Group-bound: confidence intervals for groups of variables in sparse high-dimensional regression without assumptions on the design

Abstract: It is in general challenging to provide confidence intervals for individual variables in high-dimensional regression without making strict or unverifiable assumptions on the design matrix. We show here that a "group-bound" confidence interval can be derived without making any assumptions on the design matrix. The lower bound for the regression coefficient of individual variables can be derived via linear programming. The idea also generalises naturally to groups of variables, where we can derive a one-sided confidence interval for the joint effect of a group. While the confidence intervals of individual variables are by the nature of the problem often very wide, it is shown to be possible to detect the contribution of groups of highly correlated predictor variables even when no variable individually shows a significant effect. The assumptions necessary to detect the effect of groups of variables are shown to be weaker than the weakest known assumptions to detect the effect of individual variables.