Convergence Analysis on Data-Driven Fortet-Mourier Metrics with Applications in Stochastic Optimization

Zhiping Chen, He Hu and Jie Jiang
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Zhiping Chen: School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China
He Hu: School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China
Jie Jiang: School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China

Sustainability, 2022, vol. 14, issue 8, 1-29

Abstract: Fortet-Mourier (FM) probability metrics are important probability metrics, which have been widely adopted in the quantitative stability analysis of stochastic programming problems. In this study, we contribute to different types of convergence assertions between a probability distribution and its empirical distribution when the deviation is measured by FM metrics and consider their applications in stochastic optimization. We first establish the quantitative relation between FM metrics and Wasserstein metrics. After that, we derive the non-asymptotic moment estimate, asymptotic convergence, and non-asymptotic concentration estimate for FM metrics, which supplement the existing results. Finally, we apply the derived results to four kinds of stochastic optimization problems, which either extend the present results to more general cases or provide alternative avenues. All these discussions demonstrate the motivation as well as the significance of our study.

Keywords: Fortet-Mourier metric; discrete approximation; stochastic optimization; stochastic dominance; distributionally robust (search for similar items in EconPapers)
JEL-codes: O13 Q Q0 Q2 Q3 Q5 Q56 (search for similar items in EconPapers)
Date: 2022
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