Universal Confidence Sets for Solutions of Stochastic Optimization Problems—A Contribution to Quantification of Uncertainty

Abstract

Confidence sets are well-known tools in parametric statistics. A similar concept can be successfully applied to solutions of random optimization problems, which occur if unknown quantities are replaced with estimates. The so-called universal confidence sets yield for each sample size n a conservative confidence set. The method is based on convergence properties of sequences of random closed sets. In this paper, we will show how the approach can be adapted if a decision maker has at disposal only noisy outcomes for a certain set of decisions. We consider a multivariate Fixed-Design regression model for the objective function and estimate the function by the Priestley–Chao kernel estimator. For each sample size n, a uniform confidence band for the true function will be derived which yields the main prerequisite for the construction of universal confidence sets for the optimal decisions.