Estimation of high conditional quantiles using the Hill estimator of the tail index

To implement the extremal quantile regression, one needs to have an accurate estimate of the tail index that is involved in the limit distributions of extremal regression quantiles. However, the existing quantile estimation methods are often unstable owing to data sparsity in the tails. In this paper, we propose the Hill estimator for the tail index based on regression quantiles, and construct a new estimator for high conditional quantiles through an extrapolation of the intermediate regression quantiles. In both theory and simulation, we demonstrate that the proposed estimators are more efficient than those based on the refined Pickands estimator of the tail index. The applicability of the new method is also illustrated on the Occidental Petroleum daily stock return data.