Statistical Inference For Estimating Equations And Structural Equation Models

The inference method of the estimation equation is a widely used method. Many parameter estimation methods, such as maximum likelihood method, least squares method and moment estimation are special cases of estimation equation method. Its greatest advantage is that it does not depend on any error distribution and has robustness, that is, it can still obtain reliable results when models are misspecified. The popular methods dealing with estimation equation include the generalized method of moments and the empirical likelihood method. Especially, empirical likelihood method based on estimation equation has attracted attention of many scholars. In recent years, a growing number of researchers consider empirical likelihood method based on estimation equation in the Bayesian framework, namely Bayesian empirical likelihood method (BEL). The Bayesian empirical likelihood method not only inherits the advantages of the traditional empirical likelihood method and estimation equation method, but also is convenient to combine the hierarchical structure with the additional information of parameters.We study the inference of BEL based on estimating equations and the inference of structural equation equation models, respectively. We research Bayesian local influence analysis of general estimating equations with nonignorable missing data, Bayesian variable selection based on estimating equations and Bayesian empirical likelihood estimation of quantile structural equation models based on estimating equations. Moreover, we also consider latent variable selection in structural equation models.First, we propose BEL method to estimate unknown parameters based on nonignorable missing data. We consider that BEL method with missing data depends heavily on the prior specification and missing data mechanism assumptions. It is well known that the resulting Bayesian estimations and tests may be sensitive to these assumptions and observations. To this end, a Bayesian local influence procedure is proposed to assess the effect of various perturbations to the individual observations, priors, estimating equations (EEs) and missing data mechanism in general EEs with nonignorable missing data. A perturbation model is introduced to simultaneously characterize various perturbations, and a Bayesian perturbation manifold is constructed to characterize the intrinsic structure of these perturbations. The first-and second-order adjusted local influence measures are developed to quantify the effect of various perturbations. The proposed methods are adopted to systematically investigate the tenability of nonignorable missing mechanism assumption, the sensitivity of the choice of the nonresponse instrumental variable and the sensitivity of EEs assumption, and goodness-of-fit statistics are presented to assess the plausibility of the posited EEs. Second, we consider Bayesian variable selection based on estimating equation and use shrinkage prior, i.e., Laplace prior to achieve sparse parameter estimation. Under some regular conditions, we obtain the posterior consistency. Third, we also establish quantile structure equation model(QSEM) and use the BEL method to make an estimate of parameters. We regard latent variables as missing data, use empirical likelihood function to construct posterior distribution estimation of the latent variables and incorporate the interpolation method and Gibbs algorithm to derive the parameter estimation. Finally, we develop a general SEM that latent variables are linearly regressed on themselves and it does not require specifying outcome/explanatory latent variables. A penalized likelihood method with the proper penalty function is proposed to simultaneously select latent variables and estimate coefficient matrix for determining the formulation of structural equation. We show the consistency and the oracle property of the proposed estimators. Computationally, we develop an expectation/conditional maximization (ECM) algorithm and minorization-maximization (MM) algorithm and choose the turning parameter by IC_Q criterion.