Statistical Inference Under Specific Mixed-effects Models

In statistical studies,the observed data sometimes come from subjects who are measured repeatedly over time,which is very common in different disciplines such as economics,sociology,psychology and medicine.Such data is also called repeated measurement data or longitudinal data.The classical linear model generally contains only fixed effects and error terms,so it is difficult to reflect the correlation between the measured data of the same subject at different time.However,the mixed effect model not only contains the fixed effect parameters of the regression coefficient,but also the random effects affecting a specific group of factors.What is more,if we consider the application of multiple factors in practice,the data can be well fitted,thus,this model provides a flexible and effective tool for the analysis of repeated measurement data.There have been some theoretical studies on the estimation,prediction and statistical inference of parameters in mixed effect model.In this paper,we mainly study the mean square prediction error(MSPE)estimation and parameter estimation in specific models under the framework of mixed effect.The main contents are as follows:In Chapter 1,we introduce the theoretical basis and related research status.First of all,the preliminary knowledge involved in the article is briefly introduced.Secondly,some mean square prediction error(MSPE)estimation methods under the mixed effect model are introduced,especially the Monte-Carlo Jackknife method(McJack)proposed by Jiang(2002).Thirdly,in the nonlinear mixed effect model,the iterative estimation equation method is introduced,which pave the way for the parameter estimation method proposed in this paper.Finally,a linear quantile regression model and a linear quantile mixed effect model are introduced,and the research status of its estimation methods is briefly described,which provides a basis for further studying mixed effect prediction and MSPE estimation in the following article.In Chapter 2,a linearization method is proposed to estimate the log mean square prediction error(MSPE)of the mixed effect prediction under the linear mixed effect model(LMM).It is proved that this method can guarantee the second order unbiased of the log(MSPE)estimation under some regularity conditions.At the same time,the MSPE estimation is obtained by taking an exponent,which can naturally guarantee the non-negative of the estimation.Secondly,the analysis proves that the linear mixed effect model meets the requirements of the estimation method under specific conditions,so that this method can be applied to the model.Thirdly,the numerical simulation is used to compare this method with the traditional Naive estimation and McJack estimation.The results show that this method has certain advantages in%RB and CV,and the overall performance is better than the traditional methods,especially in the computational efficiency.Finally,the method is applied to the data of fatal death of young rats in the rat nest for empirical analysis.In Chapter 3,in reality,some data set cannot be fitted in linear form.Thus,in this chapter,we mainly study a specific nonlinear mixed effect model and propose a twostep parameter estimation method under this model.The asymptotic properties of the estimation method,namely,consistency and asymptotic normality,are investigated,and a detailed proof is given.In addition,theoretical and numerical results show that when the sample size is sufficiently large,the asymptotic covariance obtained by the second step estimation is more effective than that obtained by the first step.However,the two-step estimation method have some limitations on the estimation of error variance,so we propose a special method for the estimation of error variance,and numerical simulation shows the effectiveness of this method.Since the traditional estimation methods need to approximate the likelihood function,but the two-step estimation method proposed in this chapter avoids approximation,so the calculation is more convenient and fast.Finally,this method is analyzed on the girls’ height data,Indomethicin drug concentration data and orange tree growth data.In Chapter 4,we focus on the Linear Quantile Mixed effect Model(LQMM)and study the MSPE estimation method for mixed effect prediction.Specifically,we consider the mixed effect prediction under the assumption that the random effect follows the normal distribution and the random error follows the partial Laplace distribution.At same time,a detailed derivation and analytical expression are given.Secondly,in the numerical simulation,Sumca estimation method under LQMM is compared with Naive estimation method at different quantile levels,and the results show that Sumca estimation is feasible.Finally,LQMM is used to fit and analyze the data of kidney transplantation in the hospital.In Chapter 5,we summarize the whole thesis and give a brief description of future work.