TY  - JOUR
A1  - Bachoc, Francois
A1  - Blanchard, Gilles
A1  - Neuvial, Pierre
T1  - On the post selection inference constant under restricted isometry properties
JF  - Electronic journal of statistics
N2  - Uniformly valid confidence intervals post model selection in regression can be constructed based on Post-Selection Inference (PoSI) constants. PoSI constants are minimal for orthogonal design matrices, and can be upper bounded in function of the sparsity of the set of models under consideration, for generic design matrices. In order to improve on these generic sparse upper bounds, we consider design matrices satisfying a Restricted Isometry Property (RIP) condition. We provide a new upper bound on the PoSI constant in this setting. This upper bound is an explicit function of the RIP constant of the design matrix, thereby giving an interpolation between the orthogonal setting and the generic sparse setting. We show that this upper bound is asymptotically optimal in many settings by constructing a matching lower bound.
KW  - Inference post model-selection
KW  - confidence intervals
KW  - PoSI constants
KW  - linear regression
KW  - high-dimensional inference
KW  - sparsity
KW  - restricted isometry property
Y1  - 2018
U6  - https://doi.org/10.1214/18-EJS1490
SN  - 1935-7524
VL  - 12
IS  - 2
SP  - 3736
EP  - 3757
PB  - Institute of Mathematical Statistics
CY  - Cleveland
ER  -