Asymptotic test for independence of extreme values
Let \((X_ 1,Y_ 1),(X_ 2,Y_ 2),\dots\) be a sequence of independent random vectors with a common distribution \(L\) with marginal \(R\) and \(S\) (in short, we write \(L\in{\mathcal M}(R,S))\). Let \(R\) and \(S\) be such that with suitable norming constants \(a_ n\), \(b_ n\), \(c_ n\) and \(d_ n\) the random variables \[ X(n)= (\max \{X_ 1,\dots,X_ n\}- b_ n)a_ n^{-1} \qquad \text{and} \qquad Y(n)= (\max \{Y_ 1,\dots,Y_ n\}- c_ n)d_ n^{-1} \] have nondegenerate limiting distributions \(G\) and \(H\), respectively. Assuming that \((X(n),Y(n))\) converges in distribution to some \(F\in {\mathcal M}(G,H)\) we study conditions that will assure \(F=GH\).
scientific article; zbMATH DE number 575889