Abstract: High dimensional linear regression is one of the most important models in analyzing modern data sets. Although the estimation problem is well understood, there is still a paucity of methods and fundamental theoretical results on confidence intervals for high dimensional linear regression. In this talk, I will present confidence interval results for a general linear functional. I will first construct confidence intervals of optimal expected length in the oracle setting of known sparsity level. Then, I will focus on the problem of adaptation to sparsity for the construction of confidence intervals. I will identify the regimes in which it is possible to construct adaptive confidence intervals. In terms of optimality and adaptivity, there are striking differences between linear functionals with a sparse loading and a dense loading.

In the framework of high dimensional linear models, another interesting quantity is the normalized inner product of the two regression vectors, which can represent an important concept in genetics, the genetic correlation between phenotypes. I will introduce Functional De-biased Estimator (FDE) which achieves the optimal convergence rate of estimating the genetic correlation. The FDE estimator is applied to estimate the genetic correlations among different phenotypes in a yeast data set. Finally, I will discuss an interesting connection between the aforementioned problems and provide a unified view of the proposed procedures.