In the one-way layout assuming that the underlying distribution is normal, we may execute Tukey-Kramer multiple comparisons procedure for searching all pairwise differences of locations. For the unequal sample sizes, the Tukey-Kramer (T-K) method is conservative. The T-K method is given by using the upper α point of the studentized range distribution A(t). A(t) is a lower bound for the distribution of the statistic max_<1&le;i<i'&le;k>|T_<ii'>|. We derive the distribution B(t) which gives an upper bound for the distribution of max_<1&le;i<i'&le;k>|T_<ii'>|. By using numerical double integration, we show that the value of B(t) is a little larger than that of A(t). As the result, we may verify that the conservativeness of the T-K method is small.

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Rotation of axes, which is almost routinely used in exploratory factor analysis, is not so common in principal component analysis (PCA) excepting in the fields such as climatology and psychology. In applying rotation to PCA we have to decide how to normalize the eigenvectors and which of the component coefficients or the component loadings to rotate, where the component coefficients are the coefficients for the original variables in computing the component scores and the component loadings are the coefficients for the component scores in approximating the original variables. These problems were studied numerically by applying these methods of orthogonal rotation to two actual data sets and several artificial data sets generated following factor analysis models. It was found that orthogonal rotation of component loadings tends to provide the result easier to interpret than that of the orthogonal rotation of component coefficients.

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Inference for point processes is more efficient when the exact event occurrence times are known. However, collecting exact times is sometimes difficult due to a study design or practical restrictions. Yet, even then, the grouped number of events, that is, the number of events in intervals of the observation period, is available. Having only the grouped number of recurrent events with time-dependent covariates available, we investigate the precision in estimation of a parameter representing a treatment effect. The study includes only time-dependent covariate which is a function of the cumulative number of recurrent events. The time-dependent covariate is observed at a designated design point and the observation period is divided into two or three intervals. Our results suggest that the lowest bias in estimation of the treatment effect is obtained when the design-point coincides with one of the partition points. High efficiency in parameter estimation is obtained when we take the design-point as the partition point for division of the observation period into two intervals. Additionally, in the case of the division into three intervals, high efficiency is obtained when dividing the first half of the observation period equally into two intervals and setting the design point at the second partition point.

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The paper reviews recent developments in (a) distributions on the circle and on manifolds such as sphere, torus, cylinder and disc, (b) regressions, and (c) inferences including test of symmetry, test of independence, and estimation and/or test of mean direction and concentration in von Mises-Fisher distributions. About distributions on the circle, the paper deals with the following. A symmetric Pearson type VII ditribution (t as a special case) on the circle includes the von Mises distribution as a limit. The Jones-Pewsey distribution extends the t-distribution on the circle and it contains cardioid, wrapped Cauchy, and Cartright's power-of-cosine or Minh-Farnum distributions as special cases. Relation between t- and Jones-Pewsey distributions on the sphere is discussed and an asymmetric t is proposed. The paper also reviews the development of wrapped skew Laplace, wrapped skew normal, and wrapped symmetric α-stable distributions. About distributions on manifolds such as torus, cylinder and disc, the following are discussed. A submodel of Mardia's bivariate von Mises distribution has a concise expression of the normalizing constant. Maximizing the entropy gives some models on the cylinder. One of which is a distribution whose angular marginal distribution is a wrapped Cauchy and conditional distribution is a von Mises. Another has a generalized von Mises and an exponential as conditional distributions. Similar to bivariate distributions on the torus whose marginals are specified, four dimensional distributions with specified bivariate marginals on the cylinder are proposed and are applicable to distributions of wind direction and speed observed at two sites. The Mobius transformation of a bivariate beta (Pearson type II) provides a skew distribution on the disc. Circular-circular regression model based on von Mises errors as well as wrapped Cauchy are discussed. As for inferences of models, the paper gives a review of test of symmetry and test of mean direction and/or concentration in von Mises-Fisher distributions.

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Consider a single image such that multivariate data are observed at respective pixels. Image classification is a problem of classifying pixels into several homogeneous regions by learning the feature vectors and the adjacency relationships of the pixels in the image. The classification of a pixel into one of categories is an important and fundamental problem in image pattern analysis. In this paper, we review image classification methods, Markov-random-fields (MRF)-based method as well as Spatial Boosting (SpatialBoost) which proposed by Nishii & Eguchi (2005) via statistical machine learning. Variants of SpatialBoost considered in various situations are also discussed. These methods are successfully applied to real and synthetic data, and compared with MRF-based methods.

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