# Penultimate approximation for the distribution of the excesses

• Volume: 6, page 21-31
• ISSN: 1292-8100

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## Abstract

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Let F be a distribution function (d.f) in the domain of attraction of an extreme value distribution ${H}_{\gamma }$; it is well-known that Fu(x), where Fu is the d.f of the excesses over u, converges, when u tends to s+(F), the end-point of F, to ${G}_{\gamma }\left(\frac{x}{\sigma \left(u\right)}\right)$, where ${G}_{\gamma }$ is the d.f. of the Generalized Pareto Distribution. We provide conditions that ensure that there exists, for $\gamma >-1$, a function Λ which verifies ${lim}_{u\to {s}_{+}\left(F\right)}\Lambda \left(u\right)=\gamma$ and is such that $\Delta \left(u\right)={sup}_{x\in \left[0,{s}_{+}\left(F\right)-u\left[}|{\overline{F}}_{u}\left(x\right)-{\overline{G}}_{\Lambda \left(u\right)}\left(x/\sigma \left(u\right)\right)|$ converges to 0 faster than $d\left(u\right)={sup}_{x\in \left[0,{s}_{+}\left(F\right)-u\left[}|{\overline{F}}_{u}\left(x\right)-{\overline{G}}_{\gamma }\left(x/\sigma \left(u\right)\right)|$.

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