报告题目：Bootstrap confidence sets for spectral projectors of sample covariance

报告人：Vladimir V. Ulyanov （莫斯科大学）

报告地点：知新楼B1238

报告时间：2018年10月24日16:00-17:00

报告摘要：

Let X1, . . . , Xn be i.i.d. sample in Rp with zero mean and the covariance matrix \Sigma. The problem of recovering the projector onto an eigenspace of \Sigma from these observations naturally arises in many applications. Recent technique developed by Koltchinskii and Lounici helps to study the asymptotic distribution of the distance in the Frobenius norm ||Pr – \hat{Pr}||_2 between the true projector Pr on the subspace of the r-th eigenvalue and its empirical counterpart Pr in terms of the effective rank of \Sigma. In the talk we offer a bootstrap procedure for building sharp confidence sets for the true projector Pr from the given data. This procedure does not rely on the asymptotic distribution of ||Pr – \hat{Pr}||_2and its moments. It could be applied for small or moderate sample size n and large dimension p. The main result states the validity of the proposed procedure for Gaussian samples with an explicit error bound for the error of bootstrap approximation. This bound involves some new sharp results on Gaussian comparison and Gaussian anti-concentration in high-dimensional spaces. Numeric results confirm a good performance of the method in realistic examples.

These are the joint results with V.Spokoiny (WIAS, Berlin) and A.Naumov (HSE, Moscow). See details in A. Naumov, V. Spokoiny, and V. Ulyanov, Bootstrap confidence sets for spectral projectors of sample covariance (2017). arXiv:1703.00871.

报告人简介：

Vladimir V. Ulyanov, 莫斯科大学数学系教授，长期从事概率极限理论及统计大样本理论研究。曾获苏联国家青年科学家奖（1987年）德国洪堡学者（1991-1993年）日本JSPS Research Fellowship, (1999, 2004). 曾任德国比利菲尔德大学，荷兰莱顿大学，法国巴黎五大，香港大学，新加坡国立大学，日本数理统计研究院，墨尔本大学等学校访问教授，是Member of the Bernoulli Society. 在概率论及数理统计领域发表论文50多篇。并与人合作，出版专著“Multivariate Statistics : High-Dimensional and Large-Sample Approximations”, Wiley Series in Probability and Statistics, John Wiley & Sons, 2010.

邀请人： 陈增敬教授