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The change-of-variance function for dependent data
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  • Published: December 1991

The change-of-variance function for dependent data

  • Ola Hössjer1 

Probability Theory and Related Fields volume 90, pages 447–467 (1991)Cite this article

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  • 4 Citations

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Summary

The infinitesimal stability of the asymptotic variance is considered forM-estimators of a location parameter when the nominal sample with i.i.d. data is contaminated by a possibly dependent process. It is shown that the resulting change-of-variance function can be expressed as a sum of two terms, one corresponding contamination of the univariate distribution, and one to contamination of the bivariate distributions. A change-of-variance sensitivity is introduced, the form of which is closely related to the average patch length of the outliers. Finally, optimalV-robust and mostV-robust score functions are derived. The resulting family of estimators is the same as for independent data in the general case, but the truncation point approaches zero when dependency is accounted for. For redescending score-functions, the family of estimators is changed.

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References

  • Beran, J., Künsch, H.: Location estimators for processes with long range dependence. Research report No 40, Sem. für Statist. ETH, 1985

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

    Google Scholar 

  • Bustos, O.H.: GeneralM-estimates for contaminatedpth-order autoregressive processes: Consistency and asymptotic normality. Z. Wahrscheinlichkeitstheor. Verw. Geb.59, 491–504 (1982)

    Google Scholar 

  • Fox, R., Taqqu, M.S.: Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Stat.14, 517–532 (1986)

    Google Scholar 

  • Franke, J., Hannan, E.J.: Comment on “Influence functionals for time series”. Ann. Stat.14, 822–824 (1986)

    Google Scholar 

  • Hampel, F.R.: The influence curve and its role in robust estimation. J. Am. Stat. Assoc.69, 383–393 (1974)

    Google Scholar 

  • Huber, P.J.: Robust estimation of a location parameter. Ann. Math. Stat.35, 73–101 (1964)

    Google Scholar 

  • Huber, P.J.: The behaviour of maximum likelihood estimates under nonstandard conditions. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 221–233, 1967

    Google Scholar 

  • Hössjer, O.: The change-of-variance function for dependent data. U.U.D.M. Report 1989: 18, Dept. of Mathematics, Uppsala University, 1989

  • Künsch, H.: Infinitesimal robustness for autoregressive processes. Ann. Stat.12, 843–863 (1984)

    Google Scholar 

  • Künsch, H.: Statistical aspects of self-similar processes. Bernoulli1, 67–74 (1987)

    Google Scholar 

  • Lee, C.-H., Martin, R.D.: Ordinary and properM-estimates for autoregressive-moving average models. Biometrika73, 679–686 (1986)

    Google Scholar 

  • Mandelbrot, B.B.: The fractal geometry of nature. New York: Freeman 1977

    Google Scholar 

  • Martin, R.D., Yohai, V.J.: Influence functionals for time series. Ann. Stat.14, 781–818 (1986a)

    Google Scholar 

  • Martin, R.D., Yohai, V.J.: Reply to “Influence functionals for time series”. Ann. Stat.14, 840–855 (1986b)

    Google Scholar 

  • Moustakides, G.V., Thomas, J.B.: Min-max detection of weak signals in ϕ-mixing noise. IEEE Trans. Inf. Theory30, 529–537 (1984)

    Google Scholar 

  • Noether, G.E.: On a theorem of Pitman. Ann. Math. Stat.26, 64–68 (1955)

    Google Scholar 

  • Portnoy, S.L.: Robust estimation in dependent situations. Ann. Stat.5, 22–43 (1977)

    Google Scholar 

  • Portnoy, S.L.: Further remarks on robust estimation in dependent situations. Ann. Stat.7, 224–231 (1979)

    Google Scholar 

  • Ronchetti, E., Rousseeuw, P.J.: Change-of-variance sensitivities in regression analysis. Z. Wahrscheinlichkeitstheor. Verw. Geb.68, 503–519 (1985)

    Google Scholar 

  • Rousseeuw, P.J.: A new infinitesimal approach to robust estimation. Z. Wahrscheinlichkeitstheor. Verw. Geb.56, 127–132 (1981)

    Google Scholar 

  • Rousseeuw, P.J.: Most robustM-estimators in the infinitesimal senze. Z. Wahrscheinlichkeitstheor. Verw. Geb.61, 541–555 (1982)

    Google Scholar 

  • Sadowsky, J.S.: A maximum variance model for robust detection and estimation with dependent data. IEEE Trans. Inf. Theory32, 220–226 (1986)

    Google Scholar 

  • Zamar, H.: Robustness against unexpected dependence in the location model. Stat. Probab. Lett.9, 367–374 (1990)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Uppsala University, S-75238, Uppsala, Sweden

    Ola Hössjer

Authors
  1. Ola Hössjer
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Additional information

This paper was written under support by the Swedish Board for Technical Development, contract 712-89-1073

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Hössjer, O. The change-of-variance function for dependent data. Probab. Th. Rel. Fields 90, 447–467 (1991). https://doi.org/10.1007/BF01192138

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  • Received: 28 March 1990

  • Revised: 20 June 1991

  • Issue Date: December 1991

  • DOI: https://doi.org/10.1007/BF01192138

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Keywords

  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Score Function
  • Location Parameter
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