High-Dimensional Statistical Analysis: New Developments of Theories and Methodologies

Abstract

In this paper, we introduce new developments of theories and methodologies in high-dimensional statistical analysis. Recently, Aoshima and Yata (2018a) have provided a noise model called the strongly spiked eigenvalue (SSE) model. Since the noise associated with high dimensional data is huge and non-sparse, the potential geometric structure of the data is destroyed and it is difficult to guarantee the accuracy for statistical inferences. In theory, the high-dimensional asymptotic normality that forms the basis of high-dimensional statistical analysis is not established under the SSE model. Aoshima and Yata (2018a) have developed a data transformation technique that avoids strongly spiked-noise spaces by precisely analyzing the huge noise structure. Using this method, the data is transformed into the non-strongly spiked eigenvalue (NSSE) model, which highlights the geometric structure of the latent space and enables highly accurate high-dimensional statistical inference. Aoshima and Yata (2018b) have applied this methodology to create a new theory for high-dimensional discriminant analysis. In this current paper, we explain the latest developments of high-dimensional statistical analysis while appropriately introducing literature.