Asymptotic normality of U-statistics based on trimmed samples (Q1822157)

The authors consider asymptotic normality of U-statistics based on the trimmed sample. More precisely, let \(X_ 1,...,X_ n\) be an i.i.d. sample from a df F, let \(X_{n1}\leq...\leq X_{nn}\) denote the ordered \(X_ i's\), let \(0<\alpha\), \(\beta <1/2\), and put \(n_{\alpha,\beta}=n- [\alpha n]-[\beta n]\). \(Put\) U\({}_{n\alpha \beta}=\left( \begin{matrix} n_{\alpha \beta}\\ m\end{matrix} \right)^{-1}\sum_{C_{n\alpha \beta}}h(X_{ni_ 1},...,X_{ni_ m})\) where \(h(x_ 1,...,x_ m)\) is a kernel which is asymmetric in its arguments and \(C_{n\alpha \beta}\) denotes the set of m-tuples \(\{(i_ 1,...,i_ m):\) \([\alpha n]+1\leq i_ 1<...<i_ m\leq n-[\beta n])\). It is shown that under some regularity conditions \(U_{n\alpha \beta}\) is asymptotically normal. The result is extended to the multi-sample case. The authors' results unify the flexibility of the class of U-statistics and the theory of statistics based on trimmed samples. Various examples, such as moment-type statistics and Wilcoxon-type statistics, are also shown.