The Geometry of Adaptive Confidence Sets


Abstract: We shall give general sets of necessary and sufficient 
conditions for the existence of honest confidence sets in common 
adaptation problems, such as adapting to unknown smoothness or sparsity. 
We give results for L^2-confidence balls as well as L^\infty confidence 
bands, using a sharp analysis of certain minimax testing problems, 
generalising work of Ingster and others to the situation relevant here. 
We highlight the subtle dependence of our existence results on the 
geometry of the given adaptation problem, and discuss various 
consequences for the theory of statistical inference in such models.