### Abstract

Let X_{i} be non-degenerate i.i.d. random variables with distribution function F, and let X_{n1},...,X_{nn} denote the order statistics of X_{1},...,X_{n}. 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^{-1}S_{n}X_{n,sn
}+ ∑ i=s_{n}+1 n-r_{n}X_{ni}+r_{n}X_{n,n-rn+1 } where r_{n}≥0, s_{n}≥0 and r_{n}+s_{n}≥n, is studied. The main results include a necessary and sufficient condition for asymptotic normality of the Winsorized mean under the assumption that r_{n}→∞, s_{n}→∞, r_{n}n^{-1}→0, s_{n}n^{-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 X_{1} is bounded!

Original language | English (US) |
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Pages (from-to) | 107-127 |

Number of pages | 21 |

Journal | Stochastic Processes and their Applications |

Volume | 29 |

Issue number | 1 |

DOIs | |

State | Published - 1988 |

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### Keywords

- asymptotic behaviour
- convexity condition
- robustified mean
- Winsorized mean

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Mathematics(all)
- Modeling and Simulation
- Statistics and Probability