Date of Award

12-2016

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Statistics

First Advisor

Xiao Wang

Committee Chair

Xiao Wang

Committee Member 1

Marry E. Bock

Committee Member 2

Chuanhai Liu

Committee Member 3

Lingsong Zhang

Abstract

The aim of this thesis is to systematically investigate some functional regression models for accurately quantifying the effect of functional predictors. In particular, three functional models are studied: functional linear regression model, functional Cox model, and function-on-scalar model. Both theoretical properties and numerical algorithms are studied in depth. The new models find broad applications in many areas.

For the functional linear regression model, the focus is on testing the nullity of the slope function, and a generalized likelihood ratio test based on easily implementable data-driven estimate is proposed. The quality of the test is measured by the minimal distance between the null and the alternative space that still allows a possible test. The lower bound of the minimax decay rate of this distance is derived, and test with a distance that decays faster than the lower bound would be impossible. It is shown that the minimax optimal rate is jointly determined by the reproducing kernel and the covariance kernel and our test attains this optimal rate. Later, the test is applied to the effect of the trajectories of oxides of nitrogen (NOx) on the level of ozone (O3).

In the functional Cox model, the aim is to study the Cox model with right-censored data in the presence of both functional and scalar covariates. Asymptotic properties of the maximum partial likelihood estimator is established and it is shown that the estimator achieves the minimax optimal rate of convergence under a weighted L2-risk. Implementation of the estimation approach and the selection of the smoothing parameter are discussed in detail. The finite sample performance is illustrated by simulated examples and a real application.

The function-on-scalar model concentrates on developing the simultaneous model selection and estimation technique. A novel regularization method called the Grouped Smoothly Clipped Absolute Deviation (GSCAD) is proposed. The initial problem can be transferred into a dictionary learning problem, where the GSCAD can be directly applied to simultaneously learn a sparse dictionary and select the appropriate dictionary size. Efficient algorithm is designed based on the alternative direction method of multipliers (ADMM) which decomposes the joint non-convex problem with the non-convex penalty into two convex optimization problems. Several examples are presented for image denoising and image inpainting, which are competitive with the state of the art methods.

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