Asymptotic normality of winsorized means

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3 Scopus citations

Abstract

Let Xi be non-degenerate i.i.d. random variables with distribution function F, and let Xn1,...,Xnn denote the order statistics of X1,...,Xn. In trying to robustify the sample mean as an estimator of location, several alternatives have been suggested which have the intuitive appeal of being less susceptible to outliers. Here the asymptotic distribution of one of these, the Winsorized mean, which is given by n-1SnXn,sn+ ∑ i=sn+1 n-rnXni+rnXn,n-rn+1 where rn≥0, sn≥0 and rn+sn≥n, is studied. The main results include a necessary and sufficient condition for asymptotic normality of the Winsorized mean under the assumption that rn→∞, sn→∞, rnn-1→0, snn-1→0 and F is convex at infinity. It is also shown, perhaps somewhat surprisingly, that if the convexity assumption on F is dropped then the Winsorized mean may fail to be asymptotically normal even when X1 is bounded!

Original languageEnglish (US)
Pages (from-to)107-127
Number of pages21
JournalStochastic Processes and their Applications
Volume29
Issue number1
DOIs
StatePublished - 1988

Keywords

  • Winsorized mean
  • asymptotic behaviour
  • convexity condition
  • robustified mean

ASJC Scopus subject areas

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics

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