Study On The Theory And Methods Of Parameter Estimation In Restricted Linear Model

Linear model which includes some kinds of statistical models is a most important model in mathematics and statistics, and it is wide-ranging and widespread use research branch of model statistics. Due to the disadvantage of the least squares estimate for dealing with the multicollinearity, the research into the biased estimation in the linear model is always one of the most popular issues in statistics. While the maturity of the biased estimations in the linear regression model without additional restrictions is growing, various statistical problems must be regressed with additional restrictions, which show investigative significances and applied values. Like the least squares estimate, the restricted least squares estimate is also not good to deal with the multicollinearity. How to find a better way to improve the restricted least squares estimate is important. The statisticians also face the problem of choosing the biased estimations, thus the comparison among biased estimations has certain significance of theory and practice.The thesis is concerned with statistical inferences including parameter estimation and prediction of future observations in finite populations under several statistical models.In the general linear mixed models, considering the existence of multicolliearity, the authors generalize the spectral decomposition estimation and propose the partial ridge-type spectral decomposition estimation. Through the model transformation of dimensionality reduction which is similar to the principal component estimation, we can easily get that the new estimation has stronger noise-rejection ability and some other important properties. Furthermore, by using statistical decision theoretical approach, we investigate the domination of the partial ridge-type spectral decomposition estimation and the spectral decomposition estimation, and obtain some sufficient conditions for the partial ridge-type spectral decomposition estimation to dominate the spectral decomposition estimation. From above, we derive a new estimation called unified biased spectral decomposition estimation which is unnecessary to investigate some other kinds of biased spectral decomposition estimations.For the general linear model with linear equality restrictions, we propose the restricted unified almost unbiased estimation that contain the common estimations such as restricted almost unbiased ridge estimation and restricted almost unbiased Liu estimation, which unifies all kinds of the restricted almost unbiased estimations in general. Furthermore, the sufficient conditions under which it's superior over the restricted least squares estimate in terms of mean square error and mean square error matrix are derived, respectively.For the singular linear model with linear equality restrictions from the aspect of the biased estimation, by minimzing the sum of squared residuals with restricted conditions, we obtain two kinds of the restricted biased estimations: the restricted root estimator and the generalized restricted root estimator which have the properties of avoiding the unrestricted bias of parameter vector due to the multicolliearity and have the similar forms with the estimations without restriction. We analyze theoretically the characters of two kinds of the restricted biased estimations in their bias and stabilization, and compare them with the restricted least squares estimate. The sufficient conditions under which they're superior over the restricted least squares estimate in terms of mean square error and mean square error matrix are derived, respectively, and determine the value range of the parameters. Moreover, considering the influence analysis of the Liu estimation with covariance matrix disturbance in singular linear model, the relationship between the Liu estimations is made and the influence analysis of covariance matrix disturbance is discussed. Furthermore, the issue of the influence analysis of the ridge estimation with covariance matrix disturbance in generalized linear model or data missing model with linear equality restrictions is concerned. We make the relationship of the ridge estimators among some linear model with respect to linear equality restrictions, such as Gauss-Markov model, generalized linear model and linear model with data missing. In addition, the generalized Cook distance DV to assess the disturbance influence and two computational formulas of DV are proposed, respectively.In the general linear model with stochastic linear restrictions, we introduce a new stochastic restricted estimation called the stochastic mixed ridge estimation which is a natural generalization of the mixed estimation and the ridge estimation. Necessary and sufficient conditions for the superiority of the stochastic mixed ridge estimation over the ridge estimator and the mixed estimator in the mean squared error matrix sense are derived for the two cases in which the parametric restrictions are correct and are not correct. Furthermore, we generalize the stochastic mixed ridge estimation into the stochastic weighted mixed ridge estimation naturally, and derive the expression of the stochastic weighted mixed ridge estimation and the necessary and sufficient conditions for the superiority of the stochastic weighted mixed ridge estimator over the ridge estimator and the mixed estimator in the mean squared error sense are derived. Finally, a numerical example is also given to show the theoretical results, respectively.For the general linear model with linear equality restrictions, we compare the restricted (generalized) almost unbiased ridge estimation and the restricted (generalized) almost unbiased Liu estimation, obtain the sufficient conditions for the superiority of the restricted (generalized) almost unbiased ridge estimation over the restricted (generalized) almost unbiased Liu estimation and the superiority of the restricted (generalized) almost unbiased Liu estimation over the restricted (generalized) almost unbiased ridge estimation in terms of mean squared error matrix, respectively. Furthermore, we give a numerical example and the simulation. At the same time, we deduce the restricted almost unbiased ridge estimation and the restricted almost unbiased Liu estimation are superior over the restricted ridge estimation and the restricted Liu estimation in most cases, respectively.In the seventh chapter, the problem of prediction of future observations is discussed in finite populations. Under the super-population viewpoint, our attention is to discuss the optimal and classical predictors based on the restricted ridge estimation with positive definition covariance matrix in the linear model with linear equality restrictions. Furthermore, the optimal heterogeneous and the optimal unbiased homogeneous predictors are derived in singular linear model, respectively.In the last chapter, considering the sample properties of unified biased estimation, we derive the exact general expressions for the moments of the unified biased estimation for individual regression coefficients and the first two moments of the estimation.