Bharath Kumar Sriperumbudur, Thesis Advisor/Co-Advisor Donald Richards, Committee Member Ephraim Mont Hanks, Program Head/Chair
Keywords:
Shrinkage Estimation Hilbert Space Minimax Rates Non-parametric Mean Estimators
Abstract:
In many statistical algorithms and inferential methods, the estimation of the mean function in a Hilbert space plays an important role. Examples of such algorithms and methods are principal component analysis, discriminant analysis and hypothesis testing. Mean function is often estimated by the well-known empirical average. Motivated by the Stein phenomenon we propose, in this thesis, shrinkage estimators and show them to be improved versions of the empirical average by providing oracle inequalities. We also show that the rate of convergence of these shrinkage estimators is of order n^{−1/2} and it is optimal in the minimax sense. Specifically, we establish a minimax optimal rate over the class of discrete and infinitely differentiable probability measures.
