- Thesis:
- Master in Mathematics
- Author:
- Moritz Meyer
- Title:
- Estimation of linear Poisson functionals under sparsity
- Supervisors:
- Jan JOHANNES
- Abstract:
- Statistical estimation problems involving high-dimensional data with underlying sparsity structures have gained significant attention in recent years. These problems arise in various scientific fields, including signal processing, genomics and image analysis. This thesis addresses the problem of estimating a linear functional in high-dimensional settings, where each observation follows a scaled Poisson distribution. The focus is on leveraging sparsity structures to improve estimation accuracy. We propose the group hard thresholding estimator, a method that selectively eliminates insignificant deviations from the background signal, in order to filter out noise. The theoretical analysis of the estimator focuses on its mean squared error, providing a non-asymptotic upper bound for its risk depending on the sparsity settings. Additionally, we derive minimax lower bounds for the quadratic risk over a class of sparse signals. We establish that the estimator is rate-optimal up to a logarithmic term for certain settings. Through a comprehensive simulation study, we further evaluate the estimator’s performance across a range of different model parameters.
- Reference:
- O. Collier und A. S. Dalalyan. Estimating linear functionals of a sparse family of Poisson means, Statistical Inference for Stochastic Processes 21(2):331–344, 2018.
| Statistics Group | Institute for Mathematics | Faculty of Mathematics and Computer Science | University Heidelberg |