Nobuhiro Taneichi

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...

Semiparametric estimation based on parametric modeling of the cause-specific hazard ratios in competing risks

This paper is intended as an investigation of estimating cause-specific cumulative hazard and cumulative incidence functions in a competing risks model. The proportional model in which ratios of the cause-specific hazards to the overall hazard are assumed to be constant (independent of time) is a well-known semiparametric model. We are here concerned with relaxation of the proportionality assumption. The set C of all causes are decomposed into two disjoint subsets of causes as C - C1 ∪ C2. The relative risk of cause A in the sub-causes C1 can be represented as a function defined by ratic of the cause-specific hazard of cause A to the sum of cause-specific hazards in the sub-causes C1. We call this function the risk pattern function of cause A in C1, and consider a semiparametric model in which risk pattern functions in C1 are not constant (independent of time) but those functional forms, except for finite-dimensional parameters, are known. Based on this model, semiparametric estimators are obtained, and estimated variances of them are derived by delta methods. We investigate asymptotic properties of the semiparametric estimators and compare them with the nonparametric estimators. The semiparametric procedure is illustrated with the radiation-exposed mice data set, which represents lifetimes and causes of death of mice exposed to radiation in two different environments.

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.

Redundancy Index in Canonical Correlation Analysis with Linear Constraints

The redundancy index proposed by Stewart and Love (1968) is an index to measure the degree to which one set of variables can predict another set of variables, and is associated with canonical correlation analysis. Yanai and Takane (1992) developed canonical correlation analysis with linear constraints (CCALC). In this paper we define a redundancy index in CCALC, which is based on the reformulation of CCALC by Suzukawa (1997). The index is a general measure to summarize redundancy between two sets of variables in the sense that various dependency measures can be obtained by choosing constraints suitably. The asymptotic distribution of the index is derived under normality.