Extremiles, quantiles and expectiles in the tails

Quantiles and expectiles analyze the regression model not only at the mean but also in the tails. Financial losses, medical insurance, auction bids, insurance claims and toxicity limits are all areas where it is relevant to analyze the model far from its average behavior. The extremile, another estimator designed to investigate the linear regression in the tails, is considered in this paper. The performance of these three estimators depends on the shape of the error distribution, particularly when looking at the far tails of the conditional distribution of the dependent variable. Our simulations consider skewed, contaminated and heteroscedastic error distributions. Contamination and heteroscedasticity greatly perturb the quantiles and expectiles in the tails. Extremiles present the smallest mean absolute errors in cases of contamination and heteroscedasticity as well as the smallest expected shortfall in all the experiments. The quantile estimator systematically overestimates the selected location of the dependent variable in all the experiments and yields the largest expected shortfall. Two case studies analyzing real data complete our analysis.