Distributionally Robust Conditional Quantile Prediction with Fixed Design

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.