Multivariate Extensions of Nonlinear Control Variates and Concomitants of Order Statistics

We develop multivariate extensions to two variance reduction techniques { nonlinear control variates for estimating expected values, and concomitants of order statistics for estimating a distribution (including expected values, probabilities, and percentiles). We allow the relationship between Y and the X variables to be nonlinear, and consider two cases { where the relationship is additive, and the general case. We use additive regression models and univariate numerical integration to implement the additive case, and numerical integration or Quasi-Monte Carlo methods as part of the implementation of Monte Carlo variance reduction methods.