论文标题
略微保守的bootstrap用于总和的最大值
Slightly Conservative Bootstrap for Maxima of Sums
论文作者
论文摘要
我们研究了独立随机变量总和的最大值的引导程序，这是与现代统计中许多应用相关的高度相关性的问题。由于Bootstrap的一致性是通过Chernozhukov等人的高斯近似值证明的。 （2013年），已经进行了相当多的尝试，以加强引导程序的错误，并减少引导程序一致性的样本量需求。在本文中，我们表明，当我们使推断略微保守时，可以显着改善样本量的需求，也就是说，以通过一小部分来膨胀bootstrap分位数$t_α^*$，例如$ 1 \％$至$ 1.01 \，t^*_α$。此简单的过程在适当的条件下，保守的引导程序的覆盖范围的覆盖率很快就会产生误差范围，因此，不仅可以将样本量需求降低到$ \ log p \ ll n $，而且总体融合利率也几乎是参数。此外，我们改善了标准非保守的自举的覆盖概率的错误，在数据的一般假设下，在数据的一般假设下。这些结果是为经验引导程序和具有第三次匹配的乘数引导程序的。开发了一种改进的连贯的Lindeberg插值方法，该方法最初在Deng and Zhang（2017）中提出，以得出更尖锐的比较范围，尤其是对于Maxima而言。
We study the bootstrap for the maxima of the sums of independent random variables, a problem of high relevance to many applications in modern statistics. Since the consistency of bootstrap was justified by Gaussian approximation in Chernozhukov et al. (2013), quite a few attempts have been made to sharpen the error bound for bootstrap and reduce the sample size requirement for bootstrap consistency. In this paper, we show that the sample size requirement can be dramatically improved when we make the inference slightly conservative, that is, to inflate the bootstrap quantile $t_α^*$ by a small fraction, e.g. by $1\%$ to $1.01\,t^*_α$. This simple procedure yields error bounds for the coverage probability of conservative bootstrap at as fast a rate as $\sqrt{(\log p)/n}$ under suitable conditions, so that not only the sample size requirement can be reduced to $\log p \ll n$ but also the overall convergence rate is nearly parametric. Furthermore, we improve the error bound for the coverage probability of the standard non-conservative bootstrap to $[(\log (np))^3 (\log p)^2/n]^{1/4}$ under general assumptions on data. These results are established for the empirical bootstrap and the multiplier bootstrap with third moment match. An improved coherent Lindeberg interpolation method, originally proposed in Deng and Zhang (2017), is developed to derive sharper comparison bounds, especially for the maxima.
