Extreme conditional quantile estimation for time series

Abstract

We consider the estimation of an extreme conditional quantile for a heavy‐tailed distribution in the case of a strictly stationary time series . Here, denotes the conditional quantile function of given and is small, i.e., smaller than , where is the size of the sample on which the estimation is based. In a first step, we propose a conditional extreme value index estimator in that context of time series and establish its weak convergence using the cluster method. Then, in a second step, by using a Weissman‐type construction, we propose an extreme conditional quantile estimator for which we derive also the weak convergence. The performance of our estimators is illustrated on a simulation study and compared with those of alternative estimators already proposed in the literature. Finally, our methodology is applied on a real dataset of log‐returns of the S&P 500 index and market volatility.