Research On Biased Estimation Of Parameters Of Linear Mixed Effect Model

Linear model is an important branch of mathematical statistics, which is development early,theory rich and strong applicability. It is the general term of statistical models, including linear regression model, variance analysis model and linear mixed effects model, etc. For linear mixed effects models, we are interested in the research contents mainly divided into two categories:Fixed effects and variance components, They are included in the mean and covariance matrix, respectively. For classical least squares estimate is insufficient dealing with the complex collinearity problem, the statisticians proposed many remedies. There is a commonly used method: biased estimation and a prior information about the parameter. This paper is to study the parameter estimation in linear mixed model with stochastic restrictions, discussing the superiority and related problems.For classical linear model under stochastic restrictions, we discusses the mixed estimator proposed by Theil and Goldberger(1961), and the stochastic restricted ridge regression(SRRR)estimator proposed by Ozkale(2009). This article is based on the two estimates, we discuss the sufficient and necessary conditions for the SRRRE is superior than that of mixed estimator under the mean square error criterion.For Panel data model under stochastic restrictions, we proposed the conditional ridge Between estimates, the conditional ridge Within estimates. And discussed the relationship of the least squares estimation, Between estimation and Within estimation, compare its superiority under the mean square error matrix criterion.For linear mixed effects model under stochastic restrictions, we first puts forward the conditional spectral decomposition estimate(CSDE), to calculate the mean square error(MSE). The second, combine the CSDE and ridge regression estimate,we put forward a new estimate named the conditional ridge-type spectral decomposition estimate(CRSDE). And using mean square error matrix and generalized mean square error as standard for comparing the estimates, we build the necessary and sufficient conditions for the superiority of the CRSDE over the conditional spectral decomposition estimate(CSDE). And we also given the upper and lower bounds of relative efficiency. condition and under the rule of the mean square error matrix and proves that the estimates of optimal benign. Finally, a numerical example and Monte Carlo simulation is also given to show the theoretical results.