Estimation of Extreme Conditional Expectile Via Tail Risk Transition from Intermediate Quantile

In risk management, expectile is a widely applied tail risk measure as well as Value-at-Risk. It is of great theoretical interest to estimate extreme conditional expectiles, because classic statistical approaches may introduce substantial bias when estimating extreme quantiles or expectiles due to the data sparsity on the tail region. Xu, Hou and Li (2022) introduces an approach for this estimation based on a tail equivalence transition relationship between the quantile and expectile, but without any theoretical study. In this paper, we first develop the theoretical results for the estimation based on the tail equivalence transition. Second, we propose another novel estimation method and study its asymptotic properties. Simulation studies show that both methods based on the tail equivalence transition perform well. Empirical studies applied to S&P 500 index return illustrate that our two estimation methods provide robust risk measurement tools for studying extreme tail risk.