Penultimate approximation for the distribution of the excesses
ESAIM: Probability and Statistics, Volume 6 (2002), pp. 21-31.

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 ${F}_{u}\left(x\right)$, where ${F}_{u}$ is the d.f of the excesses over $u$, converges, when $u$ tends to ${s}_{+}\left(F\right)$, 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 $\Lambda$ 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 $d\left(u\right)={sup}_{x\in \left[0,{s}_{+}\left(F\right)-u\left[}|{\overline{F}}_{u}\left(x\right)-{\overline{G}}_{\gamma }\left(x/\sigma \left(u\right)\right)|$.

DOI: 10.1051/ps:2002002
Classification: 60G70, 62G20
Keywords: generalized Pareto distribution, excesses, penultimate approximation, rate of convergence
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Worms, Rym. Penultimate approximation for the distribution of the excesses. ESAIM: Probability and Statistics, Volume 6 (2002), pp. 21-31. doi : 10.1051/ps:2002002. http://archive.numdam.org/articles/10.1051/ps:2002002/

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