wide class of stochastic process models. We also demonstrate that transformation of each column of a Latin hypercube design (LHD) based on a monotonic function can substantially improve the EEC. Two types of input transformations are considered: a quantile function of a symmetric Beta distribution chosen to optimize the EEC, and a nonparametric transformation corresponding to the quantile function of a symmetric density chosen to optimize the EEC. Numerical studies show that the proposed transformations of the LHD are efficient and effective for building robust maximum EEC designs. These designs give projections with markedly higher entropies and lower maximum prediction variances (MPV''s) at the cost of small increases in average prediction variances (APV''s) compared to state-of-the-art space-filling designs over wide ranges of covariance parameter values.     

RAO'S STATISTIC FOR THE ANALYSIS OF UNIFORM ASSOCIATION IN CROSS-CLASSIFICATIONS

In this paper we introduce a new measure for the analysis of association in cross-classifications having ordered categories. Association is measured in terms of the odd-ratios in 2 × 2 subtables formed from adjacent rows and adjacent columns. We focus our attention in the uniform association model. Our measure is based in the family of divergences introduced by Burbea and Rao [1] Burbea, J. and Rao, C. R. 1982a. On the convexity of some divergence measures based on entropy functions. IEEE Transactions on Information Theory, 28: 489–495. [Crossref], [Web of Science ®] , [Google Scholar]. Some well-known sets of data are reanalyzed and a simulation study is presented to analyze the behavior of the new families of test statistics introduced in this paper.     

Properties of Entropies of Record Values in Reliability and Life Testing Context

In this article, the entropies of record value distributions from some continuous probability models are computed and their properties are investigated, both analytically when feasible, and numerically. In an attempt to establish relationships between entropies of the parent and the corresponding record value distributions, the entropies of record value distributions associated with the uniform, exponential, Weibull, classical Pareto, normal, gamma, beta, and Cauchy distributions are considered in this article. The entropy of record value distributions associated with the uniform, exponential, Weibull, and Pareto distributions, have tractable closed forms. The entropies of record value distributions associated with the normal, gamma, beta, and Cauchy distributions do not have tractable closed forms and need further investigations. Some general conclusions are drawn in the final section.     

Mixture Representation of the Maximum Entropy Density Through Archimedean Copulas

In this article, we obtain a mixture representation of the maximum entropy density introduced by Rodrigues (2004 Rodrigues , J. ( 2004 ). An entropy model for dependent variables . Commun. Statist. Theor. Meth. 4 ( 33 ): 979 – 990 . [Google Scholar]) via Laplace approximation. This representation suggests, as in Sklar (1959 Sklar , A. ( 1959 ). Fonctions de répartition à ndimensions et marges . Publications de l 'Université de Paris 8 : 229 – 231 . [Google Scholar]), a dependence structure through Archimedean copulas independently of the specified marginal distributions. This result can be used as a natural Bayesian and non Bayesian procedure to estimate the dependence function and the marginal, separately.     

ON THE NULL DISTRIBUTIONS OF THE ENTROPY TESTS FOR THE GAUSSIAN AND INVERSE GAUSSIAN MODELS

Vasicek [1] Vasicek, O. 1976. A test for normality based on sample entropy. J. R. Statist. Soc. B, 38: 54–59.  [Google Scholar]used the “convolution of twelve uniforms” for a Monte Carlo tabulation of the 5% critical values for his entropy test for normality. We employ a superior normal generator to construct a corrected and extended tabulation for his test. Interestingly, it is shown that, the same tables can be used for implementing Mudholkar and Tian's [2] Mudholkar, G. S. and Tian, L. 1999. “An entropy characterization of the inverse Gaussian distribution and related goodness-of-fit test”. In Tech. Rep., University of Rochester Rochester, NY Submitted for publication [Google Scholar]entropy test for the composite inverse Gaussian hypothesis. The finding extends the known Gaussian, inverse Gaussian analogies.     

The resampling of entropies with the application of biodiversity

This paper discusses the bootstrap test of entropies. Since the comparison of entropies is of prime interest in applied fields, finding an appropriate way to carry out such a comparison is of utmost importance. This paper presents how resampling should be performed to obtain an accurate p-value. Although the test using a pair-wise bootstrap confidence interval (CI) has already been dealt with in few works, here the bootstrap tests are studied because it may demand quite a different resampling algorithm compared with the CI. Moreover, the multiple test is studied. The proposed tests appear to yield several appreciable advantages. The easy implementation and the power of the proposed test can be considered as advantages. Here the entropy of the discrete variable is studied. The proposed tests are examined using Monte Carlo investigations and also evaluated using various distributions.     

Asymptotic distribution of (h, φ)-entropies

In this paper, (h,φ)-entropies are presented as a generalization of φ-entropies, Havrda-Charvat entropies and the Renyi entropy among others. For this functional, asymptotic distribution for simple random sampling and stratified .sampling with proportional affixing is obtained.     

The Beta Exponential-Geometric Distribution

A new four-parameter distribution with decreasing, increasing, and upside-down bathtub failure rate called the beta exponential-geometric distribution is proposed. The new distribution, generated from the logit of a beta random variable, extends the exponential-geometric distribution of Adamidis and Loukas (1998 Adamidis , K. , Loukas , S. ( 1998 ). A lifetime distribution with decreasing failure rate . Statistics and Probability Letters 39 : 35 – 42 .[Crossref], [Web of Science ®] , [Google Scholar]) and some other distributions. A comprehensive mathematical treatment of this distribution is provided. Some expressions for the moment generating function, moments, order statistics, and Rényi entropy of the new distribution are derived. Estimation of the stress-strength parameter is also obtained. The model parameters are estimated by the maximum likelihood method and Fisher information matrix is discussed. Finally, an application to a real data set is illustrated.     

ASYMPTOTIC EFFICIENCY OF ANALYSIS OF RAO'S QUADRATIC ENTROPY

Since Rao introduced the Quadratic Entropy (QE) in 1982, results on mathematical and statistical properties of the QE and its applications in data analysis and population indices have been published in the literature. In this paper, we study the asymptotic efficiency of the analysis of Rao's quadratic entropy (ANOQE) which is a generalization of the classical analysis of variance (ANOVA). Based on the results of Liu and Rao [1] Liu, Z. J. and Rao, C. R. 1995. Asymptotic distribution of statistics based on quadratic entropy and bootstrapping. JSPI, 43: 1–18.  [Google Scholar]and Liu [2] Liu, Z. J. 1991. Bootstrapping one way analysis of Rao's quadratic entropy. Comm. Statist., 20: 1683–1702.  [Google Scholar]on asymptotic distribution and the bootstrap of the ANOQE, we derive the Bahadur's asymptotic efficiency of the ANOQE and compare efficiency of ANOQE tests based on different QE's.     

Estimating a positive normal mean

Let X ₁, X ₂, ..., X _n be a random sample from a normal population with mean μ and variance σ ². In many real life situations, specially in lifetime or reliability estimation, the parameter μ is known a priori to lie in an interval [a, ∞). This makes the usual maximum likelihood estimator (MLE) ̄ an inadmissible estimator of μ with respect to the squared error loss. This is due to the fact that it may take values outside the parameter space. Katz (1961) and Gupta and Rohatgi (1980) proposed estimators which lie completely in the given interval. In this paper we derive some new estimators for μ and present a comparative study of the risk performance of these estimators. Both the known and unknown variance cases have been explored. The new estimators are shown to have superior risk performance over the existing ones over large portions of the parameter space.