Confidence Intervals for Piecewise Normal Distributions and Stochastic Variational Inequalities

Abstract

In this paper, we first show how to obtain easy-to-compute confidence intervals for the center of a piecewise normal distribution given a sample from this distribution (assuming that the center belongs to a known affine set parallel to the common lineality space of all cones defining the piecewise normal distribution) by using certain skewed projectors on that space. We then extend this method to an asymptotic setting. Next, we apply this method to compute confidence intervals for the true solution of a stochastic variational inequality given a solution to a sample average approximation (SAA) problem for the general situation in which the asymptotic distribution of SAA solutions is piecewise normal. For stochastic complementarity problems, we obtain asymptotic normality of certain estimators of the true solution when the asymptotic distribution of the SAA solutions is piecewise normal. Funding: The research reported in this paper was supported the National Science Foundation [Grant DMS-1814894].