Data perturbations in stochastic generalized equations: statistical robustness in static and sample average approximated models

• Published in: Mathematical programming 2023-11, Vol.202 (1-2), p.135-168
• Main Authors: Guo, Shaoyan, Xu, Huifu
• Format: Article
• Language: English
• Subjects: Analysis; Approximation; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Contamination; Controllability; Empirical analysis; Estimators; Full Length Paper; Influence functions; Laws, regulations and rules; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Monte Carlo method; Monte Carlo simulation; Numerical Analysis; Perturbation; Robustness; Statistical analysis; Stochastic programming; Theoretical
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-023-01925-9

Sample average approximation which is also known as Monte Carlo method has been widely used for solving stochastic programming and equilibrium problems. In a data-driven environment, samples are often drawn from empirical data and hence may be potentially contaminated. Consequently it is legitimate to ask whether statistical estimators obtained from solving the sample average approximated problems are statistically robust, that is, the difference between the laws of the statistical estimators based on contaminated data and real data is controllable under some metrics. In Guo and Xu (Math Program 190:679–720, 2021), we address the issue for the estimators of the optimal values of a wide range of stochastic programming problems. In this paper, we complement the research by investigating the optimal solution estimators and we do so by considering stochastic generalized equations (SGE) as a unified framework. Specifically, we look into the impact of a single data perturbation on the solutions of the SGE using the notion of influence function in robust statistics. Since the SGE may have multiple solutions, we use the proto-derivative of a set-valued mapping to introduce the notion of generalized influence function (GIF) and derive sufficient conditions under which the GIF is well defined, bounded and uniformly bounded. We then move on to quantitative statistical analysis of the SGE when all of sample data are potentially contaminated and demonstrate under moderate conditions quantitative statistical robustness of the solutions obtained from solving sample average approximated SGE.