J.A. Pardo

Asymptotic approximations for the distributions of the (h‐, φ‐)‐divergence goodness‐of‐fit statistics: application to Renyi’s statistic

Read (1984) presented an asymptotic expansion for the distribution function of the power divergence statistics whose speed of convergence is dependent on the parameter of the family. Generalizes that result by considering the family of (h, φ)‐divergence measures. Considers two other closer approximations to the exact distribution. Compares these three approximations for the Renyi’s statistic in small samples.

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.

The Jensen-Shannon divergence

Abstract In this paper we investigate the Jensen-Shannon parametric divergence for testing goodness-of-fit for point estimation. Most of the work presented is an analytical study of the asymptotic differences between different members of the family proposed in goodness of fit, together with an examination of closer approximations to the exact distribution of these statistics than the commonly used chi-squared distribution. Finally the minimum Jensen-Shannon divergence estimates are introduced and compared with other well-known estimators by computer simulation.

A sequential selection method of a fixed number of fuzzy information systems based on the information energy gain

On the basis of the ‘information energy gain for fuzzy information systems’, a sequential selection method of a fixed number of fuzzy information systems is presented. The ‘information energy gain’ is a measure of the quantity of information concerning the state space, provided by a fuzzy information system X∗ when a prior probability distribution has been defined on the state space.

Use of Rényi's divergence to test for the equality of the coefficients of variation

A new family of test statistics based on Renyis divergence is introduced for the hypothesis that the coefficients of variation of k normal populations are equal. A comparative simulation study is carried out concerning the size and power of these test statistics and earlier ones. Finally, two members of the new family of tests emerge as the best from the simulation study.

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.

Sufficient fuzzy information systems

Abstract In this paper we suggest and study a new selection criterion in order to compare probabilistic information systems when the available information from them is vague, in the sense that it might be considered as fuzzy information (Tanaka, Okuda and Asai). This criterion is based on the concept of sufficient probabilistic information system established by Blackwell.

Tests for Bivariate Symmetry Against Ordered Alternatives in Square Contingency Tables

Let X and Y denote two ordinal response variables, each having I levels. When subjects are classified on both variables, there are I 2 possible combinations of classifications. Let pij= Pr(X = i, Y = j). This paper introduces a family of tests based on φ–divergence measures for testing H0: pij = pji against H1: pij ≥ pji (I≥ j); and for testing H1 against H2: pij unrestricted. A simulation study assesses some of the family of tests introduced in this paper in comparison to the likelihood ratio test.

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.

Small-sample comparisons for the Rukhin goodness-of-fit-statistics

Rukhins statistic family for goodness-of-fit, under the null hypothesis, has asymptotic chi-squared distribution; however, for small samples the chi-squared approximation in some cases does not well agree with the exact distribution. In this paper we consider this approximation and other three to get appropriate test levels in comparison with the exact level. Moreover, exact power comparisons for several values of the parameter under specified alternatives provide that the classical Pearsons statistic, obtained as a particular case of Rukhin statistic, can be improved by choosing other statistics from the family. An explanation is proposed in terms of the effects of individual cell frequencies on the Rukhin statistic.