Asymptotic Behavior of Partial Sums: A More Robust Approach Via Trimming and Self-Normalization

On the functional CLT for partial sums of truncated bounded from below random variables

“…The random variables X n are not symmetric here, but our special choice of truncating levels b n and d n allows us to use the same scheme with some obvious adjustments, so we omit the proof. (1), (2), (3), (6) and (8) then conditions (12) and (13) hold, and, as a consequence,…”

“…Many results are obtained under the symmetry assumption. See, for example, Pruitt (1988), Griffin and Pruitt (1987), Hahn and Kuelbs (1989), Haeusler and Mason (1990), Hahn et al (1991), Cuzick et al (1995) and Griffin and Qazi (2002 …”

Asymptotic Behavior of Trimmed Sums

“…For several further important contributions in trimming theory not discussed in this paper, we refer to the book [20] and the references therein. In particular, [20] discusses a number of further variants of trimming like censoring and "winsorized" trimming, see e.g., Hahn et al [19] for an asymptotic theory.…”

Asymptotic distributions of non-degenerate U-statistics on trimmed samples

“…In the Winsorized sums instead of discarding the first k 1 smallest and last k 2 largest order statistics, k ¼ k 1 þ k 2 , as in the case of the trimmed sums, we replace them with the k 1 th smallest and ðn À k 2 þ 1Þth largest order statistics, respectively. In the case m ¼ 1; hðxÞ ¼ x in (1), the study of the asymptotic distributions of trimmed and Winsorized sums is well advanced as is amply demonstrated by the results of Bickel (1965), Cheng (1992), Cso¨rg + o et al (1988), Griffin (1988), Griffin and Pruitt (1989), Hahn et al (1991), Kasahara (1993), Kasahara and Maejima (1992), Qi and Cheng (1996), Mason and Shorack (1990), Shorack (1974) and Stigler (1973). For mX2 the asymptotic distribution of such sums as ðkÞ U has received limited attention.…”