Kostas Zografos

Formulas for Rényi information and related measures for univariate distributions

In a recent paper Song [J. Stat. Plan. Infer. 93 (2001) 51] considered Renyi information of order λ and established its connection to the loglikelihood. From this relation an intrinsic distribution measure was proposed and analytic expressions of this measure and Renyi information were derived for some standard continuous distributions. In this paper, we derive analytical formulas for Renyi and Shannon entropies, as well as, for Songs measure for 26 flexible families of univariate continuous distributions. We believe that the results presented here will serve as an important reference for scientists and engineers in many areas.

Expressions for Rényi and Shannon entropies for bivariate distributions

Exact forms of Renyi and Shannon entropies are determined for 27 continuous bivariate distributions, including the Kotz type distribution, truncated normal distribution, distributions with normal and centered normal conditionals, natural exponential distribution, Freunds exponential distribution, Marshall and Olkins exponential distribution, exponential mixture distribution, Arnold and Strausss exponential distribution, McKays gamma distribution, distribution with gamma conditionals, gamma exponential distribution, Dirichlet distribution, inverted beta distribution, distribution with beta conditionals, beta stacy distribution, Cuadras and Auges distribution, Farlie Gumbel Morgenstern distribution, logistic distribution, Pearson type VII distribution, Pearson type II distribution, distribution with Cauchy conditionals, bilateral Pareto distribution, Muliere and Scarsinis Pareto distribution, distribution with Pareto conditionals and the distribution with Gumbel conditionals. We believe that the results presented here will serve as an important reference for scientists and engineers in many areas.

Minimum φ-divergence estimators with constraints in multinomial populations

Abstract This paper presents a minimum φ-divergence estimation procedure in multinomial models in which the probabilities depend on unknown parameters that are not mathematically independent but satisfy some functional relationships. This estimator is then used in a φ-divergence statistic for solving the problem of goodness-of-fit when the unknown parameters in the probabilities are not mathematically independent. The asymptotic distribution of this family of statistics is obtained under the null and contiguous alternative hypotheses. The asymptotic distribution of residuals, when the parameters are estimated using the minimum φ-divergence estimator, is also obtained.

On tests of symmetry, marginal homogeneity and quasi-symmetry in two-way contingency tables based on minimum Φ-divergence estimator with constraints

The restricted minimum φ-divergence estimator, [Pardo, J.A., Pardo, L. and Zografos, K., 2002, Minimum φ-divergence estimators with constraints in multinomial populations. Journal of Statistical Planning and Inference, 104, 221–237], is employed to obtain estimates of the cell frequencies of an I×I contingency table under hypotheses of symmetry, marginal homogeneity or quasi-symmetry. The associated φ-divergence statistics are distributed asymptotically as chi-squared distributions under the null hypothesis. The new estimators and test statistics contain, as particular cases, the classical estimators and test statistics previously presented in the literature for the cited problems. A simulation study is presented, for the symmetry problem, to choose the best function φ2 for estimation and the best function φ1 for testing.

On tests of homogeneity based on minimum ø-divergence estimator with constraints

A family of tests of homogeneity of independent multinomial populations is introduced in terms of the φ1-divergence when the parameters are estimated using the minimum φ2-divergence estimator instead of the maximum likelihood estimator. A simulation study is presented to choose the best function φ2 for estimation and the best function φ1 for testing. A new test statistic is obtained, more powerful in some cases, than the existing tests for testing homogeneity in multinomial populations.

Change-point detection in multinomial data using phi-divergence test statistics

We propose two families of maximally selected phi-divergence tests to detect a change in the probability vectors of a sequence of multinomial random variables with possibly different sizes. In addition, the proposed statistics can be used to estimate the location of the change-point. We derive the limit distributions of the proposed statistics under the no change null hypothesis. One of the families has an extreme value limit. The limit of the other family is the maximum of the norm of a multivariate Brownian bridge. We check the accuracy of these limit distributions in case of finite sample sizes. A Monte Carlo analysis shows the possibility of improving the behavior of the test statistics based on the likelihood ratio and chi-square tests introduced in Horvath and Serbinowska [7]. The classical Lindisfarne Scribes problem is used to demonstrate the applicability of the proposed statistics to real life data sets.

Conditional tests of marginal homogeneity based on ϕ-divergence test statistics

Abstract In this work, using the well-known result that symmetry is equivalent to quasi-symmetry and marginal homogeneity simultaneously holding, two families of test statistics based on ϕ -divergence measures are introduced for testing conditional marginal homogeneity assuming that quasi-symmetry holds.

Fisher's information matrix and φ−divergence for finite and optimal partitions of the sample space

In this paper we examine the behaviour of the discretized version of Fishers information matrix and φ-divergence produced from a general finite partition of the sample space. We prove that as the data information gets richer, in the sense that the data partition gets finer, both Fisher information matrix and ?-divergence increase monotonically, approaching to their continuous counterparts, respectively. In particular, the monotonicity property is proved by considering one partition as a subpartition of the other. The problem of finding an optimal partition which minimizes the loss of information of the literature, are generalized and supplemented

Preliminary test estimators in intraclass correlation model under unequal family sizes

The intraclass correlation model is well known in the literature of multivariate analysis and it is mainly used in studying familial data. This model is considered in this paper and the interest is focused on the estimation of the intraclass correlation on the basis of familial data from families which are randomly selected from two or more independent populations. The size of the families is considered unequal and the variances of the populations are considered unequal, too. In this statistical framework some preliminary test estimators are presented in a unified way and their asymptotic distribution is obtained. A decision-theoretic approach is developed to compare the estimators by using the asymptotic distributional quadratic risk under the null hypothesis of equality of the intraclass correlations and under contiguous alternative hypotheses, as well. Some interesting relationships are obtained between the estimators considered.

Ordering and selecting extreme populations by means of entropies and divergences

This paper studies the simultaneous selection of extreme populations from a set of independent populations. Two types of subset selection rules for k populations are proposed and studied. The first type selects one subset of populations that should contain the population with the smallest, and another subset of populations that should contain the population with the largest, @f-entropy. The second type selects analogously, but in terms of the extreme @f-divergences with respect a known control population. Properties of the proposed procedures are stated and studied. Examples are presented in order to illustrate the results.