The Mellin Transform in Nonparametric Statistics

This thesis deals with the nonparametric estimation for a special class of ill-posed inverse problems, 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.