Hirokazu Yanagihara

Three Approximate Solutions to the Multivariate Behrens–Fisher Problem

ABSTRACT This article provides three approximate solutions to the multivariate Behrens–Fisher problem: the F statistic, the Bartlett, as well as the modified Bartlett corrected statistics. Empirical results indicate that the F statistic outperforms the other two and five existing procedures. The modified Bartlett corrected statistic is also very competitive.

Four improved statistics for contrasting means by correcting skewness and kurtosis

This paper is concerned with removing the influence of non-normality in the classical t-statistic for contrasting means. Using higher-order expansion to quantify the effect of non-normality, four corrected statistics are provided. Two aim to correct the mean bias and two to correct the overall distribution. The classical t-statistic is also robust against non-normality when the observed variables satisfy certain structures. A special case is when the marginal distributions of the contrast are independent and identically distributed.

Tests for covariance matrices in high dimension with less sample size

In this article, we propose tests for covariance matrices of high dimension with fewer observations than the dimension for a general class of distributions with positive definite covariance matrices. In the one-sample case, tests are proposed for sphericity and for testing the hypothesis that the covariance matrix @S is an identity matrix, by providing an unbiased estimator of tr[@S^2] under the general model which requires no more computing time than the one available in the literature for a normal model. In the two-sample case, tests for the equality of two covariance matrices are given. The asymptotic distributions of proposed tests in the one-sample case are derived under the assumption that the sample size N=O(p^@d),1/2<@d<1, where p is the dimension of the random vector, and O(p^@d) means that N/p goes to zero as N and p go to infinity. Similar assumptions are made in the two-sample case.

Estimation of Varying Coefficients for a Growth Curve Model

SYNOPTIC ABSTRACT In this paper, a new approach of modelling growth curves is developed which uses time-varying coefficients. Since the mean structure of the growth curve model has many unknown parameters depending on both covariates and time trend designs, it can be difficult to understand and interpret estimated parameters. Using varying coefficient functions, the effects of covariates can be evaluated and visualized more easily. The asymptotic functional confidence intervals are derived theoretically and a procedure is proposed to test whether the effects of covariates are significant.

Bias correction of AIC in logistic regression models

Abstract This paper is concerned with the bias correction for Akaike information criterion (AIC) in logistic regression models. The AIC is an approximately unbiased estimator for a risk function based on the Kullback–Leibler information. Hence, for small to moderate sample sizes, the bias may not be negligible. Further, when at least one of the success probabilities is close to the boundary values, 0 or 1, the bias will become too large although the sample size may not be too small. By using a perturbation expansion of the maximum likelihood estimator (MLE), we propose a corrected AIC, which has an asymptotically reduced bias in logistic regression models. Further, it is shown that the corrected AIC provides a better approximation to its risk function than the uncorrected one in our simulation study and applications to real data.

Corrected versions of cross-validation criteria for selecting multivariate regression and growth curve models

This paper is concerned with cross-validation (CV) criteria for choice of models, which can be regarded as approximately unbiased estimators for two types of risk functions. One is AIC type of risk or equivalently the expected Kullback-Leibler distance between the distributions of observations under a candidate model and the true model. The other is based on the expected mean squared error of prediction. In this paper we study asymptotic properties of CV criteria for selecting multivariate regression models and growth curve models under the assumption that a candidate model includes the true model. Based on the results, we propose their corrected versions which are more nearly unbiased for their risks. Through numerical experiments, some tendency of the CV criteria will be also pointed.

Asymptotic expansion of the null distribution of test statistic for linear hypothesis in nonnormal linear model

This paper is concerned with the null distribution of test statistic T for testing a linear hypothesis in a linear model without assuming normal errors. The test statistic includes typical ANOVA test statistics. It is known that the null distribution of T converges to χ2 when the sample size n is large under an adequate condition of the design matrix. We extend this result by obtaining an asymptotic expansion under general condition. Next, asymptotic expansions of one- and two-way test statistics are obtained by using this general one. Numerical accuracies are studied for some approximations of percent points and actual test sizes of T for two-way ANOVA test case based on the limiting distribution and an asymptotic expansion.

Bias-Corrected AIC for Selecting Variables in Poisson Regression Models

In the present article, we consider the variable selection problem in Poisson regression models. Akaikes information criterion (AIC) is the most commonly applied criterion for selecting variables. However, the bias of the AIC cannot be ignored, especially in small samples. We herein propose a new bias-corrected version of the AIC that is constructed by stochastic expansion of the maximum likelihood estimator. The proposed information criterion can reduce the bias of the AIC from O(n−1) to O(n−2). The results of numerical investigations indicate that the proposed criterion is better than the AIC.

Asymptotic expansion of the null distribution of one-way ANOVA test statistic for heteroscedastic case under nonnormality

This paper deals with the null distribution of one-way ANOVA test statistic T for testing the equality of means of q nonnormal populations with unequa variances. In thwis situation , there are several test statistics Specially, we consider just this one which has been proposed by James(1951). It is known that the null distribution of T converges to when all the sample sizes from the q populations are large. we extend this result by obtaining an asymptotic expansion under a general condition. In order to calculate this expansion, we use a similar method in Fujikoshi, Ohmae and Yanagihara(1999).Numerical accuracies are studied for approximations of the precent points and actual test sizes of T based on the limiting distribution and an asymptotic expansion.

Bridging the Gap Between B-Spline and Polynomial Regression Model

Abstract B-splines are flexible smoothers and a linear curve including a constant line is a hard but meaningful in practical situations. In this paper, we propose a method to select the better of two types of models by using an information criterion in order to take both advantages. Numerical study and an example of multiple nonparametric regression analysis showed good performance of our methodology.