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Random Deletion Does Not Affect Asymptotic Normality or Quadratic Negligibility

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  • Kesten, Harry
  • Maller, R. A.
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    Abstract

    Suppose a number of points are deleted from a sample of random vectors in d. The number of deleted points may depend on the sample sizen, and on any other sample information, provided only that it is bounded in probability asn-->[infinity]. In particular, "extremes" of the sample, however defined, may be deleted. We show that this operation has no effect on the asymptotic normality of the sample sum, in the sense that the sum of the deleted sample is asymptotically normal, after norming and centering, if and only if the sample sum itself is asymptotically normal with the same norming and centering as the deleted sum. That is, the sample must be drawn from a distribution in the domain of attraction of the multivariate normal distribution. The domain of attraction concept we employ uses general operator norming and centering, as developed by Hahn and Klass. We also show that random deletion has no effect on the "quadratic negligibility" of the sample. These are conditions that are important in the robust analysis of multivariate data and in regression problems, for example.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 63 (1997)
    Issue (Month): 1 (October)
    Pages: 136-179

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    Handle: RePEc:eee:jmvana:v:63:y:1997:i:1:p:136-179

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    Related research

    Keywords: asymptotic normality random deletion of observations trimming extreme values quadratic negligibility operator norming sums and maxima of random variables sum of squares and products matrix covariance matrix;

    References

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    1. Vu, H. T. V. & Maller, R. A. & Klass, M. J., 1996. "On the Studentisation of Random Vectors," Journal of Multivariate Analysis, Elsevier, vol. 57(1), pages 142-155, April.
    2. Maller, R. A., 1993. "Quadratic Negligibility and the Asymptotic Normality of Operator Normed Sums," Journal of Multivariate Analysis, Elsevier, vol. 44(2), pages 191-219, February.
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