We obtain bounds to quantify the distributional approximation in the delta method for vector statistics (the sample mean of $n$ independent random vectors) for normal and non-normal limits, measured using smooth test functions. For normal limits, we obtain bounds of the optimal order $n^{-1/2}$ rate of convergence, but for a wide class of non-normal limits, which includes quadratic forms amongst others, we achieve bounds with a faster order $n^{-1}$ convergence rate. We apply our general bounds to derive explicit bounds to quantify distributional approximations of an estimator for Bernoulli variance, several statistics of sample moments, order $n^{-1}$ bounds for the chi-square approximation of a family of rank-based statistics, and we also provide an efficient independent derivation of an order $n^{-1}$ bound for the chi-square approximation of Pearson's statistic. In establishing our general results, we generalise recent results on Stein's method for functions of multivariate normal random vectors to vector-valued functions and sums of independent random vectors whose components may be dependent. These bounds are widely applicable and are of independent interest.

翻译：我们通过光滑检验函数度量，获得了向量统计量（$n$个独立随机向量的样本均值）在Delta方法中正态和非正态极限下分布近似的量化界。对于正态极限，我们取得了最优阶$n^{-1/2}$收敛速度的界；而对于包括二次型在内的广泛非正态极限，我们实现了更快阶$n^{-1}$收敛速度的界。我们将一般性界应用于以下场景：推导伯努利方差估计量的分布近似显式界、若干样本矩统计量的界、秩次统计量族卡方近似的$n^{-1}$阶界，并提供了Pearson统计量卡方近似$n^{-1}$阶界的高效独立推导。在建立一般性结果时，我们将近期关于多元正态随机向量函数的Stein方法推广至向量值函数及分量可能存在依赖性的独立随机向量之和。这些界具有广泛适用性且具有独立研究价值。