We assume X and Y be two independent random variables and define θ=P (X < Y ). The inference for θ can be found in various fields. This paper not only compares several methods for constructing the confidence interval for θ in a small sample but also proposes some new methods. The intervals derived by these new methods show good performance in a small sample, and their actual coverage probability is close to the nominal level. In addition, one of the biggest advantages of our methods is that it does not require complicated calculations.

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We describe a Box and Cox power-transformation to simultaneously provide additivity and homoscedasticity in regression. The two methods developed here are extensions of the power-additive transformation (PAT) discussed by Goto (1992, 1995) and Hamasaki and Goto (2005). The PAT aims to improve the additivity or linearity of some simple model represented by linear predicators. We then consider combinations of the PAT with the weighting and transform-both-sides methods. We discuss the procedures to find the maximum likelihood estimates of parameters and then consider the relationship between the methods. Also, we compare the performances of the methods through a simulation study.

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The rank statistic for a location-scale parameter is introduced to a change-point problem. A combination of the Wilcoxon and Mood statistics is extended to the change-point context. The proposed rank statistic is used to detect a change-point in a setting involving at most one change in this paper. The limiting distribution of the suggested statistic is derived under the null hypothesis (no change). The finite sample critical value of the suggested statistic is estimated by simulation studies. In addition, the accuracy of detecting a change-point is investigated by simulation studies. The method is illustrated by the analysis of various data.

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This paper proposes a dimension reduction technique in the framework of symbolic data analysis (SDA). Recent advances in technology have increased the complexity of datasets, and today, their size is much larger than it was in the past decade. Most statistical methods do not have sufficient power to analyze these datasets. SDA was proposed by Diday at the end of the 1980s and is a new approach for analyzing huge and complex data.
SDA examines “symbolic data”, which consist of concepts. A concept consists of not only values but also “higher-level units” such as an interval and a distribution. Their combination can also be represented as a kind of a concept. This implies that complex data can be formally handled in the framework of SDA. However, there are very few studies based on this simple idea. Therefore, practical methods should be developed to apply this idea to solve problems in the real world. In this study, we focus on the case in which a concept contains some subsets (the concept acts as a typical complex dataset) and develop a new method to analyze this dataset directly using SDA.
In this paper, we propose a dimension reduction technique in the framework of SDA, especially for a group structure, and introduce a numerical example.

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