A Conditional Tail Expectation Type Risk Measure for Time Series

We consider the estimation of the conditional expectation (Formula presented.), provided (Formula presented.), at extreme levels, where (Formula presented.) is a strictly stationary time series, (Formula presented.) its tail quantile function, (Formula presented.) is a positive integer and (Formula presented.) is such that (Formula presented.). We use the multivariate regular variation framework and consider initially the case of non-negative time series. A two-step method is used to propose an estimator of this risk measure: First, by introducing an estimator in the intermediate case, and then, by extrapolating outside the data by a Weissman-type construction. Under suitable assumptions, we prove the weak convergence of the estimator of this risk measure. Subsequently, we extend our approach to the case of real-valued time series by using the decomposition of the original time series into the positive and negative parts, and we prove again the weak convergence of the proposed estimator under additional assumptions. The asymptotic variance of this estimator being difficult to approximate, we show the consistency of the multiplier block bootstrap in our context and use it to construct confidence intervals for (Formula presented.). Finally, the finite sample properties of the estimator are evaluated with a simulation study, and the methodology is illustrated on a dataset of daily precipitation measurements.