We study the distribution of a general class of asymptoticallylinear statistics which are symmetric functions of $N$ independent observations. The distribution functions of these statistics are approximated by an Edgeworth expansion with a remainder of order $o(N^{-1})$. The Edgeworth expansion is based on Hoeffding's decomposition which provides a stochastic expansion into a linear part, a quadratic part as well as smaller higher order parts. The validity of this Edgeworth expansion is proved under Cram\'er's condition on the linear part, moment assumptions for all parts of the statistic and an optimal dimensionality requirement for the non linear part.
翻译:我们研究对称功能为美元独立观测的对称功能的一整类非现成线性统计的分布情况。这些统计数据的分布功能被埃格沃斯扩展的约合,其余部分为美元(N ⁇ -1})。埃格沃斯的扩展是基于霍夫丁的分解,它提供了向线性部分、四边部分和较小的顺序部分的分解。在Cram\'er对线性部分的条件、统计所有部分的瞬时假设和非线性部分的优化维度要求下,证明了这一Edgeworth扩展的正确性。