Adaptive estimation of inverse problems in presence of dependence

Ill-posed inverse problems are becoming very important in a diverse range of disciplines, including biology, climatology and economics. The vast majority of existing work on inference for statistical inverse problems impose independence and/or identical distribution (iid.) of the observations used to construct the estimators which is, however, not always realistic. Typically, a collection of daily measurements of a pollutant indicator such as NO is then obtained by splitting up a record over the last decade. Obviously in this situation an independence assumption does not reflect the data structure and the assumption of an identical distribution is at least questionable over the whole decade. Consequently, the main objective of this thesis is the study of fully data-driven curve estimators in presence of dependence in the data generating process. More precisely, taking classical non-parametric density estimation and regression with random design problems as a starting point we show that in regression models with endogenous or functional covariates a thresholded linear Galerkin type estimator with fully data-driven choice of the dimension parameter combining model selection and Lepski’s method can attain minimax-optimal rates for iid. observations when dismissing the independence assumption and assuming weak dependence characterised by a sufficiently fast decay of the mixing conditions.