Nonparametric Curve Estimation in the Presence of Nuisance Functions

University Of Texas At Dallas, Richardson TX

Investigators

Abstract

The proposal focuses on developing statistical methodology, theory and methods of adaptive nonparametric curve estimation in the presence of nuisance functions. It is motivated by biological and medical applications. The investigator studies four main classes of considered problems. Each is classified by how an estimator can match the performance of an oracle knowing underlying nuisance functions. They are: (a) Estimator can match oracle. An example is a regression problem with a smooth design density being nuisance function. (b) Estimator can match oracle if complementary observations are available. Examples are deconvolution and regression with measurement errors in predictors where complementary observations are needed to estimate distribution of measurement error. (c) Estimator cannot match oracle. An example is estimation of the density of regression error when a nuisance regression function is not sufficiently smooth. (d) A mixture of the above-formulated settings. It is proposed to develop a general theory of adaptive estimation for the aforementioned classes of statistical problems with particular applications to: missing, stratified and censored data, hidden components, mixed multivariate models involving continuous and nominal or ordinal categorical variables, and time series. Theoretical results are tested and applied to the analysis of ChIP-on-chip microarrays and ultra-fast fMRI. The primary focus of the research is to create adaptive statistical procedures which can work in the presence of nuisance functions. This research is motivated by and tested on well-understood applications in the statistical analysis of: (i) ChIP-on-chip microarrays used to find regulatory protein binding sites in a bacterial genome. Interactions between protein and DNA are fundamental to life. They facilitate and mediate gene expression, DNA replication and repair. The proposed statistical analysis of ChIP-on-chip microarrays points on exact location of protein-DNA binding sites. Because the statistical analysis does not require measuring of nuisance functions, it makes microarray experiments cheaper, faster and more accurate. (ii) Ultra-fast functional magnet resonance images, which help in understanding aging and brain diseases such as Alzheimer's and Parkinson's Diseases. Ultra-fast fMRI is an exciting new technology for studying brain functions with the temporal resolution of 50 milliseconds. This resolution sheds light on both neurons and physiological activities in the brain. Proposed statistical analysis, which is robust to nuisance functions, can denoise emodynamic responses, study cognitive functions like memory, speech and emotion, and create a map of physiological activities of the human brain.