- Thesis:
- Master in Mathematics
- Author:
- Janine Steck
- Title:
- Fourier-type density estimation in a tomography problem
- Supervisors:
- Sergio Brenner Miguel
- Jan JOHANNES
- Abstract:
- In this thesis, we study non-parametric kernel density estimation of a multidimensional unknown probability density function f in a tomography problem. In this particular case, we only have access to indirect independent and identically distributed observations, whose common density is proportional to the Radon transform. The Fourier approach for density estimation is presented to derive the estimator of f in the idealised tomography problem. First, we study the Fourier approach and show the minimax-optimality for the mean squared error and the mean integrated squared error over the Fourier–Sobolev spaces, which characterise the regularity of the unknown density by the decay of its Fourier transform. Further, we show the minimax-optimality for the mean squared error of the density estimator corresponding to the tomography problem and compare the convergence rate with that of the Fourier approach. Typically, the convergence rates depend on unknown smoothing parameters. As a result, we introduce the data-driven choice of smoothing parameters for both approaches. At the end, our theoretical results are illustrated by Monte-Carlo simulations of several examples for the tomography approach.
- Reference:
- A. Abhishek and A. Sakshi. Adaptive estimation of a function from its Exponential Radon Transform in presence of noise, arXiv:2011.06887, 2020
| Statistics Group | Institute for Mathematics | Faculty of Mathematics and Computer Science | University Heidelberg |