Operations Research Seminar - "A conditional tail expectation type risk measure for time series" - Armelle Guillou, Strasbourg University

We consider the estimation of the conditional expectation E(Xh|X0 > QX(1 − p)) at extreme levels, where (Xt)t∈Z is a strictly stationary β−mixing time series, QX its associated quantile function, p ∈ (0, 1) and h a positive integer. We use the multivariate regular variation framework and start to consider the case of non-negative time series. A two-step method is used in order 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 Weissmantype 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. Some simulations are provided in order to illustrate the performance of our estimator.