Confidence Regions for Stochastic Variational Inequalities

The sample average approximation (SAA) method is a basic approach for solving stochastic variational inequalities (SVI). It is well known that under appropriate conditions the SAA solutions provide asymptotically consistent point estimators for the true solution to an SVI. It is of fundamental inter...

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Bibliographic Details
Published in:Mathematics of operations research 2013-08, Vol.38 (3), p.545-568
Main Authors: Lu, Shu, Budhiraja, Amarjit
Format: Article
Language:English
Subjects:
Approximation theory
central limit theorems in Banach spaces
Confidence intervals
confidence regions
Inequalities (Mathematics)
large deviations
Operations research
Statistical analysis
statistical inference
Stochastic models
Stochastic processes
stochastic variational inequalities
Studies
variational inequalities
Variational principles
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Summary:The sample average approximation (SAA) method is a basic approach for solving stochastic variational inequalities (SVI). It is well known that under appropriate conditions the SAA solutions provide asymptotically consistent point estimators for the true solution to an SVI. It is of fundamental interest to use such point estimators along with suitable central limit results to develop confidence regions of prescribed level of significance for the true solution. However, standard procedures are not applicable because the central limit theorem that governs the asymptotic behavior of SAA solutions involves a discontinuous function evaluated at the true solution of the SVI. This paper overcomes such a difficulty by exploiting the precise geometric structure of the variational inequalities and by appealing to certain large deviations probability estimates, and proposes a method to build asymptotically exact confidence regions for the true solution that are computable from the SAA solutions. We justify this method theoretically by establishing a precise limit theorem, apply it to complementarity problems, and test it with a linear complementarity problem.
ISSN:0364-765X
1526-5471
DOI:10.1287/moor.1120.0579