Consider the following three important problems in statistical inference, namely, constructing confidence intervals for (1) the error of a high-dimensional ( @math ) regression estimator, (2) the linear regression noise level, and (3) the genetic signal-to-noise ratio of a continuous-valued trait (related to the heritability). All three problems turn out to be closely related to the little-studied problem of performing inference on the @math -norm of the signal in high-dimensional linear regression. We derive a novel procedure for this, which is asymptotically correct when the covariates are multivariate Gaussian and produces valid confidence intervals in finite samples as well. The procedure, called EigenPrism, is computationally fast and makes no assumptions on coefficient sparsity or knowledge of the noise level. We investigate the width of the EigenPrism confidence intervals, including a comparison with a Bayesian setting in which our interval is just 5 wider than the Bayes credible interval. We are then able to unify the three aforementioned problems by showing that the EigenPrism procedure with only minor modifications is able to make important contributions to all three. We also investigate the robustness of coverage and find that the method applies in practice and in finite samples much more widely than just the case of multivariate Gaussian covariates. Finally, we apply EigenPrism to a genetic dataset to estimate the genetic signal-to-noise ratio for a number of continuous phenotypes.
The residual variance and the proportion of explained variation are important quantities in many statistical models and model fitting procedures. They play an important role in regression diagnostics and model selection procedures, as well as in determining the performance limits in many problems. In this paper we propose new method-of-moments-based estimators for the residual variance, the proportion of explained variation and other related quantities, such as the l2 signal strength. The proposed estimators are consistent and asymptotically normal in high-dimensional linear models with Gaussian predictors and errors, where the number of predictors d is proportional to the number of observations n; in fact, consistency holds even in settings where d n → ∞. Existing results on residual variance estimation in high-dimensional linear models depend on sparsity in the underlying signal. Our results require no sparsity assumptions and imply that the residual variance and the proportion of explained variation can be consistently estimated even when d>n and the underlying signal itself is nonestimable. Numerical work suggests that some of our distributional assumptions may be relaxed. A real-data analysis involving gene expression data and single nucleotide polymorphism data illustrates the performance of the proposed methods.