Statistical Inference Of Nonlinear Structural Equation Model Based On Partial Normal Distribution

Structural equation model is used widely in economics, behavioral science, medicine and sociology and so on, it is an important statistical analysis tool to analyze and research the inner link between manifest variable and latent variable at home and abroad. Statistical software currently used to analysis of structural equation models are:the LISREL, AMOS, EQS6and WinBUGS. With the development of science and technology and data complexity, the rise of the structural equation model also obtained fast development, the initial confirmatory factor analysis model for the development of nonlinear structural equation model, multilevel structural equation model, mixed structural equation model, with missing data and orderly classification of nonlinear structural equation model, and the distribution of the manifest variables to obey exponential family structure equation model and so on. From these studies promoted the development of the structural equation model objectively, enriched the content of the multivariate statistical analysis.In recent years, because of the complexity of the data and the data is often not obey the normal distribution, it is very difficulty to research, therefore, the skew normal distribution is developed well. The skew normal distribution is used widely, such as Azzalinin and Capitanio are given in the application of multivariate statistical analysis, Capitanio presented its application in graph model. Based on the skew normal distribution, this paper studies the nonlinear structural equation model of maximum likelihood estimation, the local influence analysis and Bayesian analysis. Now the summary of the main research content of this article is as follows:(1) To study the nonlinear structural equation model based on the skew normal distribution of the maximum likelihood estimation problem. Because of the complexity of the model, the calculation of E step need calculate higher integral, and some analytic expressions have no expressions, therefore, in this paper, with the MH algorithm, the conditional expectation of complete data logarithmic likelihood function is approximated by sample mean; While the M step is achieved by using the method of conditional maximization. In addition, the article also further establishes a modified EM algorithm based on random representation, and the convergence of algorithm is faster and more effectiveness than the classic EM algorithm.(2)According to the nonlinear structural equation model sets up a local influence measurement method to evaluate the sensitivity of the model for small perturbations based on the skew normal distribution. Using the EM algorithm of MCMC technology to get the model parameters of maximum likelihood estimation based on the skew normal distribution, and more, this article based on Q-function sets up a local influence measurement and introduces five different perturbation models and corresponding algorithm.(3)According to the nonlinear structural equation model based on the skew normal distribution sets up a Bayesian analysis method. In this paper, the Bayesian estimation of the model has been got by the Gibbs sampling and MH algorithm, and the Bayesian factor for model comparison has been obtained by the path sampling.Above all, based on the skew normal distribution, this paper gets the maximum likelihood estimation of nonlinear structural equation model by the EM algorithm, studies the problems of local influence measure with the hierarchical representation of the skew normal distribution, uses Gibbs sampling and MH algorithm to discuss the Bayesian analysis. The study further promotes and develops the theory of structural equation model.