Asymptotic normality of winsorized means

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 X₁,...,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^-1S_nX_n,sn+ ∑ i=s_n+1 n-r_nX_ni+r_nX_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_nn^-1→0, s_nn^-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!