Optimally combined estimation for tail quantile regression

Abstract

Quantile regression offers a convenient tool to access the relationship between a response and covariates in a comprehensive way and it is appealing especially in applications where interests are on the tails of the response distribution. However, due to data sparsity, the finite sample estimation at tail quantiles often suffers from high variability. To improve the tail estimation efficiency, we consider modeling multiple quantiles jointly for cases where the quantile slope coefficients tend to be constant at the tails. We propose two estimators, the weighted composite estimator that minimizes the weighted combined quantile objective function across quantiles, and the weighted quantile average estimator that is the weighted average of quantile-specific slope estimators. By using extreme value theory, we establish the asymptotic distributions of the two estimators at the tails, and propose a procedure for estimating optimal weights. We show that the optimally weighted estimators improve the efficiency over equally weighted estimators, and the efficiency gain depends on the heaviness of the tail distribution. The performance of the proposed estimators is assessed through a simulation study and the analysis of precipitation downscaling data.