Extreme expectile estimation for heavy-tailed time series

 Expectiles define a least squares analogue of quantiles which have later received substantial attention in actuarial and financial risk management contexts. Unlike quantiles, expectiles define coherent risk measures and are determined by 

tail expectations rather than tail probabilities; unlike the popular Expected Shortfall, they define elicitable risk measures. The behaviour and estimation of extreme expectiles based on independent and identically distributed heavy-tailed data has recently been considered in a number of papers. The stationary but weakly dependent case has, however, been much less addressed. We investigate here the theoretical and practical behaviour of two extreme expectile estimators in a strictly stationary framework context that contains most of the ARMA, ARCH and GARCH models with heavy-tailed innovations that are of interest in financial applications. We focus on the construction of asymptotic confidence intervals adapted to the dependence framework. The methods are showcased in a numerical simulation study and a real financial data example.