- Talk (.pdf):
- “Kolloquium für Statistik”, Institute of Applied Mathematics, Ruprecht-Karls-Universität in Heidelberg, Germany
- Presented by:
- Jan JOHANNES
- Title:
- Statistics of Inverse Problems: Motivations, methods and theory
- Abstract:
- Statistical ill-posed inverse problems are becoming increasingly
important in a diverse range of disciplines, including geophysics,
astronomy, medicine and economics. Roughly speaking, in all of these
applications the observable signal g = Tf is a transformation of the
functional parameter of interest f under a linear operator
T. Statistical inference on f based on an estimation of g which
usually necessitates an inversion of T is thus called an inverse
problem. Moreover, by ill-posed we mean that the transformation T is
not stable, i.e., T has not a continuous inverse. In most
applications, however, both the signal g and the inherent
transformation T are not known in practise, although they can be
estimated from the data. Consequently, a statistical inference has to
take into account that a random noise is present in both the estimated
signal and the estimated operator.
Minimax-optimal estimation and adaptation. Typical questions in this context are the nonparametric estimation of the functional parameter f. It is well-known that in terms of its risk the attainable accuracy of an estimation procedure is essentially determined by the conditions imposed on f and the operator T which are, for example, expressed in the form f ∈ F and T ∈ Τ for suitable chosen classes F and T. Minimax-optimality of an estimator is then usually shown by establishing both an upper and a lower bound of the maximal risk over given classes F and Τ. In many cases the proposed estimation procedures rely on the choice of at least one tuning parameter, which in turn, crucially influences the attainable accuracy of the constructed estimator. Its optimal choice, however, follows often from a classical squared-bias-variance trade-off and relies on an a-priori knowledge about the classes F and Τ, which is usually inaccessible in practise. This motivates its data-driven choice in the context of nonparametric statistics since its very beginning in the fifties of the last century.
In this presentation special attention is given to three models and their extensions, namely nonparametric instrumental regression under endogeneity, functional linear regression and density deconvolution, each of them leading naturally to a statistical ill-posed inverse problem. We propose an estimator of the functional parameter f which is based on a dimension reduction and additional thresholding. It is shown that the estimator can attain minimax optimal rates of convergence. The estimator, however, requires an optimal choice of a dimension parameter and we investigate its fully data-driven choice which combines model selection and Lepski’s method. It is shown that the data-driven estimator can attain minimax optimal rates of convergence, and this over a variety of classes of functions F and operators Τ.
| Statistics Group | Institute for Mathematics | Faculty of Mathematics and Computer Science | University Heidelberg |