Constructing fractional factorial designs with given properties is one of important topics in design of experiments. In this paper, we give some results based on the technique of computational algebraic methods. For each fractional factorial design with given property, we characterize its polynomial indicator function and enumerate solutions of the algebraic equation for the coefficients of the corresponding indicator functions. By this approach, we can enumerate all the fractional factorial designs with given orthogonality for any size in theory. However, problems of computational feasibility arise in actual computations.

We consider the estimation of multivariate normal mean. A large class of minimax generalized Bayes estimators is derived by Fourdrinier et al. (1998, Annals of Statistics). In the proof, the correlation inequality for two monotone function is quite effective. However, some prior densities corresponds to monotonicity with opposite direction. As a result, the range of minimaxity is not large. In this paper, we prove an extended correlation inequality with the penalty and enlarge a class of minimax generalized Bayes estimators using the inequality.

In this paper, we introduce high-dimensional statistical analysis for the Strongly Spiked Eigenvalue (SSE) model. We deal with statistical inference for non-sparse high-dimensional data, such as genomic data, especially equality test of high-dimensional covariance matrices and quadratic discriminant analysis. In statistical inference, the key to guaranteeing theoretical accuracy is the high-dimensional asymptotic normality of the statistic. However, high-dimensional asymptotic normality does not generally hold for high-dimensional data belonging to the SSE model. New techniques are needed to derive asymptotic distributions of statistics and to handle the data instead of high-dimensional asymptotic normality. We describe ideas on how to approach the SSE model and guarantee high accuracy, as well as the latest developments in high-dimensional statistical analysis.

We construct robust inference theories for time series models under non-regular settings such as innite variance and long-range dependence. In such non-regular models, the rates of convergence and limit distributions of typical statistics depend on unknown nuisance parameters, making it challenging to establish statistical inference methods based on these distributions. To address these problems, we employ robust methodologies such as self-normalization, self-weighting, and empirical likelihood approach. Under mild conditions of dependence structure and higher-order moments of the models, we construct robust statistics which have desired limit distributions. In addition, we apply the proposed method to change point analysis and tests of causality.