Statistical inference for extreme extremile in heavy-tailed heteroscedastic regression model

As a least squares analogue of quantiles, extremiles define a coherent risk measure determined by weighted expectations instead of tail probabilities. Estimating extremiles of heavy-tailed variables in a regression framework is a challenging task, especially for dependent cases. This paper develops some methods for the estimation of extreme conditional extremiles in the framework of heteroscedastic regression model with heavy-tail noises, specifically, direct and indirect methods based on the conditional extremile estimators for the residuals. We also construct corresponding bias-reduced estimators and investigate their asymptotic properties compared to the original versions. Our mathematical assumptions are satisfied in the mean-variance regression model and heteroscedastic single-index model, which makes it possible to apply our result in a series of important examples. We demonstrate our results through a simulation study and real sets of insurance and financial data analyses.