Some model-independent parameter-based estimation methods

In this project, we propose two novel approaches to promote the estima- tion accuracy of some popularly used methodologies in the literature. Both re- spectively define estimators of model dependent parameter of interest in terms of changing the values of model-independent parameters. In one approach, a set of initial estimators is constructed according to different values of model- independent parameter; and then by using an asymptotically linear represen- tation of the initial estimators, a linear regression is established against the values of model-independent parameter to form a least squares type composite estimator. This composite estimator can be applied to obtain a more accu- rate estimation for diverse models such as quantile regression models; a faster convergence rate than the existing optimal rate of nonparametric regression es- timation; and smaller estimation bias in variable selection. The other approach can handle the computational burden and instability that any classical method such as nonlinear least squares, quasi-likelihood suffers from, (caused by solving too many estimating equations when the dimension of predictor vector is high). When applied to generalized linear and semiparametric transformation mod- els, this second method is also very simply implemented, and is robust against any distribution of error term. Under a necessary and sufficient condition, it can transform variable selection problem for nonlinear or even semiparamet- ric models to that for linear models. To enhance the estimation accuracy, we can also define the least squares type composite estimator in terms of a set of model-independent transformations for the response.

Clearly, this research project investigates some fundamental problems in estimation theory, and expect the new methodologies and theories developed in the course of the project will have a lasting impact on statistical science.