On some estimates based on sample behavior near high level excursions (Q1326312)

Let \(\{\xi_ j\}\) be a stationary sequence of weakly dependent random variables and let \(M_ n^{(k)}\) be the \(k\)-th largest value of \(\xi_ j\), \(1 \leq j \leq n\). The estimation of the parameters of the asymptotic distribution of \(M_ n^{(k)}\) is considered using a procedure motivated by a limit theorem pertaining to the point process \(\sum_ j \delta_{(j/n,n \overline F(\xi_ j))}\). A number of statistical issues concerning the procedure, including how to select the tuning parameters, are addressed. The second problem that we consider is the estimation of the filter of a moving average process with heavy tails. In particular, the investigation covers the moving average stable process. Motivated by ideas of \textit{H. Rootzén} [Ann. Probab. 6, 847-869 (1978; Zbl 0394.60025)], our estimator uses information contained in the sample behavior of the process near the largest excursion.