论文标题
新的Edgeworth型扩展具有有限样品保证
New Edgeworth-type expansions with finite sample guarantees
论文作者
论文摘要
我们建立了高阶非反应扩张，以差异为I.I.D的概率分布之间的差异。欧几里得空间中的随机向量。派生的边界在两类集合上是均匀的：所有欧几里得球的集合和所有半空间的集合。这些结果允许考虑所考虑分布的高阶力矩或累积物的影响；获得的误差项取决于样本量和明确的尺寸。在非常普遍的条件下，新的不平等现象的近似值优于正常近似的准确性。在某些对随机汇总概率分布的对称性假设下，所获得的结果在维度和样本量之间的比率方面是最佳的。我们开发用于建立非沉淀的高阶扩展的新技术本身可能很有趣。使用新的高阶不等式，我们研究了非参数bootstrap近似的准确性，并在可能的模型错误指定下提出了引导得分测试。本文的结果还包括一般椭圆置信区域的显式误差界限，以期为随机汇总的期望值，以及高斯抗浓缩不平等的最佳性，而不是所有欧几里得球的集合。
We establish higher-order nonasymptotic expansions for a difference between probability distributions of sums of i.i.d. random vectors in a Euclidean space. The derived bounds are uniform over two classes of sets: the set of all Euclidean balls and the set of all half-spaces. These results allow to account for an impact of higher-order moments or cumulants of the considered distributions; the obtained error terms depend on a sample size and a dimension explicitly. The new inequalities outperform accuracy of the normal approximation in existing Berry-Esseen inequalities under very general conditions. Under some symmetry assumptions on the probability distribution of random summands, the obtained results are optimal in terms of the ratio between the dimension and the sample size. The new technique which we developed for establishing nonasymptotic higher-order expansions can be interesting by itself. Using the new higher-order inequalities, we study accuracy of the nonparametric bootstrap approximation and propose a bootstrap score test under possible model misspecification. The results of the paper also include explicit error bounds for general elliptic confidence regions for an expected value of the random summands, and optimality of the Gaussian anti-concentration inequality over the set of all Euclidean balls.
