Rank Conditional Coverage and Confidence Intervals in High-Dimensional Problems

Confidence interval procedures used in low-dimensional settings are often inappropriate for high-dimensional applications. When many parameters are estimated, marginal confidence intervals associated with the most significant estimates have very low coverage rates: They are too small and centered at biased estimates. The problem of forming confidence intervals in high-dimensional settings has previously been studied through the lens of selection adjustment. In that framework, the goal is to control the proportion of noncovering intervals formed for selected parameters. In this article, we approach the problem by considering the relationship between rank and coverage probability. Marginal confidence intervals have very low coverage rates for the most significant parameters and high rates for parameters with more boring estimates. Many selection adjusted intervals have the same behavior despite controlling the coverage rate within a selected set. This relationship between rank and coverage rate means that the parameters most likely to be pursued further in follow-up or replication studies are the least likely to be covered by the constructed intervals. In this article, we propose rank conditional coverage (RCC) as a new coverage criterion for confidence intervals in multiple testing/covering problems. The RCC is the expected coverage rate of an interval given the significance ranking for the associated estimator. We also propose two methods that use bootstrapping to construct confidence intervals that control the RCC. Because these methods make use of additional information captured by the ranks of the parameter estimates, they often produce smaller intervals than marginal or selection adjusted methods. These methods are implemented in R (R Core Team, 2017 R Core Team (2017), R: A Language and Environment for Statistical Computing, Vienna, Austria: R Foundation for Statistical Computing. Google Scholar]) in the package rcc available on CRAN at https://cran.r-project.org/web/packages/rcc/index.html. Supplementary material for this article is available online.