The Generalized Ridge And Principal Components Estimate And Its Optimality

The parameter estimation is the subject of extensive research in mathematical statistics. The most commonly used estimate is the least square estimate. Since Gauss raised the least square method in the beginning of the last century, the least square estimate has been widely used. However, when variables increase and multi-collinearity exists, the least square estimate show considerable instability. For this, the statisticians have done a lot of improvement work. In the past fifty years, statisticians have proposed various new estimates. The most common one is unbiased estimate class, such as the principal components estimate, the ridge estimate and the ridge and principal components estimate. When multi-collinearity exists, in the means square error sense, they are all better than the least square estimate. In this paper we aim to study the generalized ridge and principal components estimate, and the point is its good properties under the balanced loss function.In this thesis, we first outline the current research of parameter estimate. In chapter two, we introduce some basic knowledge of matrix, linear model, and some biased estimate. In chapter three, we research the generalized ridge and principal components estimate under Gauss-Markov model, and discuss its basic properties. In chapter four, we research its admissibility under the balanced loss function. In chapter five, we discuss its good properties under the balanced loss function. To show it better than the least square estimate, the principal components estimate as well as the ridge estimate, we summarize three important theorems respectively, and give the detail proof of the conclusions. Besides, we cite Webster—Gunst—Mason data to show its good performance.