Fitting high-dimensional statistical models often requires the use of non-linear parameter estimation procedures. As a consequence, it is generally impossible to obtain an exact characterization of the probability distribution of the parameter estimates. This in turn implies that it is extremely challenging to quantify the associated with a certain parameter estimate. Concretely, no commonly accepted procedure exists for computing classical measures of uncertainty and statistical significance as confidence intervals or @math -values for these models. We consider here high-dimensional linear regression problem, and propose an efficient algorithm for constructing confidence intervals and @math -values. The resulting confidence intervals have nearly optimal size. When testing for the null hypothesis that a certain parameter is vanishing, our method has nearly optimal power. Our approach is based on constructing a de-biased' version of regularized M-estimators. The new construction improves over recent work in the field in that it does not assume a special structure on the design matrix. We test our method on synthetic data and a high-throughput genomic data set about riboflavin production rate.
@cite_31: We propose a method for constructing p-values for general hypotheses in a high-dimensional linear model. The hypotheses can be local for testing a single regression parameter or they may be more global involving several up to all parameters. Furthermore, when considering many hypotheses, we show how to adjust for multiple testing taking dependence among the p-values into account. Our technique is based on Ridge estimation with an additional correction term due to a substantial projection bias in high dimensions. We prove strong error control for our p-values and provide sufficient conditions for detection: for the former, we do not make any assumption on the size of the true underlying regression coefficients while regarding the latter, our procedure might not be optimal in terms of power. We demonstrate the method in simulated examples and a real data application.
@cite_32: The purpose of this paper is to propose methodologies for statistical inference of low-dimensional parameters with high-dimensional data. We focus on constructing confidence intervals for individual coefficients and linear combinations of several of them in a linear regression model, although our ideas are applicable in a much broad context. The theoretical results presented here provide sufficient conditions for the asymptotic normality of the proposed estimators along with a consistent estimator for their finite-dimensional covariance matrices. These sufficient conditions allow the number of variables to far exceed the sample size. The simulation results presented here demonstrate the accuracy of the coverage probability of the proposed confidence intervals, strongly supporting the theoretical results.
In other words, the guarantees of @cite_32 @cite_31 can be suboptimal by a factor as large as @math . Equivalently, in order for the coefficient @math to be detectable with appreciable probability, it needs to be larger than the overall @math error. Here we will propose a test that --for random designs-- achieves significance level @math and power @math for @math .