Pretest and Stein-type Shrinkage Estimators in Linear and Generalized Partial Linear Models

Abstract

In this thesis, we consider two estimation problems of the regression parameters in generalized partial linear regression model and multiple linear regression model with many covariates. We consider the situation where some of the regression parameters may be suspected to satisfy some restrictions and the nonparametric part is considered as nuisance. We first propose some novel and improved methods to estimate the regression coefficients of generalized partial linear models (GPLM). This model extends the generalized linear model by adding a nonparametric component. Like parametric models, variable selection is important in the GPLM to single out the inactive covariates. Instead of deleting inactive covariates, we use them as an auxiliary information. We define two models, the unrestricted model includes all the covariates whereas the restricted one includes the active covariates only. We then combine these two model estimators optimally to form the pretest and shrinkage estimators. We study the asymptotic properties to derive the asymptotic biases and risks of the estimators. We show that the asymptotic risks of the shrinkage estimators are strictly less than that of the full model estimators. A simulation study is conducted to assess the performance of the proposed estimators. We then apply our proposed methods to analyze a real credit scoring data. Both simulation study and real data example corroborate with the theoretical result. Optimal design plays an important role in achieving good estimation of the parameters. Motivated by this fact, we propose another novel method to further improve the pretest and shrinkage estimators. The results are very promising. Apart from the modeling and post-modeling procedures, pre-modeling stage plays a key role in achieving efficient estimators of the parameters. The optimal combinations of values of inputs which are normally numeric must be chosen before running an experiment. We consider the most popular D-optimality criterion and construct the optimal design using a class of algorithms. We then generate the data according to the optimal design and finally obtain our pretest and shrinkage estimators in multiple linear regression models. Our studies evidently show that our proposed estimators using optimal design theory outperform the estimators without using optimal design.