Yuri Sekiya

Improvement of approximations for the distributions of multinomial goodness-of-fit statistics under nonlocal alternatives

Cressie and Read (J. Roy. Statist. Soc. B 46 (1984) 440-464) introduced the power divergence statistics, Ra, as multinomial goodness-of-fit statistics. Each Ra has a limiting noncentral chi-square distribution under a local alternative and has a limiting normal distribution under a nonlocal alternative. Taneichi et al. (J. Multivariate Anal. 81 (2002) 335-359) derived an asymptotic approximation for the distribution of Ra under local alternatives. In this paper, using multivariate Edgeworth expansion for a continuous distribution, we show how the approximation based on the limiting normal distribution of Ra under nonlocal alternatives can be improved. We apply the expansion to the power approximation for Ra. The results of numerical investigation show that the proposed power approximation is very effective for the likelihood ratio test.

A FORMULA FOR NORMALIZING TRANSFORMATION OF SOME STATISTICS

ABSTRACT On the basis of Konishis study of a normalizing transformation (Konishi [1]), a concrete normalizing transformation is derived. Some applications of the proposed normalizing transformation are shown, and performance of the transformation in the applications is numerically investigated.

A power approximation of the test of homogeneity for multinomial populations based on a normalizing transformation

Cressie and Read(1984) introduced multino~riial goodness-of-fit statistics based on a, class of divergence measures Ia between discrete distributions. It was also introduced that a class of statistics Ra for the test of homogeneity for multino~riial populations based on Ia (Read and Cressie(1988)). This class includes Pearsons X2 statistic (when a = 1) and the log-likelihood ratio statistic (when a = 0). All Ra have the same chi-squared limiting null distribution. The power of the class is ordinary approximated from a noncentral chi-squared distribution that is also the same for all a. Applying the power approximation theories for the multinomial goodness-of-fit test developed by Broffitt and Randles(1977) and Drost et al.(l989), Taneichi and Sekiya(1995) proposed three approximations to the power of Ra that vary with the statistic chosen. In this paper we propose a. new approximation to the power of Ra The new approximation is a normal approximation based on a normalizing transformation of the statistic...

Improved transformed deviance statistic for testing a logistic regression model

In logistic regression models, we consider the deviance statistic (the log likelihood ratio statistic) D as a goodness-of-fit test statistic. In this paper, we show the derivation of an expression of asymptotic expansion for the distribution of D under a null hypothesis. Using the continuous term of the expression, we obtain a Bartlett-type transformed statistic D@? that improves the speed of convergence to the chi-square limiting distribution of D. By numerical comparison, we find that the transformed statistic D@? performs much better than D. We also give a real data example of D@? being more reliable than D for testing a hypothesis.

Approximations of the Distributions of Test Statistics for Homogeneity of a Product Multinomial Model

Statistics R a based on power divergence can be used for testing the homogeneity of a product multinomial model. All R a have the same chi-square limiting distribution under the null hypothesis of homogeneity. R 0 is the log likelihood ratio statistic and R 1 is Pearsons X 2 statistic. In this article, we consider improvement of approximation of the distribution of R a under the homogeneity hypothesis. The expression of the asymptotic expansion of distribution of R a under the homogeneity hypothesis is investigated. The expression consists of continuous and discontinuous terms. Using the continuous term of the expression, a new approximation of the distribution of R a is proposed. A moment-corrected type of chi-square approximation is also derived. By numerical comparison, we show that both of the approximations perform much better than that of usual chi-square approximation for the statistics R a when a ≤ 0, which include the log likelihood ratio statistic.

Improvements of goodness-of-fit statistics for sparse multinomials based on normalizing transformations

We consider multinomial goodness-of-fit tests for a specified simple hypothesis under the assumption of sparseness. It is shown that the asymptotic normality of the PearsonX2 statistic (Xk2) and the log-likelihood ratio statistic (Gk2) assuming sparseness. In this paper, we improve the asymptotic normality ofXk2 andGk2 statistics based on two kinds of normalizing transformation. The performance of the transformed statistics is numerically investigated.