On some estimates based on sample behavior near high level excursions

Summary Let { _j } be a stationary sequence of weakly dependent random variables and letM _n ^(k) be thek-th largest value of _j , 1jn. The estimation of the parameters of the asymptotic distribution ofM _n ^(k) is considered using a procedure motivated by a limit theorem pertaining to the point process . 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 Rootzén (1978), our estimator uses information contained in the sample behavior of the process near the largest excursion.Research supported by AFOSR Contract No. 91-0030, NAVY-ONR Grant No. N00014-92-J-1007, and NSF Grant No. 9107507