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 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 selection procedures. In addition to that, the problem of estimating an unknown parameter that determines the sparsity pattern of the data is addressed.