12  Adaptive estimation

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Next time I teach the course I will add a note here on adaptive estimation! The main point is that, even though we have found the minimax risk in a few situations, for example in the Normal means model with \(\boldsymbol{\theta}\) lying in a general or Sobolev ellipsoid, we cannot compute the Pinsker weights, which are defined in terms unknown parameters \(a_1,a_2,\dots\) or \(\beta\).

An estimator which is able to achieve the minimax risk asymptotically as \(n \to \infty\) no matter the values of the unknown parameters is called an adaptive (in the minimax sense) estimator. The James–Stein estimator is one such estimator. It is minimax adaptive in the Normal means model with \(\boldsymbol{\theta}\) lies in a ball in \(\mathbb{R}^n\) of finite radius centered at the origin.

See Section 3.7 of Tsybakov (2008).

Tsybakov, Alexandre B. 2008. Introduction to Nonparametric Estimation. Springer Science & Business Media.