Hostname: page-component-77c89778f8-m8s7h Total loading time: 0 Render date: 2024-07-25T05:06:06.713Z Has data issue: false hasContentIssue false
  • English
  • Français

Poisson approximation of intermediate empirical processes

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

E. Kaufmann  and
R.-D. Reiss
E. Kaufmann*
Affiliation:
University of Siegen
R.-D. Reiss
Affiliation:
University of Siegen
*
∗ Postal address for both authors: FB 6, Uni-Gh Siegen, Hölderlinstr. 3, 5900 Siegen, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the asymptotic behaviour of empirical processes truncated outside an interval about the (1 – s(n)/n)-quantile where s(n) → ∞ and s(n)/n → 0 as the sample size n tends to ∞. It is shown that extreme value (Poisson) processes and, alternatively, the homogeneous Poisson process may serve as approximations if certain von Mises conditions hold.

Keywords

POISSON PROCESSESVARIATIONAL AND HELLINGER DISTANCE

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 4 , December 1992 , pp. 825 - 837
Copyright
Copyright © Applied Probability Trust 1992 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Work partially supported by Deutsche Forschungsgemeinschaft.

References

[1] Anderson, C. W. (1971) Contributions to the Asymptotic Theory of Extreme Values. , University of London.Google Scholar
[2] Cooil, B. (1984) Limiting multivariate distributions of intermediate order statistics. Ann. Prob. 13, 469–477.Google Scholar
[3] Davison, A. C. and Smith, R. L. (1990) Models for exceedances over high threshold. J. R. Statist. Soc. B52, 393–442.Google Scholar
[4] Falk, M. (1990) A note on uniform asymptotic normality of intermediate order statistics. Ann. Inst. Statist. Math. 41, 19–29.Google Scholar
[5] Falk, M, and Reiss, R.-D. (1992) Poisson approximation of empirical processes. Statist. Prob. Lett. 14, 39–48.CrossRefGoogle Scholar
[6] Matthes, K., Kerstan, J. and Mecke, J. (1978) Infinitely Divisible Point Processes. Wiley, Chichester.Google Scholar
[7] Reiss, R.-D. (1989) Approximate Distributions of Order Statistics: With Applications to Nonparametric Statistics. Springer-Verlag, New York.Google Scholar
[8] Resnick, S. I. (1987) Extreme Values, Regular Variation, and Point Processes. Springer-Verlag, New York.Google Scholar
[9] Smith, R. L. (1982) Uniform rates of convergence in extreme value theory. Adv. Appl. Prob. 14, 600–622.Google Scholar