Wavelet Estimation: Minimax Theory and Application

• Activity Number: 340
• Type: Contributed
• Date/Time: Tuesday, August 5, 2014 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Ekaterina Smirnova*+ and Sam Efromovich
• Companies: and University of Texas at Dallas
• Keywords: Adaptation ; Local minimax ; Oracle ; Regression ; Neuroplasticity

Abstract

New theoretical results on wavelet estimation, concerning new minimax rates and adaptive estimation are presented. Theoretical results and the estimators are used for the analysis of fMRI images obtained in the study of neuroplasticity. It is well known in the wavelet theory that for a Besov space of order $\alpha$ the minimax rate of the mean integrated squared error (MISE) convergence is $n^{-2\alpha/(2\alpha+1)}$. At the same time, it is also known that a fixed function from the Besov space can be estimated with a faster rate. To bridge these known rates of the MISE convergence, a new minimax approach, based on a sequence of shrinking Besov subspaces, is proposed. This approach allows us to establish both faster rates than the classical minimax ones and propose a new adaptive estimator which is sharp minimax under the both approaches. Using the more accurate adaptive wavelet estimator for the analysis of fMRI images, which are known to have small signal-to-noise ratios, has allowed us to make new interesting discoveries in how neurons communicate.