A main motivation for the study of nonparametric models is that they do not impose potentially unrealistic finite-dimensional, or parametric, a priori restrictions. The minimax paradigm has revealed that the statistical performance of optimal nonparametric procedures depends heavily on structural properties of the parameter to be estimated and does not typically scale at the universal rate encountered in classical parametric models. This dependence arises typically through the choice of tuning parameters which require choices of usually unknown aspects of the function fto be estimated, for instance, its smoothness r and the corresponding bound on the Besov norm. The question arises as to how fully automatic procedures that do not require the specification of such parameters can perform from a minimax point of view and whether procedures exist that ‘adapt’ to the unknown values of r,B. We shall show in this chapter that full adaptation is possible in many testing and estimation problems and that mild losses occur for some adaptive testing problems. In contrast, the theory of adaptive confidence sets – and, more generally, the problem of adaptive uncertainty quantification – is more intricate, and the price for adaptation can be severe unless some additional structural assumptions on the parameter space are imposed. We shall explicitly characterise the parameter regions in nonparametric models where this discrepancy between estimation and uncertainty quantification arises and reveal the underlying relationship to certain nonparametric hypothesis-testing problems.

The theory of adaptive inference in infinite-dimensional models reveals fundamental, and in this form previously unseen, information-theoretic differences between the three main pillars of statistics, that is, between estimation, testing and the construction of confidence sets. The insights drawn from the results in this chapter belong to the most intriguing statistical findings of the nonparametric theory, showcasing the genuine challenges of statistical inference in infinite dimensions. To meet this challenge, a class of ‘self-similar’ functions will be introduced, for which a unified theory of estimation, testing and confidence sets can be demonstrated to exist.