Cramér type large deviations for Studentized U-statistics
Let \(U_ n=\left( \begin{matrix} n\\ 2\end{matrix} \right)^{-1}\sum \sum_{1\leq i<j\leq n}h(X_ i,X_ j)\) denote a U-statistic with kernel h of degree 2, based on n independent and identically distributed random variables \(X_ 1,X_ 2,...,X_ n\). Assume that E h(X\({}_ 1,X_ 2)=\theta\) and let the variance of \(E[h(X_ 1,X_ 2)| X_ 1]\) be positive. If, in addition, \(E| h(X_ 1,X_ 2)-\theta |^ p\leq K^ pp^{\gamma p}\) for \(p=1,2,..\). and some constants K and \(\gamma\geq 0\) not depending on p, then \[ P(\sqrt{n}(U_ n-\theta)/S_ n>x)=[1-\Phi (x)](1+o(1)) \] uniformly in the range \(0\leq x\leq o(n^{\alpha})\), with \(\alpha =2^{-1}(3+2\gamma)^{-1}\) for \(\gamma\leq 2\) and \(\alpha =2^{-1}(4\gamma -1)^{-1}\) for \(\gamma\geq 2\). The quantity \(n^{- 1}S^ 2_ n\) is the jackknife estimator of the variance of \(U_ n\) and \(\Phi\) denotes the standard normal distribution function. For the important special case of the Student t-statistic (take \(h(x,y)=2^{- 1}(x+y))\) the authors obtain a somewhat better range for \(\alpha\).
scientific article; zbMATH DE number 4001182