某些渐近卡方分布统计量的收敛速度：基于Stein方法
The rate of convergence of some asymptotically chi-square distributed statistics by Stein's method
AI总结：
本文利用Stein方法为渐近卡方统计量（如Friedman统计量和幂散度统计量）建立$O(n^{-1})$收敛速度界，并猜想该速度对任意参数成立。
摘要：
我们基于近期关于多元正态随机变量函数的Stein方法的研究，推导了某些渐近卡方分布统计量收敛速度的界。我们获得了一些一般性的界，并为光滑检验函数的$n^{-1}$阶收敛速度建立了简单的充分条件。这些一般性的界被应用于Friedman统计量（用于比较$n$次试验中的$r$个处理）以及幂散度统计量族（用于$n$次试验和$r$个分类的拟合优度检验），其索引参数为$λ\in\mathbb{R}$（Pearson统计量对应于$λ=1$）。对于任意处理数$r\geq2$，我们获得了Friedman统计量收敛速度的$O(n^{-1})$界。对于任意$r\geq2$，当$λ$为正整数或大于5的任意实数时，我们也获得了幂散度统计量收敛速度的$O(n^{-1})$界。我们猜想$O(n^{-1})$的收敛速度对任意$λ\in\mathbb{R}$均成立。
英文摘要：
We build on recent works on Stein's method for functions of multivariate normal random variables to derive bounds for the rate of convergence of some asymptotically chi-square distributed statistics. We obtain some general bounds and establish some simple sufficient conditions for convergence rates of order $n^{-1}$ for smooth test functions. These general bounds are applied to Friedman's statistic for comparing $r$ treatments across $n$ trials and the family of power divergence statistics for goodness-of-fit across $n$ trials and $r$ classifications, with index parameter $λ\in\mathbb{R}$ (Pearson's statistic corresponds to $λ=1$). We obtain a $O(n^{-1})$ bound for the rate of convergence of Friedman's statistic for any number of treatments $r\geq2$. We also obtain a $O(n^{-1})$ bound on the rate of convergence of the power divergence statistics for any $r\geq2$ when $λ$ is a positive integer or any real number greater than 5. We conjecture that the $O(n^{-1})$ rate holds for any $λ\in\mathbb{R}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
