In this paper, we consider the two-sample test for high-dimensional data. In the high-dimensional statistical analysis, it is crucial to construct theories and methodologies depending onthe high-dimensional covariance structure. In particular, the high-dimensional noise has harmful effects upon the theory and methodology quite severely. In order to overcome the difficulty, the key is how to handle the high-dimensional spiked noise. In this paper, we deal with a typical high-dimensional noise model called the uni-SSE (strongly spiked eigenvalue)model. We propose a new test procedure that can be established without assuming the equality of the first eigenspacesbetween two classes. We show that the proposed test procedure enjoys consistency properties both for the size and power asymptoticallywhen the dimension goes to infinity while the sample sizes are fixed. We check the performance of the proposed test procedure by numerical simulations. Finally, we give a demonstration using microarray data sets and explain how to select a suitable test procedure.

Suppose we have unreplicated responses on fractional factorial designsof multilevel factors. In this setting, the response functions arerepresented by polynomial functions, and the identifiability of the parameter can be characterized by the Gröbner basis theory. This argument is one of the classical topics on the computational algebraic statistics introduced by Pistone and Wynn (1996). In Pistone and Wynn (1996), the concept of the confounding of factors, which is an important concept in the theory of the design of experiments, is defined as the ideal membership problem to the set of polynomials vanishing on the design points (i.e., the design ideal). In this framework, some of the concepts such as confoundingand resolution, which are usually considered in the settings of regular fractional factorial designs of two-level or three-level factors, can be generalized to general fractional factorial designs.In this paper, based on the computational algebraic methods, the relationship between the ANOVA models and the polynomial models on the designs is considered. The ANOVA models are usually considered in the setting of balanced multi-way layout, and are equivalent to the polynomial models in the two-level cases. On the other hand, the correspondence is not completely clear in the general settings. In this paper, we define the ANOVA model in the setting of unbalanced multi-way layout. We also propose some ideas to characterize the ANOVA models as the polynomial models, and how to apply the computational algebraic methods to the ANOVA models.