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Qualitative robustness of estimators on stochastic processes

Author

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  • Katharina Strohriegl

    (Universität Bayreuth)

  • Robert Hable

    (Technische Hochschule Deggendorf)

Abstract

A lot of statistical methods originally designed for independent and identically distributed (i.i.d.) data are also successfully used for dependent observations. Still most theoretical investigations on robustness assume i.i.d. pairs of random variables. We examine an important property of statistical estimators—the qualitative robustness in the case of observations which do not fulfill the i.i.d. assumption. In the i.i.d. case qualitative robustness of a sequence of estimators is, according to Hampel (Ann Math Stat 42:1887–1896, 1971), ensured by continuity of the corresponding statistical functional. A similar result for the non-i.i.d. case is shown in this article. Continuity of the corresponding statistical functional still ensures qualitative robustness of the estimator as long as the data generating process satisfies a certain convergence condition on its empirical measure. Examples for processes providing such a convergence condition, including certain Markov chains or mixing processes, are given as well as examples for qualitatively robust estimators in the non-i.i.d. case.

Suggested Citation

  • Katharina Strohriegl & Robert Hable, 2016. "Qualitative robustness of estimators on stochastic processes," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(8), pages 895-917, November.
    • Handle: RePEc:spr:metrik:v:79:y:2016:i:8:d:10.1007_s00184-016-0582-z
    DOI: 10.1007/s00184-016-0582-z
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    References listed on IDEAS

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    1. Steinwart, Ingo & Hush, Don & Scovel, Clint, 2009. "Learning from dependent observations," Journal of Multivariate Analysis, Elsevier, vol. 100(1), pages 175-194, January.
    2. Doukhan, Paul & Louhichi, Sana, 1999. "A new weak dependence condition and applications to moment inequalities," Stochastic Processes and their Applications, Elsevier, vol. 84(2), pages 313-342, December.
    3. Hable, Robert & Christmann, Andreas, 2011. "On qualitative robustness of support vector machines," Journal of Multivariate Analysis, Elsevier, vol. 102(6), pages 993-1007, July.
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    Cited by:

    1. Wei Wang & Huifu Xu & Tiejun Ma, 2020. "Quantitative Statistical Robustness for Tail-Dependent Law Invariant Risk Measures," Papers 2006.15491, arXiv.org.

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