Rates of uniform convergence of extreme order statistics
The main result of the paper is a sharp upper bound of the variational distance between the distribution of an extreme order statistic and an extreme value distribution. The concept of generalized Pareto distributions is crucial for the derivation and the formulation of the upper bound. Let f be the underlying density and \(w_ n\) the density of a shifted generalized Pareto distribution which corresponds to the approximating extreme value distribution G. Then the upper bound of the remainder term of the approximation is of the form \[ O[\int (\frac{a_ nf(a_ nx+b_ n)}{w_ n}-1)^ 2dG]^{1/2} \] where \(a_ n>0\) and the real \(b_ n\) are normalizing constants. Of equal importance is the extension of the one-dimensional result to the joint distribution of extreme order statistics. Several examples highlight the applicability of the results mentioned above.
scientific article; zbMATH DE number 3978154
scientific article