Let \(\{X_ j\}^{n_ 1}_{j=1}\) and \(\{Y_ k\}^{n_ 2}_{k=1}\) be independent samples from \(F_ 1\) and \(F_ 2\), respectively. \(\{X_ j\}\) and \(\{Y_ k\}\) are assumed to be mutually independent. Let \(\phi (x_ 1,...,x_{m_ 1};y_ 1,...,y_{m_ 2})\) be a Borel measurable function which is symmetric in the elements within each of its two sets of arguments (x and y). The following generalized U-statistic with kernel \(\phi\) can be chosen as an unbiased estimator for \(Q=E \phi\)
U\({}_{n_ 1n_ 2}=\left( \begin{matrix} n_ 1\\ m_ 1\end{matrix} \right)^{-1}\left( \begin{matrix} n_ 2\\ m_ 2\end{matrix} \right)^{- 1}\sum_{1\leq \alpha_ 1<...<\alpha_{m_ 1}\leq n_ 1,1\leq \beta_ 1<...<\beta_{m_ 2}\leq n_ 2}\phi (X_{\alpha_ 1},...,X_{\alpha_{m_ 1}};Y_{\beta_ 1},...,Y_{\beta_{m_ 2}}).\)
The purpose of this note is to construct jackknife and bootstrap methods for \(U_{n_ 1n_ 2}\). Some asymptotic properties are discussed. These methods work for other multivariate symmetric statistics.