Extremal local linear quantile regression for nonlinear dependent processes

Estimating extreme conditional quantiles accurately in the presence of data sparsity in the tails is a challenging and important problem. While there is existing literature on quantile analysis, limited work has been done on capturing nonlinear relationships in dependent data structures for extreme quantile estimation. They propose a novel estimation procedure that combines the local linear quantile regression method and extreme value theory. They develop a new enhanced Hill estimator for the conditional extreme value index, constructed based on the local linear quantile estimators at a sequence of quantile levels. That approach allows for data-adaptive weights assigned to different quantiles, providing flexibility and potential for enhancing estimation efficiency. Furthermore, they propose an estimator for extreme conditional quantiles by extrapolating from the intermediate quantiles. Their methodology enables both point and interval estimation of extreme conditional quantiles for processes with an α-mixing dependence structure. They derive the Bahadur representation of the intermediate quantile estimators within the local linear extreme-quantile framework and establish the asymptotic properties of their proposed estimators. Simulation studies and real data analysis are conducted to demonstrate the effectiveness and performance of their methods.