In this series of lectures, we will start with considering exponential inequalities, including concentration inequalities, for the deviation of averages from their mean. We furthermore present some notions from approximation theory, because this enables us to assess the modulus of continuity of empirical processes. We introduce e.g., Vapnik Chervonenkis dimension: a combinatorial concept (from learning theory) of the "size" of a collection of sets or functions. As statistical applications, we study consistency and exponential inequalities for empirical risk minimizers, and asymptotic normality in semi-parametric models. We moreover examine regularization and model selection.
The following list contains further literature. The books can be downloaded from the ETH library; please see the provided link (you need to have a VPN connection running if you want to download them from home).
- A. W. van der Vaart and J. A. Wellner. Weak convergence and empirical processes : with applications to statistics. Springer Series in Statistics. Springer, New York, 1996, [Online].
- R. Vershynin. High-dimensional probability : an introduction with applications in data science. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2018, [Online].