Almost sure convergence of the stable tail empirical dependence function in multivariate extreme statistics
Consider a bivariate i.i.d. sequence \((X_1Y_1)\), \((X_2,Y_2),\dots\) with joint d.f. \(F(x,y)\). Assume that \(F\) has a ``stable tail dependence function'' \(l(x,y)\), i.e. for \(x,y>0\), \[ l(x,y):= \lim_{t\downarrow 0} t^{-1} \{1-F(Q_1(tx), Q_2(ty))\} \] exists, where \(Q_i(x)= \sup\{y: 1-F_i(x)\geq x\}\), \(0\leq x\leq 1\) \((i=1,2)\). The author's main result proves that \[ l_n(x,y)= k^{-1} \sum_{j=1} I(X_j\geq X_{n-[kx]+1,n} \text{ or } Y_j\geq Y_{n-[ky]+1,n}), \] \(1\leq k\leq n\), \(k= k(n)\), \(k/\log\log n\to\infty\), but \(k/n\to 0\) (as \(n\to\infty\)), is a strongly consistent estimator of \(l(x,y)\), uniformly on \([0,T]\times [0,T]\) for any \(T>0\). This extends an earlier result of \textit{X. Huang} [Statistics of bivariate extreme values. Ph.D. Diss., Erasmus Univ., Rotterdam (1992)] who e.g. proved convergence in probability. A possible generalization to a higher-dimensional setting is also discussed.
stable tail empirical dependence function
scientific article; zbMATH DE number 1063728