ON SOME ESTIMATES BASED ON SAMPLE BEHAVIOR NEAR HIGH-LEVEL EXCURSIONS

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 less-than-or-equal-to j less-than-or-equal-to 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 SIGMA(j)delta(j/n, nF(xij)BAR). 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 in Rootzen (1978), our estimator uses information contained in the sample behavior of the process near the largest excursion.