Mean Estimation in High Dimension

Explore a 44-minute lecture on mean estimation in high dimensions presented by Gabor Lugosi for the International Mathematical Union. Delve into the statistical problem of estimating the mean of a random vector using independent, identically distributed data. Examine recent advances in this classical problem, with a focus on high-dimensional aspects. Learn about distributions with second moments, confidence bounds, sub-Gaussian bounds, and empirical means with heavy tails. Investigate multivariate distributions, sub-Gaussian mean estimators, and high-dimensional median of means. Discover median-of-means tournaments, sub-Gaussian estimates, and multivariate trimmed means. Study general norms and direction-dependent accuracy in both Gaussian and non-Gaussian cases. Access accompanying slides for visual aids and further understanding of the presented concepts.

Intro
estimating the mean
distributions with a second moment
confidence bounds
sub-Gaussian bounds
empirical mean-heavy tails
bibliographic remarks
multivariate distributions
sub-gaussian mean estimators
high-dimensional median of means
multivariate median of means
median-of-means tournament
sub-gaussian estimate
multivariate trimmed means
general norms
direction-dependent accuracy - Gaussian case To obtain guidance, we inspect the enpirical meat for Gaussian
direction-dependent accuracy - main result
questions