Hostname: page-component-848d4c4894-p2v8j Total loading time: 0 Render date: 2024-05-06T19:47:57.712Z Has data issue: false hasContentIssue false
  • English
  • Français

Consistency of Hill's estimator for dependent data

Part of: Nonparametric inference Limit theorems Parametric inference Stochastic processes Distribution theory

Published online by Cambridge University Press:  14 July 2016

Sidney Resnick  and
Cătălin Stărică
Sidney Resnick*
Affiliation:
Cornell University
Cătălin Stărică*
Affiliation:
Cornell University
*
Postal address of both authors: School of ORIE, Cornell University Ithaca, NY 14853, USA.
Postal address of both authors: School of ORIE, Cornell University Ithaca, NY 14853, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider a sequence of possibly dependent random variables having the same marginal distribution F, whose tail 1−F is regularly varying at infinity with an unknown index − α < 0 which is to be estimated. For i.i.d. data or for dependent sequences with the same marginal satisfying mixing conditions, it is well known that Hill's estimator is consistent for α−1 and asymptotically normally distributed. The purpose of this paper is to emphasize the central role played by the tail empirical process for the problem of consistency. This approach allows us to easily prove Hill's estimator is consistent for infinite order moving averages of independent random variables. Our method also suffices to prove that, for the case of an AR model, the unknown index can be estimated using the residuals generated by the estimation of the autoregressive parameters.

Keywords

REGULAR VARIATIONTIME SERIESTAIL EMPIRICAL PROCESSHEAVY TAILSINFINITEORDER MOVING AVERAGESAUTOREGRESSIVE PROCESSES

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G10: Stationary processes 60G35: Signal detection and filtering 60G57: Random measures 62E17: Approximations to distributions (nonasymptotic) 62F10: Point estimation 62G30: Order statistics; empirical distribution functions
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 1 , March 1995 , pp. 139 - 167
Copyright
Copyright © Applied Probability Trust 1995 

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

∗

Partially supported by NSF Grant DMS-9100027 at Cornell University. Some support was also received from NSA Grant 92G-116.

∗∗

Supported by NSF Grant DMS-9100027 at Cornell University.

References

[1] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[2] Bingham, N., Goldie, C. and Teugels, J. (1987) Regular Variation. Cambridge University Press.Google Scholar
[3] Brockwell, P. and Davis, R. (1991) Time Series: Theory and Methods, 2nd edn. Springer-Verlag, New York.CrossRefGoogle Scholar
[4] Brockwell, P. and Davis, R. (1991) ITSM, an Interactive Time Series Modelling Package for the PC. Springer-Verlag, New York.Google Scholar
[5] Cline, D. (1983) Infinite series of random variables with regularly varying tails. Technical Report 83-24, Institute of Applied Mathematics and Statistics, University of British Columbia.Google Scholar
[6] Davis, R. and Resnick, S. (1985) Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Prob. 13, 179–195.CrossRefGoogle Scholar
[7] Davis, R. and Resnick, S. (1986) Limit theory for sample covariance and correlation functions. Ann. Statist. 14, 533–558.Google Scholar
[8] Deheuvels, P. and Mason, D. M. (1991) A tail empirical process approach to some nonstandard laws of the iterated logarithm. J. Theoret. Prob. 4, 53–85.CrossRefGoogle Scholar
[9] Feigin, P. D. and Resnick, S. (1992) Estimation for autoregressive processes with positive innovations. Stoch. Models 8, 479–498.Google Scholar
[10] Feigin, P. D. and Resnick, S. (1994) Limit distributions for linear programming time series estimators. Stoch. Proc. Appl. 51, 135–165.CrossRefGoogle Scholar
[11] Hill, B. (1975) A simple approach to inference about the tail of a distribution. Ann. Statist. 3, 1163–1174.Google Scholar
[12] Hsing, T. (1991) On tail estimation using dependent data. Ann. Statist. 19, 1547–1569.Google Scholar
[13] Kallenberg, O. (1983) Random Measures, 3rd edn. Akademie-Verlag, Berlin.Google Scholar
[14] Knight, K. (1989) Order selection for autoregressions. Ann. Statist. 17, 824–840.Google Scholar
[15] Mason, D. (1982) Laws of large numbers for sums of extreme values. Ann. Prob. 10, 754–764.Google Scholar
[16] Mason, D. (1988) A strong invariance theorem for the tail empirical process. Ann. Inst. H. Poincaré 24, 491–506.Google Scholar
[17] Mikosch, T., Gadrich, T., Klüppelberg, C. and Adler, R. (1993) Estimation for infinite variance ARMA models. Ann. Statist. Google Scholar
[18] Resnick, S. (1986) Point processes, regular variation and weak convergence. Adv. Appl. Prob. 18, 66–138.Google Scholar
[19] Resnick, S. (1987) Extreme Values, Regular Variation and Point Processes. Springer-Verlag, Berlin.Google Scholar
[20] Rootzén, H., Leadbetter, M. and De Haan, L. (1990) Tail and quantile estimation for strongly mixing stationary sequences. Preprint, Econometric Institute.Google Scholar