Strong domain of attraction of extreme generalized order statistics (Q1424689)

For the ordinary order statistics it is known that under the von Mises conditions the distribution function \(F\) belongs to the strong domain of attraction of an extreme value distribution \(G\), i.e. the convergence with respect to the variational distance holds true \(\sup_{B}| P\{a_{n}^{-1}(X_{n,n}-b_{n})\in B\}-G(B)|\to0\) as \(n\to\infty\), where \(\sup\) is taken over all Borel sets \(B\subset R\). The author proves that the von Mises conditions imply that the underlying distribution function \(F\) belongs to the strong domain of attraction of a generalized extreme value distribution. Moreover, it turns out that the strong domains of attractions of joint distributions of the fixed number of generalized extreme order statistics can be characterized by means of the corresponding result for generalized maxima. The author determines the asymptotic joint distribution of (upper and lower) extreme generalized order statistics, and shows that the Hill estimator based on extreme generalized order statistics is asymptotically normal.