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_14: In the sparse linear regression setting, we consider testing the significance of the predictor variable that enters the current lasso model, in the sequence of models visited along the lasso solution path. We propose a simple test statistic based on lasso fitted values, called the covariance test statistic, and show that when the true model is linear, this statistic has an Exp(1) asymptotic distribution under the null hypothesis (the null being that all truly active variables are contained in the current lasso model). Our proof of this result for the special case of the first predictor to enter the model (i.e., testing for a single significant predictor variable against the global null) requires only weak assumptions on the predictor matrix X. On the other hand, our proof for a general step in the lasso path places further technical assumptions on X and the generative model, but still allows for the important high-dimensional case p > n, and does not necessarily require that the current lasso model achieves perfect recovery of the truly active variables. Of course, for testing the significance of an additional variable between two nested linear models, one typically uses the chi-squared test, comparing the drop in residual sum of squares (RSS) to a χ12 distribution. But when this additional variable is not fixed, and has been chosen adaptively or greedily, this test is no longer appropriate: adaptivity makes the drop in RSS stochastically much larger than χ12 under the null hypothesis. Our analysis explicitly accounts for adaptivity, as it must, since the lasso builds an adaptive sequence of linear models as the tuning parameter λ decreases. In this analysis, shrinkage plays a key role: though additional variables are chosen adaptively, the coefficients of lasso active variables are shrunken due to the l1 penalty. Therefore, the test statistic (which is based on lasso fitted values) is in a sense balanced by these two opposing properties—adaptivity and shrinkage—and its null distribution is tractable and asymptotically Exp(1).
@cite_14 develop a test for the hypothesis that a newly added coefficient along the LASSO regularization path is irrelevant. This however does not allow to test arbitrary coefficients at a given value of @math , which is instead the problem addressed in this paper. These authors further assume that the current LASSO support contains the actual support @math and that the latter has bounded size.