Takis Papaioannou

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.

New parametric measures of information

In this paper methods are presented for obtaining parametric measures of information from the non-parametric ones and from information matrices. The properties of these measures are examined. The one-dimensional parametric measures which are derived from the non-parametric are superior to Fishers information measure because they are free from regularity conditions. But if we impose the regularity conditions of the Fisherian theory of information, these measures become linear functions of Fishers measure.

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.

Information in experiments and sufficiency

Abstract We define measures of information contained in an experiment which are by-products of the parametric measures of Fisher, Vajda, Mathai and Boekee and the non-parametric measures of Bhattacharyya, Renyi, Matusita, Kagan and Csiszar. We use these measures to compare sufficient experiments according to Blackwells definition. In particular, we prove that if δ X and δ Y are two experiments and δ X ≥ δ Y then l X ≥ l y for all of the above measures.

Asymmetry Models for Contingency Tables

Abstract For the quasi-symmetry (QS) model, applicable to square contingency tables with commensurable classification variables, it is proved that under certain conditions, it is the closest model to symmetry in terms of the Kullback-Leibler distance. Replacing the Kullback-Leibler distance by f-divergence we introduce a generalized quasi-symmetry model, the QS[f], and develop interpretational aspects for its parameters. QS is a special case of QS[f], whereas the most characteristic of the newly introduced QS-type models is the Pearsonian QS model. We compute maximum likelihood estimates of the parameters of the Pearsonian QS model and compare it, through examples and simulation studies, to the classical QS model in terms of goodness of fit and of the powers of the tests for marginal homogeneity conditional on the QS and Pearsonian QS models.

On two forms of fisher's measure of information

ABSTRACT Fishers information number is the second moment of the “score function” where the derivative is with respect to x rather than Θ. It is Fishers information for a location parameter, and also called shift-invariant Fisher information. In recent years, Fishers information number has been frequently used in several places regardless of parameters of the distribution or of their nature. Is this number a nominal, standard, and typical measure of information? The Fisher information number is examined in light of the properties of classical statistical information theory. It has some properties analogous to those of Fishers measure, but, in general, it does not have good properties if used as a measure of information when Θ is not a location parameter. Even in the case of location parameter, the regularity conditions must be satisfied. It does not possess the two fundamental properties of the mother information, namely the monotonicity and invariance under sufficient transformations. Thus the Fisher information number should not be used as a measure of information (except when Θ a location parameter). On the other hand, Fishers information number, as a characteristic of a distribution f(x), has other interesting properties. As a byproduct of its superadditivity property a new coefficient of association is introduced.

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).

Information and random censoring

Fisher and divergence type measures of information in the area of random censoring are introduced and compared with the measures of Hollander, Proschan, and Sconing. The basic properties of statistical information theory are established for these measures of information. The winners are the classical measures of information.

Rank correlation inequalities with missing data

New inequalities are obtained for the Spearman rank correlation coefficient, its associated sum of squares of rank differences and the Kendall tau coefficient when one or more ranks or individuals are missing or added to the data. New bounds using the Daniels and Durbin-Stuart inequalities are also derived.

Flexible designs for phase II comparative clinical trials involving two response variables

The aim of phase II clinical trials is to determine whether an experimental treatment is sufficiently promising and safe to justify further testing. The need for reduced sample size arises naturally in phase II clinical trials owing to both technical and ethical reasons, motivating a significant part of research in the field during recent years, while another significant part of the research effort is aimed at more complex therapeutic schemes that demand the consideration of multiple endpoints to make decisions. In this paper, our attention is restricted to phase II clinical trials in which two treatments are compared with respect to two dependent dichotomous responses proposing some flexible designs. These designs permit the researcher to terminate the clinical trial when high rates of favorable or unfavorable outcomes are observed early enough requiring in this way a small number of patients. From the mathematical point of view, the proposed designs are defined on bivariate sequences of multi-state trials, and the corresponding stopping rules are based on various distributions related to the waiting time until a certain number of events appear in these sequences. The exact distributions of interest, under a unified framework, are studied using the Markov chain embedding technique, which appears to be very useful in clinical trials for the sample size determination. Tables of expected sample size and power are presented. The numerical illustration showed a very good performance for these new designs.