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Abstract

This thesis deals with the nonparametric estimation for a special class of ill-posed inverse prob- lems, the so-called multiplicative measurement error models. In these models, the observations of the unknown, to be estimated quantity is only accessible with a multiplicative measurement error. As a consequence, the instability of the reconstruction depends on the distribution of the error by effecting the ill-posedness of the underlying inverse problem. The theory of Mellin transform al- lows to express the influence of the error distribution on the instability of the reconstruction and to reduce the estimation of the unknown quantity to a regularized estimation of its unknown Mellin transform. The proposed estimation strategies will be evaluated in terms of a mean weighted(- integrated) squared risk. Aside from being an introduction to the theory of Mellin transforms and multiplicative convolu- tions, this thesis is structuered in three topics. In the first part, we consider global density estimation under multiplicative measurement error. After a comparison between direct and noisy observations, we study several families of error dis- tributions, the multivariate case and the influence of dependence structures in the data. Here in each case we will propose an estimation strategy, discuss its minimax-optimality and consider data- driven choices of smoothing parameters. The theoritcal expected behavior of the estimators are illustrated through Monte-Carlo simulations. In the second part, we study global survival function estimation, which is, alongside the density of a distribution, a frequently considered characterization of a distribution. We once again propose an estimation method, prove its minimax-optimality and discuss data-driven choices of smoothing parameters. Furthermore, we analyse the stability of the estimator for Bernoulli-shift processes and visualize it using a Monte-Carlo simulation. The third part considers the estimation of the evaluation of an linear functional under multiplicative measurement errors. The point evaluation of the density, the survival function and the cumulative distribution function, to mention only a few, can be intrepreted as an evaluation of a linear func- tional. This allows the simultaneous analysis of these different estimation problems and the com- parison of the ill-posedness of the underlying inverse problems. A minimax-optimal estimation strategy as well as a data-driven choice of the smoothing parameters are presented and analyzed.