Confidence regions of stochastic variational inequalities: error bound approach

Abstract

ABSTRACT In this paper, we aim to build confidence regions of the true solution to the stochastic variational inequalities problem (SVIP) when the sample average approximation (SAA) scheme is implemented. A new approach based on error bound conditions admitted by the SVIP is proposed. This so-called error bound approach provides an upper bound of the distance between SAA solutions and the true solution set through the distance between the SAA function and the true counterpart at the SAA solutions. Certain statistical tools such as central limit theorem and Owen's empirical likelihood theorem are then employed to construct the asymptotic confidence regions of the solutions to SVIP. In particular, if the SVIP admits a global error bound condition, the non-asymptotic (uniform) confidence regions of the solutions are also approachable. Different from the conventional normal map approach, our error bound approach does not require any information regarding the derivative of the solution mapping with respect to perturbations of involved functions in SVIP. For constructing component-wise confidence regions, the validity of the error bound approach is guaranteed for those cases where the functions own separable structures.