We will present the notion of Stein kernel, which provides generalizations of the integration by parts, a.k.a. Stein's formula, for the normal distribution. We will first focus on dimension one, where under good conditions the Stein kernel has an explicit formula. We will see that the Stein kernel appears naturally as a weighting of a Poincaré type inequality and that it enables precise concentration inequalities, of the Mills' ratio type. In a second part, we will work in higher dimensions, using in particular Max Fathi's construction of a Stein kernel through the so-called "moment maps" transportation. This will allow us to describe the performance of some shrinkage and thresholding estimators, beyond the classical assumption of Gaussian (or spherical) data.
This presentation is mostly based on joint works with Max Fathi, Larry Goldstein, Gesine Reinert and Jon Wellner.
Underlying papers: https://arxiv.org/abs/1804.03926 and https://arxiv.org/abs/2004.01378
Personal website of Adrien Saumard