Risk measures of a financial position are, from an empirical point of view, mainly based on quantiles. Replacing quantiles with their least squares analogues, called expectiles, has recently received increasing attention. The novel expectile-based risk measures satisfy all coherence requirements. We revisit their extreme value estimation for heavy-tailed distributions. First, we estimate the underlying tail index via weighted combinations of top order statistics and asymmetric least squares estimates. The resulting expectHill estimators are then used as the basis for estimating tail expectiles and Expected Shortfall. The asymptotic theory of the proposed estimators is provided, along with numerical simulations and applications to actuarial and financial data.
Asymmetric least squares; Coherent risk measures; Expected shortfall; Expectile; Extrapolation; Extremes; Heavy tails; Tail index;
- C13: Estimation: General
- C14: Semiparametric and Nonparametric Methods: General
Abdelaati Daouia, Stéphane Girard, and Gilles Stupfler, “ExpectHill estimation, extreme risk and heavy tails”, TSE Working Paper, n. 18-953, September 2018.
Abdelaati Daouia, Stéphane Girard, and Gilles Stupfler, “ExpectHill estimation, extreme risk and heavy tails”, Journal of Econometrics, vol. 221, n. 1, March 2021, pp. 97–117.