论文标题
高斯近似的有效浓度
Efficient Concentration with Gaussian Approximation
论文作者
论文摘要
样品平均值的浓度不平等，例如由于伯恩斯坦和Hoeffding引起的浓度不平等，对任何样本量有效，但过于保守，产生了不必要的宽敞的置信区间。中心极限定理（CLT）提供了具有最佳宽度的渐近置信区间，但对于所有样本量，这些间隔都是无效的。为了解决这种张力，我们开发了具有渐近最佳尺寸，有限样本有效性和高斯亚下衰减的新的可计算浓度不平等现象。这些界限可以为任何样本量和有效的经验浆果界限的有效置信区间的构建，并不需要先验知识的人口差异。我们通过使用零偏置耦合以及Stein的可交换对方法将非均匀的Kolmogorov和Wasserstein距离紧密地结合到高斯，从而得出了不平等。
Concentration inequalities for the sample mean, like those due to Bernstein and Hoeffding, are valid for any sample size but overly conservative, yielding confidence intervals that are unnecessarily wide. The central limit theorem (CLT) provides asymptotic confidence intervals with optimal width, but these are invalid for all sample sizes. To resolve this tension, we develop new computable concentration inequalities with asymptotically optimal size, finite-sample validity, and sub-Gaussian decay. These bounds enable the construction of efficient confidence intervals with correct coverage for any sample size and efficient empirical Berry-Esseen bounds that require no prior knowledge of the population variance. We derive our inequalities by tightly bounding non-uniform Kolmogorov and Wasserstein distances to a Gaussian using zero-bias couplings and Stein's method of exchangeable pairs.
