Takashi Seo

An asymptotic approximation for EPMC in linear discriminant analysis based on two-step monotone missing samples

In this paper, we consider the expected probabilities of misclassification (EPMC) in the linear discriminant function (LDF) based on two-step monotone missing samples and derive an asymptotic approximation for the EPMC with an explicit form for the considered LDF. For this purpose, we also provide some results of the expectations for the inverted Wishart matrices in this paper. Finally, we conduct the Monte Carlo simulation for evaluating our result.

Testing Equality of Two Mean Vectors with Two-Step Monotone Missing Data

SYNOPTIC ABSTRACT We consider the problem of testing for two normal mean vectors when the data have two-step monotone pattern missing observations. Under the assumption that the population covariance matrices are equal, we obtain two test statistics for this problem: a generalized Hotellings T2 test statistic and the likelihood ratio test statistic. We propose the approximate upper percentiles of these statistics. The accuracy of the approximation is investigated by Monte Carlo simulation. The proposed method is illustrated using an example.

Testing Equality of Mean Vectors in Two Sample Problem with Missing Data

In this article, we propose tests for equality of mean vectors in two samples problem including missing data from multivariate normal population without condition of missing patterns. By using the idea of Srivastava (1985), we derive test statistics and simultaneous confidence intervals when covariance matrices are not equal. Finally, we investigate the behavior of the distribution of test statistics by Monte Carlo simulations.

The Asymptotic Approximation of EPMC for Linear Discriminant Rules Using a Moore-Penrose Inverse Matrix in High Dimension

We consider the discriminant rule in a high-dimensional setting, i.e., when the number of feature variables p is comparable to or larger than the number of observations N. The discriminant rule must be modified in order to cope with singular sample covariance matrix in high-dimension. One way to do so is by considering the Moor-Penrose inverse matrix. Recently, Srivastava (2006) proposed maximum likelihood ratio rule by using Moor-Penrose inverse matrix of sample covariance matrix. In this article, we consider the linear discriminant rule by using Moor-Penrose inverse matrix of sample covariance matrix (LDRMP). With the discriminant rule, the expected probability of misclassification (EPMC) is commonly used as measure of the classification accuracy. We investigate properties of EPMC for LDRMP in high-dimension as well as the one of the maximum likelihood rule given by Srivastava (2006). From our asymptotic results, we show that the classification accuracy of LDRMP depends on new distance. Additionally, our asymptotic result is verified by using the Monte Carlo simulation.

Asymptotic Expansion of the Distribution of the Studentized Linear Discriminant Function Based on Two-Step Monotone Missing Samples

This article proposes an asymptotic expansion for the Studentized linear discriminant function using two-step monotone missing samples under multivariate normality. The asymptotic expansions related to discriminant function have been obtained for complete data under multivariate normality. The result derived by Anderson (1973) plays an important role in deciding the cut-off point that controls the probabilities of misclassification. This article provides an extension of the result derived by Anderson (1973) in the case of two-step monotone missing samples under multivariate normality. Finally, numerical evaluations by Monte Carlo simulations were also presented.

Simultaneous Confidence Intervals Among k Mean Vectors in Repeated Measures with Missing Data

SYNOPTIC ABSTRACT In this article, we consider a test for the equality of k mean vectors in the intraclass correlation model with monotone missing data. We derive simultaneous confidence intervals for all pairwise comparisons and for comparisons with a control by using the idea in Koizumi and Seo (2009). Finding distributions of T2max type statistics exactly is extremely difficult even if the complete data are obtained. Hence we discuss the approximation to the upper percentage point of T2max type statistic by using Bonferronis inequality. Finally, the accuracy and conservativeness for procedures proposed in this article are evaluated by Monte Carlo simulation. Two examples are also given.

Asymptotic Expansion for Sampling Distribution and Sample Size in Statistical Inference III—The Modified Likelihood Ratio Λ-Test in the Noncentral Case

ABSTRACT This is the third article on the subject stated in the title, treating the asymptotic expansion formula for the OC function of the modified Λ-test. According to the idea and approaching procedure in our first article (Siotani et al., 1995), the reference domain of parameters for making the expansion formula valid was set up. Then an approximate upper bound on the absolute error of the treated expanded OC function was constructed numerically and experimentally over the reference domain in such a way that it provides the guarantee for the practical validity of the expanded OC function and method of determining an effective sample size for designing the test procedure satisfying some requirements on the power of the test.

On the asymptotic distribution of T2-type statistic with two-step monotone missing data

In this article, we consider the asymptotic distribution of Hotelling’s T2-type test statistic when a two-step monotone missing data set is drawn from a multivariate normal population under a large-sample asymptotic framework. In particular, asymptotic expansions for the distribution and upper percentiles are derived using a perturbation method up to the terms of order n−1, where n = N-2 and N denotes the total sample size. Furthermore, making use of Fujikoshi’s transformations, we also have Bartlett-type corrections of the test statistic considered in this article. Finally, we investigate the performance of the proposed approximation to the upper percentiles and Bartletttype correction for the test statistic by conducting Monte Carlo simulations for some selected parameters.

On simultaneous confidence interval estimation for the difference of paired mean vectors in high-dimensional settings

Abstract To test whether two populations have the same mean vector in a high-dimensional setting, Chen and Qin (2010, Ann. Statist.) derived an unbiased estimator of the squared Euclidean distance between the mean vectors and proved the asymptotic normality of this estimator under local assumptions about the mean vectors. In this study, their results are extended without assumptions about the mean vectors. In addition, asymptotic normality is established in the class of general statistics including Chen and Qin’s statistics and other important statistics under general moment conditions that cover both Chen and Qin’s moment condition and elliptical distributional assumption. These asymptotic results are applied to the construction of simultaneous intervals for all pair-wise differences between mean vectors of k ≥ 2 groups. The finite-sample and dimension performance of the proposed methods is also studied via Monte Carlo simulations. The methodology is illustrated using microarray data.

Testing equality of mean vectors in a one-way MANOVA with monotone missing data

ABSTRACT In this study, testing the equality of mean vectors in a one-way multivariate analysis of variance (MANOVA) is considered when each dataset has a monotone pattern of missing observations. The likelihood ratio test (LRT) statistic in a one-way MANOVA with monotone missing data is given. Furthermore, the modified test (MT) statistic based on likelihood ratio (LR) and the modified LRT (MLRT) statistic with monotone missing data are proposed using the decomposition of the LR and an asymptotic expansion for each decomposed LR. The accuracy of the approximation for the Chi-square distribution is investigated using a Monte Carlo simulation. Finally, an example is given to illustrate the methods.