Random Deletion Does Not Affect Asymptotic Normality or Quadratic Negligibility

Abstract

Suppose a number of points are deleted from a sample of random vectors in Rd. 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→∞. 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.