Penultimate approximation for the distribution of the excesses

Rym Worms

ESAIM: Probability and Statistics (2002)

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

Abstract

top
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

top

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

top
  1. [1] A. Balkema and L. de Haan, Residual life time at great age. Ann. Probab. 2 (1974) 792-801. Zbl0295.60014MR359049
  2. [2] C.M. Goldie, N.H. Bingham and J.L. Teugels, Regular variation. Cambridge University Press (1987). Zbl0617.26001MR898871
  3. [3] J.P. Cohen, Convergence rates for the ultimate and penultimate approximations in extreme-value theory. Adv. Appl. Prob. 14 (1982) 833-854. Zbl0496.62019MR677559
  4. [4] R.A. Fisher and L.H.C. Tippet, Limiting forms of the frequency of the largest or smallest member of a sample. Proc. Cambridge Phil. Soc. 24 (1928) 180-190. Zbl54.0560.05JFM54.0560.05
  5. [5] M.I. Gomes, Penultimate limiting forms in extreme value theory. Ann. Inst. Stat. Math. 36 (1984) 71-85. Zbl0561.62015MR752007
  6. [6] I. Gomes and L. de Haan, Approximation by penultimate extreme value distributions. Extremes 2 (2000) 71-85. Zbl0947.60019MR1772401
  7. [7] M.I. Gomes and D.D. Pestana, Non standard domains of attraction and rates of convergence. John Wiley & Sons (1987) 467-477. Zbl0618.62023MR900238
  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). 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.