In a one-way analysis of variance model, robust versions based on R-estimators are proposed for single-step multiple comparisons procedures discussed by Tukey (1953), Dunnett (1955), and Scheffé (1953). The robust procedures are two methods based on joint ranks and pairwise ranks. It is shown that the two methods are asymptotically equivalent. Although we fail to construct simultaneous tests based on linear joint ranks, we are able to propose simultaneous tests based on the R-estimators. Robustness for asymptotic properties is discussed. The accuracy of asymptotic approximation is investigated.

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Applying principal component analysis as a substitute for factor analysis, we often adopt the following greater-than-one rule to decide the number of factors, k: Take the number of eigenvalues of the correlation matrix that is greater than one. Another approach to deciding k is based on the scree graph. In the present paper, the adequacy of these rules for one-factor cases is discussed. On the basis of obtained results, some recommendations for data analysis are given. Our approach to this study is based on the analytical expressions of eigenvalues under some simple but practical cases. In deriving theoretical results, we use a representation of a polynomial in terms of a remainder sequence. This technique is useful for finding the sign of polynomials under inequality constraints, so the idea is also introduced.

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This paper proposes a test for the normalization of cointegrating vectors. Our test is constructed using the unrestricted maximum likelihood estimator and then it may be seen as a Wald-type test. The test statistic is shown to be asymptotically bounded above by a chi-square distribution with one degree of freedom (χ₁²) and then we can conduct a conservative test using critical values of χ₁².

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The so-called Stein problem is addressed in the estimation of a mean vector of a multivariate normal distribution with a known covariance matrix. For general prior distributions with sphericity, the paper derives conditions on priors under which the resulting generalized Bayes estimators are minimax relative to the usual quadratic loss. It is also shown that the conditions can be expressed based on the inverse Laplace transform of the general prior. Stein's super-harmonic condition is derived from the general conditions. Finally, the priors are characterized for the admissibility.

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We investigate the canonical correlation of the principal components from two populations, and attain the limiting distribution using the perturbation expansion of the canonical correlation estimate. We discuss the numerical accuracy of the limiting distribution.

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This paper deals with a test of equality of mean vectors of several heteroscedastic multivariate populations. We derive not only the asymptotic expansion up to N⁻¹ of the nonnull distribution of James's (1954) statistic, but also those of two corrected statistics due to Cordeiro and Ferrari (1991) and Kakizawa (1996). The derivation we considered here is based on the differential operator method developed in Kakizawa and Iwashita (2005).

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Let Λ_i, 1 ≤ i ≤ l be simple patterns, i.e., finite sequences of outcomes from a set Γ = {b₁, b₂, ..., b_m} and let Λ be a compound pattern (a set ofl distinct simple patterns). In this paper, we study joint distributions of the waiting time until the r-th occurrence of the compound pattern Λ, and the numbers of each simple pattern observed at that time in the multistate Markov dependent trials. We provide methods for deriving the probability generating functions of the joint distributions under two types of counting schemes (non-overlap counting and overlap counting) for the compound pattern Λ. Besides, the present work is useful in elucidating the primary difference between non-overlap counting and overlap counting. As applications, when Λ is a set of runs, the corresponding joint distributions are investigated and a practical example is mentioned. Also, the Chen-Stein approximation is derived for the waiting time distribution, and its asymptotic behaviour is discussed. Finally, we address the parameter estimation in the waiting time distributions of the compound pattern along with problems of identifiability.

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We propose a nonparametric procedure to test the hypothesis that the j-th largest eigenvalues of a covariance matrix are equal in multipopulation. We apply the Mood test by using the principal component scores and deal the equality of eigenvalues with the equality of variance. We investigate the significance level and the power of test by simulation and show that this nonparametric test is useful for symmetric populations.

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We make a conjugate analysis for the five location-dispersion families including the normal, the transformed gamma and the von Mises distributions. The five families are introduced through the requirement for the existence of conjugate prior densities. We show in a unified way that a Pythagorean relationship holds with respect to posterior risks, which clarifies the optimality of the posterior mode under a Kullback-Leibler loss. An explicit form of the posterior mode is given, and a type of linearity is observed. We construct an empirical Bayes estimator of a location vector explicitly.

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