Optimal Component Selection in High Dimensions
Xin Cui · 2014
It is now the modern trend and reality in various fields of life and science that the data sets to be analyzed are high dimensional, and the number of observations is much smaller than their dimension.As the classical statistical methods are not designed to deal with big or high-dimensional data, the problem of developing new methods of high-dimensional statistical analysis is very important.In many statistical applications such as analysis of microarray data, signal recovery, and functional magnetic resonance imaging, the focus is often on identifying and estimating a relatively few significant components from a high-dimensional vector.In this thesis, we study the problem of component or variable selection in a normal mixture model based on a single high-dimensional observation.The goal is to examine the possibilities and limitations of optimal component selection in a two-point normal mixture model and, whenever possible, to construct optimal (nonimprovable) selection procedures.In addition to that, the problem of estimating an unknown parameter that determines the sparsity pattern of the data is addressed.The main theoretical findings of the thesis obtained in Chapter 1, and partly in Chapter 2, are supported by the simulation study; the results of the simulation study are presented in Chapter 3. The main results of the thesis, Theorems 2-5, are new.