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On some estimates based on sample behavior near high level excursions
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  • Published: September 1993

On some estimates based on sample behavior near high level excursions

  • Tailen Hsing1 

Probability Theory and Related Fields volume 95, pages 331–356 (1993)Cite this article

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Summary

Let {ξ j } be a stationary sequence of weakly dependent random variables and letM (k)n be thek-th largest value of ξ j , 1≦j≦n. The estimation of the parameters of the asymptotic distribution ofM (k)n is considered using a procedure motivated by a limit theorem pertaining to the point process\(\sum\nolimits_j {\delta _{(j/n, n\bar F(\xi _j ))} } \). A number of statistical issues concerning the procedure, including how to select the tuning parameters, are addressed. The second problem that we consider is the estimation of the filter of a moving average process with heavy tails. In particular, the investigation covers the moving average stable process. Motivated by ideas in Rootzén (1978), our estimator uses information contained in the sample behavior of the process near the largest excursion.

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References

  • Adler, R.J.: The geometry of random fields. New York: Wiley 1981

    Google Scholar 

  • Billingsley, P.: Convergence of probability measures. New York: Wiley 1968

    Google Scholar 

  • Brockwell, P., Davis, R.A.: Simple consistent estimation of the coefficients of a linear filter. Stochastic Processes Appl28, 47–59 (1988)

    Google Scholar 

  • Brockwell, P., Davis R.A.: Time series: Theory and methods. Berlin Heidelberg New York: Springer 1991

    Google Scholar 

  • Davis, R.A., Resnick, R.I.: Limit theory for moving averages of random varibles with regularly varying tail probabilities. Ann Probab.13, 179–195 (1985)

    Google Scholar 

  • Dekkers, A.L.M., Haan, L.de: An the estimation of the extreme-value index and large quantile estimation. Ann. Stat.17, 1795–1832 (1989)

    Google Scholar 

  • Dekkers, A.L.M., Einmahl, J.H.J., Haan, L.de: A moment estimator for the index of an extremevalue distribution. Ann. Stat.17, 1833–1855 (1989)

    Google Scholar 

  • Feller, W.: An introduction to probability theory and its applications. New York: Wiley 1971

    Google Scholar 

  • Hsing, T.: On the characterization of certain point processes. Stochastic Processes Appl.26, 297–316 (1987)

    Google Scholar 

  • Hsing, T.: On the weak convergence of extreme order statistics. Stochastic Processes Appl.29, 155–169 (1988)

    Google Scholar 

  • Hsing, T.: Estimating the parameters of rare events. Stochastic Processes Appl.37, 117–139 (1991a)

    Google Scholar 

  • Hsing, T.: Estimating tail index using dependent data. Ann. Stat.19, 1547–1569 (1991b)

    Google Scholar 

  • Hsing, T.: Estimations based on high level excursions. Technical Report, Department of Statistics, Texas A&M University 1991c

  • Hsing, T.: Extremal index estimation for weakly dependent stationary sequences. (Preprint, 1991d)

  • Hsing, T.: Leadbetter, M.R.: On the excursion random measures of stationary processes. (Preprint, 1991)

  • Hsing, T., Hüsler, J., Leadbetter, M.R.: On the exceedance point process for a stationary sequence. Probab. Theory Relat. Fields78, 97–112 (1988)

    Google Scholar 

  • Ibragimov, I.A., Linnik, Y.V.: Independent and stationary sequences of random variables. Groningen: Wolters-Noordhoff 1969

    Google Scholar 

  • Kallenberg, O.: Random measures. New York: Academic Press 1983

    Google Scholar 

  • Leadbetter, M.R.: Extremes and local dependence in a stationary sequence. Z. Wahrscheinlichkeitstheor. Verw. Geb.65, 291–306 (1983)

    Google Scholar 

  • Leadbetter, M.R., Hsing, T.: On exceedance random measures for stationary processes. Stochastic Processes Appl.36, 231–243 (1990)

    Google Scholar 

  • Leadbetter, M.R., Rootzén, H.: Extremal theory for stochastic processes. Ann. Probab.16, 431–478 (1988)

    Google Scholar 

  • Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and related properties of random sequences and processes. Berlin Heibelberg New York: Springer 1983

    Google Scholar 

  • Leadbetter, M.R., Weissman, I., de Haan, L., Rootzén, H.: On clustering of high values in statistically stationary series. Technical Report no. 253, Center for Stochastic Processes, Department of Statistics, University of North Carolina 1989

  • Matthes, K., Kerstan, J., Mecke, J.: Infinitely divisible point processes. New York: Wiley 1978

    Google Scholar 

  • Mori, T.: Limit distributions of two-dimensional point processes generated by strong mixing sequences. Yokohama Math. J.25, 155–168 (1977)

    Google Scholar 

  • Nandagopolan, L.: Multivariate extreme value theory and extremal index estimation (Dissertation) Technical Report no. 315, Center for Stochastic Processes, Department of Statistics, University of North Carolina 1990

  • O'Brien, G.L.: Extreme values for stationary and Markov sequences. Ann. Probab.15, 281–289 (1987)

    Google Scholar 

  • Peligrad, M.: Recent advances in the central limit theorem and its weak invariance principle for mixing sequences of random variables (A survey). In: Eberlein, E., Taqqu, M.S. (eds.) Dependence in probability and statistics, pp. 193–224. Boston Basel Stuttgart: Birkhäuser 1986

    Google Scholar 

  • Pickands, J., III: Statistical inference using extreme order statistics. Ann. Stat.3, 119–131 (1975)

    Google Scholar 

  • Resnick, S.I.: Extreme values, regular variation, and point processes. Berlin Heidelberg New York: Springer 1987

    Google Scholar 

  • Rootzén, H.: Extremes of moving averages of stable processes. Ann. Probab.6, 847–869 (1978)

    Google Scholar 

  • Rootzén, H.: Extreme value theory for moving average processes. Ann Probab.14, 612–652 (1986)

    Google Scholar 

  • Rootzén, H.: Maxima and exceedances of stationary Markov chains. Adv. Appl. Probab.20, 371–390 (1988)

    Google Scholar 

  • Rootzén, H., Leadbetter, M.R., de Haan, L.: Tail and quantile estimation for strongly mixing stationary sequences. Technical Report no. 292, Center for Stochastic Processes, University of North Carolina at Chapel Hill 1990

  • Samorodnitsky, G., Taqqu, M.: Stable non-Gaussian random processes. (Forthcoming, 1993)

  • Smith, R.L.: Estimating tails of probability distributions. Ann. Stat.15, 1174–1207 (1987)

    Google Scholar 

  • Smith, R.L.: Extreme value analysis of environmental times series. Stat. Sci.4, 367–393 (1989)

    Google Scholar 

  • Smith, R.L., Weissman, I.: Estimating the extremal index. (Preprint, 1991)

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Author information

Authors and Affiliations

  1. Department of Statistics, Texas A&M University, 77843, College Station, TX, USA

    Tailen Hsing

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  1. Tailen Hsing
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Additional information

Research supported by AFOSR Contract No. 91-0030, NAVY-ONR Grant No. N00014-92-J-1007, and NSF Grant No. 9107507

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Hsing, T. On some estimates based on sample behavior near high level excursions. Probab. Th. Rel. Fields 95, 331–356 (1993). https://doi.org/10.1007/BF01192168

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  • Received: 20 May 1991

  • Revised: 18 August 1992

  • Issue Date: September 1993

  • DOI: https://doi.org/10.1007/BF01192168

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Keywords

  • Stochastic Process
  • Probability Theory
  • Limit Theorem
  • Mathematical Biology
  • Point Process
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