英文摘要： Apart from the discrete and continuous data, there exist various complicated multivariate response data in practice, for example, ordinal categorical data and ranking data that are often-used in psychometric, educational, biological, medical and economical sciences. Based on the single-index dimension-reduction technique, we will develop a series of semi-parametric multivariate statistical models and methods for analyzing those multivariate data in the proposed project. In detail, we will adopt multivariate single-index-type models, multivariate single-index-type probit models and generalized multivariate single-index-type models, to analyze the multivariate continuous responses, the multivariate ordinal categorical data and other data including the ranking data, respectively. Besides those, we will also develop single-index-type structural equation models to model the covariance structure or the conditional covariance structure of those complicated multivariate data. There are at least two innovations in the proposed project: 1) We not only extend the single-index-type models from univariate response to multivariate responses, but also introduce the single-index technique to the research area of structural equation models; 2) In additional to the case that all of the covariates in the indices are the observable variables, we also consider the case that the unobservable (latent) variable(s) corresponding to ordinal categorical and ranking variable(s) acts as the covariate(s) to satisfy the practice needs. For the latter, we first investigate the identifiability conditions of the models. We will make use of Bayesian approach of free-knot splines (i.e., the nonparametric functions in the model are approximated by splines but the numbers and locations of knots are treated as random variables) to make inference about the proposed models via sampling from the joint posterior. The advantage of Bayesian method is that a multivariate response model can be converted into a series of univariate response models by deriving the fully conditional posteriors, which greatly facilitate the sequent analysis. To obtain efficient and fast-convergent algorithms, we will apply acceleration techniques such as generalized Gibbs sampler, alternating subspace-spanning resampling, and partially collapsed Gibbs sampler to our methods of analysis. The research results of the proposed project will lay a solid foundation for the methodological research on the analysis of the complicated multivariate response data.
英文关键词： Semiparametric models;Multivariate statistical analysis;Single-index models;Latent variable;Structural equation models