Penultimate approximation for the distribution of the excesses

Rym Worms

ESAIM: Probability and Statistics (2010)

  • 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 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 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 (2010): 21-31. <http://eudml.org/doc/104290>.

@article{Worms2010,
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 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 $G_\{\gamma\}(\frac\{x\}\{\sigma(u)\})$, 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 Λ 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.; generalized Pareto distribution; rate of convergence},
language = {eng},
month = {3},
pages = {21-31},
publisher = {EDP Sciences},
title = {Penultimate approximation for the distribution of the excesses},
url = {http://eudml.org/doc/104290},
volume = {6},
year = {2010},
}

TY - JOUR
AU - Worms, Rym
TI - Penultimate approximation for the distribution of the excesses
JO - ESAIM: Probability and Statistics
DA - 2010/3//
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 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 $G_{\gamma}(\frac{x}{\sigma(u)})$, 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 Λ 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.; generalized Pareto distribution; rate of convergence
UR - http://eudml.org/doc/104290
ER -

References

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  6. I. Gomes and L. de Haan, Approximation by penultimate extreme value distributions. Extremes2 (2000) 71-85.  
  7. M.I. Gomes and D.D. Pestana, Non standard domains of attraction and rates of convergence. John Wiley & Sons (1987) 467-477.  
  8. J. Pickands III, Statistical inference using extreme order statistics. Ann. Stat.3 (1975) 119-131.  
  9. J.-P. Raoult and R. Worms, Rate of convergence for the Generalized Pareto approximation of the excesses (submitted).  
  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).  

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