Why minimax is not that pessimistic

TL;DR

This paper challenges the traditional view of minimax optimality in nonparametric statistics, showing that the minimax risk often reflects typical performance rather than worst-case scenarios, especially for classical estimators in Besov spaces.

Contribution

It demonstrates that the minimax risk aligns with generic performance for classical estimators in Besov spaces, questioning the pessimistic interpretation of minimax analysis.

Findings

Minimax risk often matches the typical performance in Besov spaces.

Classical estimators achieve the minimax rate on almost every function in the space.

The traditional worst-case perspective may be overly pessimistic for practical estimation.

Abstract

In nonparametric statistics an optimality criterion for estimation procedures is provided by the minimax rate of convergence. However this classical point of view is subject to controversy as it requires to look for the worst behaviour reached by an estimation procedure in a given space. The purpose of this paper is to show that this is not justified as the minimax risk often coincides with a generic one. We are here interested in the rate of convergence attained by some classical estimators on almost every, in the sense of prevalence, function in a Besov space.