K. Zografos

Divergence statistics: sampling properties and multinomial goodness of fit and divergence tests

φ-divergence .statistics are obtained by either replacing both distributions involved in the argument of the φ -divergence measure by their sample estimates or replacing one distribution and considering the other as given. The sampling properties of estimated divergence-type measures are investigated. Approximate means and variances are derived and asymptotic distributions are obtained. Tests of goodness of fit of observed frequencies to expected ones and tests of equality of divergences based on two or more multinomial samples are constructed.

Survival exponential entropies

The multivariate survival function of a random vector X is used to define a broad class of entropy measures. Several properties of the proposed class are studied and explicit expressions of the measures derived for specific probabilistic models. The cumulative residual entropy, introduced by Rao et al. and Wang et al., is a particular case of the proposed class of measures.

f-Dissimilarity of Several Distributions in Testing Statistical Hypotheses

Various problems in statistics have been treated by the decision rule, based on the concept of distance between distributions. The aim of this paper is to give an approach for testing statistical hypotheses, using a general class of dissimilarity measures among k ≥ 2 distributions. The test statistics are obtained by the replacement, in the expression of the dissimilarity measure, of the unknown parameters by their maximum likelihood estimators. The asymptotic distributions of the resulting test statistics are investigated and the results are applied to multinomial and multivariate normal populations.

Informational distances and related statistics in mixed continuous and categorical variables

Abstract A general class of dissimilarity measures among k⩾2 distributions and their sample estimators are considered, for mixed continuous and categorical variables. The distributional properties are studied for the location model and the asymptotic distributions are investigated, in the general parametric case. The asymptotic distributions of the resulting statistics are used in various settings, to test statistical hypotheses.

Discrete approximations to the Csiszár, Renyi, and Fisher measures of information

The problem of loss of information due to the discretization of data and its estimate is studied for various measures of information. The results of Ghurye and Johnson (1981) are generalized and supplemented for the Csiszar and Renyi measures of information as well as for Fishers information matrix.

A maximum entropy characterization of symmetric Kotz type and Burr multivariate distributions

In this paper a maximum entropy characterization is presented for Kotz type symmetric multivariate distributions as well as for multivariate Burr and Pareto type III distributions. Analytical formulae for the Shannon entropy of these multivariate distributions are also derived.

A necessary test of fit of specific elliptical distributions based on an estimator of Song's measure

In a recent paper, Zografos [K. Zografos, On Mardias and Songs measures of kurtosis in elliptical distributions, J. Multivariate Anal. 99 (2008) 858-879] has obtained general formulas for Songs measure for the elliptic family of distributions, and he introduced and studied its sample analogue. In this paper, based on the empirical estimator of this measure, we present a test to verify if the data are distributed according to a specific elliptical (spherical) distribution. In this context, the asymptotic distribution of the proposed statistic under the null hypothesis of specific spherical distributions is obtained. The proposed statistic also provides us with a procedure for testing multivariate normality. In order to evaluate the convergence of the proposed statistic to its limiting distribution, under the null hypothesis, a simulation study is performed to analyze the behavior of the percentiles of the proposed statistic in some special cases of spherical distributions. Moreover, a Monte Carlo study is carried out on the performance of the test statistic as a necessary test of fit of specific spherical distributions. In this framework, the type I error rates as well as the power of the test are studied. Finally, a well-known data set is used to illustrate the method developed in this paper.

On a measure of dependence based on fisher's information matrix

A class of measures of dependence between two random vectors is defined, in terms of the canonical correlations obtained from Fishers information matrix. Some basic properties are proved for this class of measures. Examples are given to illustrate that the class gives good measures, under normal models. Interesting measures are also arise for bivariate models where the correlation coefficient does not exist for some values of the parameters of the model.

Limiting properties of some measures of information

In this paper we investigate the limiting behaviour of the measures of information due to Csiszár, Rényi and Fisher. Conditions for convergence of measures of information and for convergence of Radon-Nikodym derivatives are obtained. Our results extend the results of Kullback (1959,Information Theory and Statistics, Wiley, New York) and Kirmani (1971,Ann. Inst. Statist. Math.,23, 157–162).

Asymptotic distributions of estimated f-dissimilarity between populations in stratified random sampling

Asymptotic distributions of the f-dissimilarity statistic, arising in the estimation of dissimilarity between populations, are investigated in a stratified random sampling set-up from the populations considered. Applications of these results in testing homogeneity of populations and equality of dissimilarities are discussed.