Titel:
The Mellin transform in Nonparametric Statistics
Beteiligte:
Brenner Miguel, Sergio Filipe
[Verfasser:in]
Erschienen:
Heidelberg University: HeiDok, 2023
Sprache:
Englisch
DOI:
https://doi.org/10.11588/heidok.00033855
Entstehung:
Anmerkungen:
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Beschreibung:
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 ...