Statistical Inference For High Dimensional Linear Models
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Statistical Inference for High-Dimensional Linear Models
High-dimensional linear models play an important role in the analysis of modern data sets. Although the estimation problem has been well understood, there is still a paucity of methods and theories on the inference problem for high-dimensional linear models. This thesis focuses on statistical inference for high-dimensional linear models and consists of the following three parts. 1. The first part of the thesis considers confidence intervals for linear functionals in high-dimensional linear regression. We first establish the convergence rates of the minimax expected length for confidence intervals. Furthermore, we investigate the problem of adaptation to sparsity for the construction of confidence intervals and identify the regimes in which it is possible to construct adaptive confidence intervals. 2. In the second part of the thesis, we consider point and interval estimation of the lq loss of a given estimator in high-dimensional linear regression. For the class of rate-optimal estimators, we establish the minimax rates for estimating their lq losses, the minimax expected length of confidence intervals for their lq losses and the possibility of adaptivity of confidence intervals for their lq losses. 3. In the third part of the thesis, we consider the problem in the framework of high-dimensional instrumental variable regression and construct confidence intervals for the treatment effect in the presence of possibly invalid instrumental variables. We develop a novel selection procedure, Two-Stage Hard Thresholding (TSHT) to select valid instrumental variables and construct honest confidence intervals for the treatment effect using the selected instrumental variables.
Statistics for High-Dimensional Data
Author: Peter Bühlmann
language: en
Publisher: Springer Science & Business Media
Release Date: 2011-06-08
Modern statistics deals with large and complex data sets, and consequently with models containing a large number of parameters. This book presents a detailed account of recently developed approaches, including the Lasso and versions of it for various models, boosting methods, undirected graphical modeling, and procedures controlling false positive selections. A special characteristic of the book is that it contains comprehensive mathematical theory on high-dimensional statistics combined with methodology, algorithms and illustrations with real data examples. This in-depth approach highlights the methods’ great potential and practical applicability in a variety of settings. As such, it is a valuable resource for researchers, graduate students and experts in statistics, applied mathematics and computer science.
Partially Linear Models
Author: Wolfgang Härdle
language: en
Publisher: Springer Science & Business Media
Release Date: 2000-09-14
In the last ten years, there has been increasing interest and activity in the general area of partially linear regression smoothing in statistics. Many methods and techniques have been proposed and studied. This monograph hopes to bring an up-to-date presentation of the state of the art of partially linear regression techniques. The emphasis is on methodologies rather than on the theory, with a particular focus on applications of partially linear regression techniques to various statistical problems. These problems include least squares regression, asymptotically efficient estimation, bootstrap resampling, censored data analysis, linear measurement error models, nonlinear measurement models, nonlinear and nonparametric time series models.