Moment Selection And Prediction For Complex Data Models

Over the past decade,variable selection has been becoming the fundamental and hot topic of statistical analysis.Statisticians have proposed many variable selection methods such as AIC,BIC,DIC,LASSO and Adaptive LASSO,which has been applied to the linear model,the Cox model,the generalized linear model and the partially linear model,and so on.However,the issues for moment selection and variable selection as well as the prediction problem after variable selection of the complex data models such as the unconditional moment models,the dynamic panel data model,the nested error regression model,are not well investigated.This thesis first studies the moment selection and parameter estimation problem of high-dimensional unconditional moment conditions,and then investigates the simultaneous moment selection and model selection problem of growing-dimensional unconditional moment models.Under the nested error regression model,moreover,this thesis also focuses on the mean estimation and variance/MSE estimation of small areas by applying the variable selection techniques.Those investigations not only develop new methods for the issue of moment selection but also extend the fields of moment selection.Thus,both theoretical results and real data applications confirm the merits of this thesis.Specially,the main work of the thesis are summarized as follows.On the one hand,for high-dimensional unconditional moment models,we first propose a Fantope projection and selection(FPS)approach to distinguish the informative and uninformative moments in high-dimensional unconditional moment conditions.Second,for the selected unconditional moment conditions,we present a generalized empirical likelihood(GEL)approach to estimate unknown parameters.The proposed method is computationally feasible,and can efficiently avoid the well-known ill-posed problem of GEL approach in the analysis of high-dimensional unconditional moment conditions.Under some regularity conditions,we show the consistency of the selected moment conditions,the consistency and asymptotic normality of the proposed GEL estimator.Two simulation studies and a real example are conducted to investigate the finite sample performance of the proposed methodologies.On the other hand,generalized method of moments(GMM)and generalized empirical likelihood(GEL)coupled with moment conditions have been widely appliedto a broad class of statistical problems.It is known in the literature that GMM and GEL approaches encounter difficulties when handling problems having high-dimensional model parameter and moment conditions.To address the challenges,we construct a novel penalized Fantope projection minimum deviation(FPMD)criterion function.Based on FPMD function,we propose a new penalized FPMD estimator by applying two penalty functions respectively regularizing the model parameters and the associated weighted matrix in the optimizations of FPMD function.We have the following two findings:(i)penalizing the model parameters can effectively reduce the number of model parameters;(ii)by penalizing the weighted matrix involved in the FPMD to encourage sparsity,we show that dimension reduction in the number of moment conditions can be effectively derived without impact on the validity and consistency of the resultant estimators,and such a reduction in the dimensionality of moment conditions can be interpreted as a moment selection mechanism.Under some regularity conditions,such as properly limiting the diverging rate of the number of parameters and moment conditions,we show the consistence and the asymptotic normality of the proposed penalized FPMD estimator.Three simulation studies demonstrate the numerical advantages of our method over the existing ones.The performance of the proposed method is also illustrated via a real data analysis.Finally,we turn to investigate the small area mean prediction problem after variable selection.To this end,we propose an extended nested error regression model.We assume that the effects of some of the explanatory variables are fixed but have multiple centers in the newly defined model.Based on the new extended linear mixed regression model,we propose a multiply directional penalty function,and then use it to fit the model and estimate the coefficients simultaneously.After proper choosing penalty function and tuning parameter,we then propose two new small area mean estimators and construct estimators of the mean square error.Simulations based on artificial and realistic finite populations show that the new estimators can be efficient.Furthermore,the confidence intervals based on the new methods have accurate coverage probabilities.