Multiple testing and variable selection along the path of the least angle regression

Abstract We investigate multiple testing and variable selection using the Least Angle Regression (LARS) algorithm in high dimensions under assumption of Gaussian noise. LARS is known to produce a piecewise affine solution path with change points referred as knots path. The key our results an expression closed form exact joint law $K$-tuple conditional on variables selected by LARS, so-called post-selection knots. Numerical experiments demonstrate perfect fit findings. This paper makes three main contributions. First, we build procedures entering model along general design case when noise level can be unknown. These are Generalized $t$-Spacing tests prove that they have non-asymptotic (i.e. Type I error exactly controlled). extends work [ 31] where spacing test works for consecutive variance. Second, introduce new procedure after may this has unknown level. Third, control false discovery rate orthogonal assumption. Monte-Carlo simulations real data experiment provided illustrate case. Of independent interest, equivalent formulation based recursive function.