Distributionally Robust Conditional Quantile Prediction with Fixed Design

Meng Qi (), Ying Cao () and Zuo-Jun (Max) Shen ()
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Meng Qi: Department of Industrial Engineering and Operations Research, University of California, Berkeley, Berkeley, California 94704
Ying Cao: Department of Industrial Engineering and Operations Research, University of California, Berkeley, Berkeley, California 94704
Zuo-Jun (Max) Shen: College of Engineering, University of California, Berkeley, Berkeley, California 94704; Faculty of Engineering and Faculty of Business and Economics, University of Hong Kong, Hong Kong

Management Science, 2022, vol. 68, issue 3, 1639-1658

Abstract: Conditional quantile prediction involves estimating/predicting the quantile of a response random variable conditioned on observed covariates. The existing literature assumes the availability of independent and identically distributed (i.i.d.) samples of both the covariates and the response variable. However, such an assumption often becomes restrictive in many real-world applications. By contrast, we consider a fixed-design setting of the covariates, under which neither the response variable nor the covariates have i.i.d. samples. The present study provides a new data-driven distributionally robust framework under a fixed-design setting. We propose a regress-then-robustify method by constructing a surrogate empirical distribution of the noise. The solution of our framework coincides with a simple yet practical method that involves only regression and sorting, therefore providing an explanation for its empirical success. Measure concentration results are obtained for the surrogate empirical distribution, which further lead to finite-sample performance guarantees and asymptotic consistency. Numerical experiments are conducted to demonstrate the advantages of our approach.

Keywords: quantile prediction; data-driven newsvendor; distributionally robust optimization; Wasserstein distance (search for similar items in EconPapers)
Date: 2022
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