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摘要翻译：
我们导出了高维随机向量和的最大值的高斯近似结果。具体地，我们建立了极大值分布近似于具有与原始向量相同协方差矩阵的高斯随机向量之和的极大值分布的条件。当随机向量的维数($P$)与样本容量($N$)相比较大时，该结果适用；事实上，$P$可以比$N$大得多，而不限制这些向量坐标的相关性。我们还证明了具有未知协方差矩阵的随机向量之和的最大值的分布可以通过将原始向量乘以I.I.D得到的条件高斯随机向量之和的最大值的分布一致地估计。高斯乘法器。这是高斯乘法器（或野生）引导过程。在这里，$P$也可以很大，甚至比$N$大得多。这些分布近似，无论是高斯近似还是条件高斯近似，对原始最大值的分布都产生了高质量的近似，并且近似误差往往以样本大小的多项式递减，因此在许多应用中都很有兴趣。我们演示了我们的高斯近似和乘法器引导如何用于现代高维估计、多重假设检验和自适应规范检验。所有这些结果都包含逼近误差的非渐近界。
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英文标题：
《Gaussian approximations and multiplier bootstrap for maxima of sums of
high-dimensional random vectors》
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作者：
Victor Chernozhukov, Denis Chetverikov, Kengo Kato
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最新提交年份：
2018
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分类信息：
一级分类：Mathematics 数学
二级分类：Statistics Theory 统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Economics 经济学
二级分类：Econometrics 计量经济学
分类描述：Econometric Theory, Micro-Econometrics, Macro-Econometrics, Empirical Content of Economic Relations discovered via New Methods, Methodological Aspects of the Application of Statistical Inference to Economic Data.
计量经济学理论，微观计量经济学，宏观计量经济学，通过新方法发现的经济关系的实证内容，统计推论应用于经济数据的方法论方面。
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一级分类：Mathematics 数学
二级分类：Probability 概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Statistics 统计学
二级分类：Statistics Theory 统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
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英文摘要：
We derive a Gaussian approximation result for the maximum of a sum of high-dimensional random vectors. Specifically, we establish conditions under which the distribution of the maximum is approximated by that of the maximum of a sum of the Gaussian random vectors with the same covariance matrices as the original vectors. This result applies when the dimension of random vectors ($p$) is large compared to the sample size ($n$); in fact, $p$ can be much larger than $n$, without restricting correlations of the coordinates of these vectors. We also show that the distribution of the maximum of a sum of the random vectors with unknown covariance matrices can be consistently estimated by the distribution of the maximum of a sum of the conditional Gaussian random vectors obtained by multiplying the original vectors with i.i.d. Gaussian multipliers. This is the Gaussian multiplier (or wild) bootstrap procedure. Here too, $p$ can be large or even much larger than $n$. These distributional approximations, either Gaussian or conditional Gaussian, yield a high-quality approximation to the distribution of the original maximum, often with approximation error decreasing polynomially in the sample size, and hence are of interest in many applications. We demonstrate how our Gaussian approximations and the multiplier bootstrap can be used for modern high-dimensional estimation, multiple hypothesis testing, and adaptive specification testing. All these results contain nonasymptotic bounds on approximation errors.
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PDF链接：
https://arxiv.org/pdf/1212.6906
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