New estimators for the extremal index and other cluster characteristics
Let \(\{X_n \}\) be a stationary sequence of rv's satisfying the asymptotic independence condition \(D(u_n)\) of \textit{M.R. Leadbetter, G. Lindgren} and \textit{H. Rootzen} [Extremes and related properties of random sequences and processes. (1983; Zbl 0518.60021)]. Let \(\{X^*_n \}\) be the associated sequence of independent rv's. As known, \[ P( max_{1\leq k\leq n}X_k \leq u_n ) \rightarrow G^{\theta}(x) \Longleftrightarrow P( max_{1\leq k\leq n} X^*_k \leq u_n ) \rightarrow G(x), \] as \(n\rightarrow \infty\). Thus, the extremal index \(0 \leq \theta \leq 1\) may be seen as a measure of dependence in extreme values. Estimating \(\theta\) is equivalent to identifying independent clusters. Different understanding of clusters leads to different estimators for \(\theta\). The run estimator of \(\theta\), \(\theta(u,m) = P( max_{2\leq k\leq m}X_k \leq u \mid X_1 >u)\), based on the characterization of \textit{G.L. O'Brien} [see Ann. Probab. 15 281--291 (1987; Zbl 0619.60025)], takes independent clusters to be exceedances of \(u\) that are separated by \(m-1\) consecutive values which do not exceed \(u\). The authors' two-thresholds extension \(\theta(u,m,c), \quad c<u \), of the runs approach has clusters terminating if a sufficiently small (i.e., \( \leq c \) ) value of the process is observed between two exceedances of the threshold \(u\). The authors consider two types of dependence: induced by either autoregression or by volatility. On different examples they illustrate the advantages of their new method for identifying clusters. Furthermore, the authors apply the method to real data for the Wooster daily minimum temperatures and the FTSE100 daily returns series.
scientific article; zbMATH DE number 2102873