Topics in high dimensional statistical estimation and inference

This thesis deals with three problems. The first two of the problems are related in that they are concerned with estimation of correlation and precision matrix in spectral norm. These two problems are tackled in Chapters 2, 3. The third problem is the construction of chi-squared type test for groups of variables in high dimensional linear regression. In Chapter 2, we study concentration in spectral norm of nonparametric estimates of correlation matrices. We study two nonparametric estimates of correlation matrices in Gaussian copula models and prove that when both the number of variables and sample size are large, the spectral error of the nonparametric estimators is of no greater order than that of the latent sample covariance matrix, at least when compared with some of the sharpest known error bounds for the later. As an application, we establish the minimax optimal rate in the estimation of high-dimensional bandable correlation matrices via tapering off of these nonparametric estimators. An optimal convergence rate for sparse principal component analysis is also established. In Chapter 3, we study the sparse precision matrix estimation procedure in the same Gaussian copula model as in Chapter 2. We employ the scaled Lasso procedure for inversion of nonparametric correlation matrix estimates based on Kendall’s tau. We prove the optimal rate of convergence in estimation of sparse precision matrices under the weaker condition of bound on the spectral norm of the precision matrix. Chapter 4 deals with confidence regions and approximate chi-squared tests for variable groups in high-dimensional linear regression. We develop a scaled group Lasso for efficient chi-squared-based statistical inference of variable groups. We prove that the proposed methods capture the benefit of group sparsity under proper conditions, for statistical inference of the noise level and variable groups, large and small. Oracle inequalities are provided for the scaled group Lasso in prediction and several estimation losses, and for the group Lasso as well in a weighted mixed loss. Some simulation results are also provided in support of the theory.Ph.D.Includes bibliographical referencesby Ritwik Mitr