Shrinkage estimation of nonlinear models and covariance matrix

Abstract

Recent developments in shrinkage estimation are remarkable. Capable of shrinking some coefficients to exactly 0, the penalized approach combines continuous shrinkage with automatic variable selection. This thesis investigates the methodology and application of the shrinkage estimation from the following three aspects. 1. We propose to employ the adaptive LASSO approach in threshold variable selection of smooth threshold autoregressive (STAR) model. Moreover, by penalizing the direction of the coefficient vector in this nonlinear model, the threshold variable is more accurately selected. 2. We propose a novel principal varying coefficient model which discovers the possible linear dependence structure amongst the varying coefficients thus reducing the actual number of nonparametric functions and having better estimation efficiency. 3. We consider a new way of covariance matrix estimation through variate clustering. Both theoretical and numerical properties are studied in the above three aspects.