Statistical Estimation of Composite Risk Functionals and Risk Optimization Problems

Abstract

We address the statistical estimation of composite functionals whichmay be nonlinear in the probability measure. Our study is motivated bythe need to estimate coherent measures of risk, which become increasinglypopular in finance, insurance, and other areas associated with optimization under uncertainty and risk. We establish central limit formulae forcomposite risk functionals. Furthermore, we discuss the asymptotic behavior of optimization problems whose objectives are composite risk functionals and we establish a central limit formula of their optimal valueswhen an estimator of the risk functional is used. While the mathematicalstructures accommodate commonly used coherent measures of risk, theyhave more general character, which may be of independent interest.