Minimax Goodness-of-Fit Testing in Ill-Posed Inverse Problems with Partially Unknown Operators

Clément Marteau (1) , Theofanis Sapatinas (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 Department of Mathematics and Statistics [Cyprus]

We consider a Gaussian sequence model that contains ill-posed inverse problems as special cases. We assume that the associated operator is partially unknown in the sense that its singular functions are known and the corresponding singular values are unknown but observed with Gaussian noise. For the considered model, we study the minimax goodness-of-fit testing problem. Working with certain ellipsoids in the space of squared-summable sequences of real numbers, with a ball of positive radius removed, we obtain lower and upper bounds for the minimax separation radius in the non-asymptotic framework, i.e., for fixed values of the involved noise levels. Examples of mildly and severely ill-posed inverse problems with ellipsoids of ordinary-smooth and super-smooth sequences are examined in detail and minimax rates of goodness-of-fit testing are obtained for illustrative purposes.

Mots clés: Gaussian sequence model; minimax goodness-of-fit testing; minimax signal detection; ill-posed inverse problems

Domaines: Statistiques [math.ST]

Clément Marteau, Theofanis Sapatinas. Minimax Goodness-of-Fit Testing in Ill-Posed Inverse Problems with Partially Unknown Operators. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2017, vol. 53 (4), pp.1675-1718. ⟨10.1214/16-AIHP768⟩. ⟨hal-01136521v2⟩