Penultimate approximation for the distribution of the excesses

Rym Worms

ESAIM: Probability and Statistics (2002)

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

Abstract

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Let F be a distribution function (d.f) in the domain of attraction of an extreme value distribution H γ ; 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 γ ( x σ ( u ) ) , where G γ is the d.f. of the Generalized Pareto Distribution. We provide conditions that ensure that there exists, for γ > - 1 , a function Λ which verifies lim u → s + ( F ) Λ ( u ) = γ and is such that Δ ( u ) = sup x ∈ [ 0 , s + ( F ) - u [ | F ¯ u ( x ) - G ¯ Λ ( u ) ( x / σ ( u ) ) | converges to 0 faster than d ( u ) = sup x ∈ [ 0 , s + ( F ) - u [ | F ¯ u ( x ) - G ¯ γ ( x / σ ( u ) ) | .

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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