Penultimate approximation for the excesses (Q2741011)

Let \(F\) be a distribution function (d.f.) in the maximum domain of attraction of an extreme value distribution with the parameter \(\gamma\). If \(F_u\) is the d.f. of the excesses over \(u\), and \(u\) tends to the right end-point of \(F\), then the d.f. \(F_u\) of the excesses over \(u\) tends to a corresponding d.f. of the generalized Pareto distribution. In a previous paper (2000), the author studied the asymptotic behaviour of the ``ultimate'' approximation of \(F_u\). The present paper gives appropriate conditions that ensure the asymptotic behaviour of the ``penultimate'' approximation \(F_u\) of the excesses, and proves that its rate of convergence is faster than the convergence rate of the ``ultimate'' approximation.