A Conditional Tail Expectation Type Risk Measure for Time Series
针对极端水平下的时间序列，提出一种条件尾部期望风险度量的估计方法，通过两步法和Weissman型外推构造估计量，并证明其弱收敛性，最后用模拟和日降水量数据验证。
ABSTRACT We consider the estimation of the conditional expectation , provided , at extreme levels, where is a strictly stationary time series, its tail quantile function, is a positive integer and is such that . 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 . 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.