# 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 ${F}_{u}\left(x\right)$, 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 $d\left(u\right)={sup}_{x\in \left[0,{s}_{+}\left(F\right)-u\left[}|{\overline{F}}_{u}\left(x\right)-{\overline{G}}_{\gamma }\left(x/\sigma \left(u\right)\right)|$.

## How to cite

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Worms, Rym. "Penultimate approximation for the distribution of the excesses." ESAIM: Probability and Statistics 6 (2002): 21-31. <http://eudml.org/doc/245324>.

@article{Worms2002,
abstract = {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 $F_u(x)$, where $F_u$ is the d.f of the excesses over $u$, converges, when $u$ tends to $s_+(F)$, the end-point of $F$, to $G_\{\gamma \}(\frac\{x\}\{\sigma (u)\})$, where $G_\{\gamma \}$ is the d.f. of the Generalized Pareto Distribution. We provide conditions that ensure that there exists, for $\gamma &gt;-1$, a function $\Lambda$ which verifies $\lim _\{u \rightarrow s_+(F)\} \Lambda (u) =\gamma$ and is such that $\Delta (u)= \sup _\{x \in [0,s_+(F)-u[\} |\bar\{F\}_u(x) - \bar\{G\}_\{\Lambda (u)\} (x/ \sigma (u))|$ converges to $0$ faster than $d(u)=\sup _\{x \in [0,s_+(F)-u[\} |\bar\{F\}_u(x) - \bar\{G\}_\{\gamma \}(x/ \sigma (u))|$.},
author = {Worms, Rym},
journal = {ESAIM: Probability and Statistics},
keywords = {generalized Pareto distribution; excesses; penultimate approximation; rate of convergence},
language = {eng},
pages = {21-31},
publisher = {EDP-Sciences},
title = {Penultimate approximation for the distribution of the excesses},
url = {http://eudml.org/doc/245324},
volume = {6},
year = {2002},
}

TY - JOUR
AU - Worms, Rym
TI - Penultimate approximation for the distribution of the excesses
JO - ESAIM: Probability and Statistics
PY - 2002
PB - EDP-Sciences
VL - 6
SP - 21
EP - 31
AB - 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 $F_u(x)$, where $F_u$ is the d.f of the excesses over $u$, converges, when $u$ tends to $s_+(F)$, the end-point of $F$, to $G_{\gamma }(\frac{x}{\sigma (u)})$, where $G_{\gamma }$ is the d.f. of the Generalized Pareto Distribution. We provide conditions that ensure that there exists, for $\gamma &gt;-1$, a function $\Lambda$ which verifies $\lim _{u \rightarrow s_+(F)} \Lambda (u) =\gamma$ and is such that $\Delta (u)= \sup _{x \in [0,s_+(F)-u[} |\bar{F}_u(x) - \bar{G}_{\Lambda (u)} (x/ \sigma (u))|$ converges to $0$ faster than $d(u)=\sup _{x \in [0,s_+(F)-u[} |\bar{F}_u(x) - \bar{G}_{\gamma }(x/ \sigma (u))|$.
LA - eng
KW - generalized Pareto distribution; excesses; penultimate approximation; rate of convergence
UR - http://eudml.org/doc/245324
ER -

## References

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8. [8] J. Pickands III, Statistical inference using extreme order statistics. Ann. Stat. 3 (1975) 119-131. Zbl0312.62038MR423667
9. [9] J.-P. Raoult and R. Worms, Rate of convergence for the Generalized Pareto approximation of the excesses (submitted). Zbl1044.60041
10. [10] R. Worms, Vitesse de convergence de l’approximation de Pareto Généralisée de la loi des excès. Preprint Université de Marne-la-Vallée (10/2000).
11. [11] R. Worms, Vitesses de convergence pour l’approximation des queues de distributions Ph.D. Thesis Université de Marne-la-Vallée (2000).

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