Data -analytic modeling for high -dimensional statistical problems.

摘要

With advancement of technology and huge investment in various forms of data gathering, high dimensional problems have become increasingly important in quantitative research of many disciplines. Large sample sizes with high dimensions are important characteristics of many challenging problems in science, engineering and medicine. To reduce possible modeling biases, many flexible models are introduced at the initial phase of study. It poses a new theoretical and methodological challenge to study whether methods used for finite-dimensional problems are still useful in high-dimensional problems. This forms the theme of this thesis. In this thesis, three inter-related topics on high-dimensional statistical modeling are studied, which are detailed below.;A class of variable selection procedures for parametric models via nonconcave penalized likelihood is proposed by Fan and Li (2001) to simultaneously estimate parameters and select important variables. They demonstrate that this class of procedures has an oracle property when the number of parameters is finite. However, in most model selection problems, the number of parameters should be large, and grow with the sample size. In this thesis, some asymptotic properties of the nonconcave penalized likelihood are established for situations in which the number of parameters tends to infinity as the sample size increases. Under regularity conditions, we have established an oracle property and the asymptotic normality of the penalized likelihood estimators. Furthermore, the consistency of the sandwich formula of the covariance matrix is demonstrated. Nonconcave penalized likelihood ratio statistics are discussed, and their asymptotic distributions under the null hypothesis are obtained by imposing some mild conditions on the penalty functions. The asymptotic results are augmented by a simulation study and the newly developed methodology is illustrated by an analysis of the court case on the sexual discrimination of salary. (Abstract shortened by UMI.).