# Penultimate approximation for the distribution of the excesses

ESAIM: Probability and Statistics (2002)

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

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topWorms, 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 $\gamma >-1$, 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 $\gamma >-1$, 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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- [9] J.-P. Raoult and R. Worms, Rate of convergence for the Generalized Pareto approximation of the excesses (submitted). Zbl1044.60041
- [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] R. Worms, Vitesses de convergence pour l’approximation des queues de distributions Ph.D. Thesis Université de Marne-la-Vallée (2000).

## Citations in EuDML Documents

top- Jean Diebolt, Armelle Guillou, Rym Worms, Asymptotic behaviour of the probability-weighted moments and penultimate approximation
- Jean Diebolt, Armelle Guillou, Rym Worms, Asymptotic behaviour of the probability-weighted moments and penultimate approximation
- Julien Worms, Rym Worms, Estimation of second order parameters using probability weighted moments
- Julien Worms, Rym Worms, Estimation of second order parameters using probability weighted moments

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