Wilks confidence regions for empirical weighted quantiles

Quantiles are fundamental tools in statistics and risk analysis. While asymptotic and finite-sample results for standard empirical quantiles are well established, analogous results for weighted quantiles remain scarce. In this paper, we establish a comprehensive asymptotic theory for weighted quantiles. We derive a multivariate central limit theorem for multiple perturbed weighted quantiles. This result yields, as corollaries, (i) a multivariate CLT for weighted empirical quantiles, (ii) an asymptotically distribution-free confidence interval for weighted quantiles in the spirit of Wilks’ method, and (iii) Wilks confidence bounds for the weighted expected shortfall. Our theoretical contributions are also supported by numerical experiments, which code is publicly available at https://github.com/michael-allouche/confidence-region-weighted-quantile . • Wilks confidence regions for empirical weighted quantiles. • Multivariate central limit theorem for weighted quantiles. • Distribution-free confidence interval for univariate weighted quantile. • Confidence bounds for the weighted expected shortfall. • Scenarios with the heaviest tails are in average further away from the target significance level.