Date of Award

2008

Publication Type

Dissertation

Degree Name

Ph.D.

Department

Mathematics and Statistics

First Advisor

Syed Ejaz Ahmed

Keywords

Pure sciences, LASSO estimators, Parametric linear models, Shrinkage

Rights

info:eu-repo/semantics/openAccess

Creative Commons License

Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.

Abstract

The theory of pretest (Bancroft (1944)) and James-Stein (James and Stein (1961)) type shrinkage estimation has been quite well known for the last five decades though its application remains limited. In this dissertation, some contributions to different types of parametric and semiparametric linear models based on shrinkage and preliminary test estimation methods are made which improve on the maximum likelihood estimation method. The objective of this dissertation is to study the properties of improved estimators of the parameter of interest in parametric and semiparametric linear models and compare these estimators with the least absolute shrinkage and selection operator (Tibshirani (1996)) estimator. Chapter two contains a study of the properties of the shrinkage estimators of the parameters of interest in a Weibull regression model where the survival time may be subject to fixed censoring and the regression parameters are under linear restrictions. Asymptotic properties of the suggested estimators are established using the notion of asymptotic distributional risk. Bootstrapping procedures are used to develop confidence intervals. An extensive simulation study is conducted to assess the performance of the suggested estimators for moderate and large samples. In chapter three, we consider generalized linear models for binary and count data. Here, we propose James-Stein type shrinkage estimators, a pretest estimator and a Park and Hastie estimator. We demonstrate the relative performances of shrinkage and pretest estimators based on the asymptotic analysis of quadratic risk functions and it is found that the shrinkage estimators outperform the maximum likelihood estimator uniformly. On the other hand, the pretest estimator dominates the maximum likelihood estimator only in a small part of the parameter space, which is consistent with the theory. A Monte Carlo simulation study has been conducted to compare shrinkage, pretest and Park and Hastie type estimators with respect to the maximum likelihood estimator through relative efficiency. In chapter four, we consider a partial linear model where the vector of coefficients β in the linear part can be partitioned as (β 1, β2) where β1 is the coefficient vector for main effects and β2 is a vector for “nuisance” effects. In this situation, inference about β1 may benefit from moving the least squares estimate for the full model in the direction of the least squares estimate without the nuisance variables, or from dropping the nuisance variables if there is evidence that they do not provide useful information (pre-testing). We investigate the asymptotic properties of Stein-type and pretest semiparametric estimators under quadratic loss and show that, under general conditions, a Stein-type semiparametric estimator improves on the full model conventional semiparametric least squares estimator. The relative performance of the estimators is examined using asymptotic analysis of quadratic risk functions and it is found that the Stein-type estimator outperforms the full model estimator uniformly. On the other hand, the pretest estimator dominates the least squares estimator only in a small part of the parameter space, which is consistent with the theory. We also consider an absolute penalty type estimator for partial linear models and give a Monte Carlo simulation comparison of shrinkage, pretest and the absolute penalty type estimators.

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