Topics in Nonparametric Statistics: Faster Minimax Rates, Large-p-Small-n Cross-Correlation Matrices, Survival Analysis

University Of Texas At Dallas, Richardson TX

Investigators

Abstract

The project focuses on three main statistical activities motivated by medical, engineering and insurance applications. The first one is the optimal denoising and decomposition of signals and images. Analysis of functional magnet resonance images (fMRI) in the human brain is a particular example of application. Loosely speaking, fMRI is a noisy time series of images which reflects the level of oxygen in blood during measurements. Because this is about dealing with oxygen in blood, the time series contains respiratory, cardiac and neural response to functional stimuli. The proposed statistical procedure will allow doctors and bioengineers to separate and denoise these components, with potential applications in developing new methods for diagnosis and treatment of Alzheimer's and Parkinson's Diseases. The second topic is the estimation of large cross-covariance/correlation matrices for noisy signals, with the number of elements in millions and sample sizes of just several hundreds. This is a familiar problem in statistical analysis of Chip-on-chip microarrays used to study bacteria. Another application is the study of neural plasticity, which is the ability of the brain to recognize neural pathways based on new experience and change in learning. This will allow physicians to create new methods for early diagnosis and treatment of stroke which is the 4th leading cause of death in the US. The third topic is adaptive and efficient estimation of hazard rate and survival function from indirect observations, including new methodology of sequentially controlled experiments and protocols. This research is motivated by new methods of radiation and drug therapy for lung and breast cancers as well as by innovative technologies of waste-water treatment and the actuarial problem of developing adaptive life tables. The project focuses on three objectives. (1) Advance knowledge and understanding of nonparametric curve estimation to develop a general theory of shrinking local minimax estimation that allows statisticians to get a new benchmark for the quality of estimation and generate a family of more accurate estimators. Preliminary results indicate that new efficient estimators can be proposed either via mimicking oracles or via aggregation of different estimators in frequency domain. Most challenging and rewarding results are expected for multivariate curves where new rates can remedy the familiar curse of multidimensionality. Applications in statistical analysis of microarrays, fMRI, radiation and drug therapy of cancer cells, and innovative technologies of waste-water treatment are expected.(2) Develop new methods of inference for large cross-covariance/correlation matrices for noisy signals, based on wavelet methods and exponential inequalities that can be applied to dependent and non-Gaussian observations. The main application is the study of neural plasticity of the human brain. This allows doctors and bioengineers to observe changes in neural pathways based on new experience and change in learning with applications to treatment of stroke and other brain diseases.(3) Improve nonparametric theory and create efficient methods of hazard rate and survival function estimation for indirect observations, including a new methodology of sequentially controlled experiments and protocols.