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.
Acknowledgements
We would like to thank the editor for organizing an effective review and two anonymous referees for insightful comments and constructive suggestions which help us significantly to consolidate the paper. The research is supported by the NSFC grants #11971090, #11971220, Fundamental Research Funds for the Central Universities under grant DUT19LK24 and Guangdong Basic and Applied Basic Research Foundation 2019A1515011152.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Notes
1 The uniqueness and Lipschitz of is with respect to the perturbation of the function
in SVIP (Equation3
(3)
(3) ).
2 For a given point, the difference of the SAA function and the true one can be taken as a random vector.
3 denotes the B-derivative, see [Citation16, Page 547] for details.
4 denotes the Banach space of continuously differentiable mappings
.
5 The ball is defined in the Banach space of continuously differentiable mappings equipped with the norm defined as in [Citation16, (9) page 4].
6 A solution to complementarity problem (Equation16
(16)
(16) ) is said to be non-degenerate if
.