Latent Single-Index Models And Nonparametric Model Based On Martingale Difference Errors

In this paper,we mainly analyze two types of data:ordinal categorical data and martingale difference errors data.Accordingly,the purpose in this paper is the study of Bayesian analysis and the large sample property for the nonparametric estimator under two kinds of model,i.e.,the latent single-index model and the nonparametric regression model,respectively.We split it into three parts as follows:Firstly,we propose a latent single-index model with the ordinal data to evaluate the effect of latent covariates to the latent response variables,and develop a Bayesian method with free-knot splines to analyze the proposed model.Since the traditional spline methods cannot be directly applied to approximate the unknown link func-tion.We consider a modified version to address this problem by transforming the index into the unit interval via a continuously cumulative distribution function and then constructing the spline bases on the unit interval.To obtain a rapidly convergent algorithm,we make use of the marginalization and parameter expansion and reparam-eterization techniques,improve the movement step of Bayesian splines with free-knots so that all the knots can be relocated each time,and design a generalized Gibbs sam-pling step.We check the performance of the proposed model and estimation method by a simulation study.As an application,we use our model and estimation method to analyze a real dataset.Secondly,we further develop multivariate single-index models with factor struc-ture for multivariate ordinal categorical variables,to assess the effects of the latent covariates on the latent responses and explore the covariance structure of the latent responses.Based on Bayesian framework,we make statistical inference for the pro-posed model and obtain the estimator of factor structure parameters,index coefficient vectors,covariance matrix of latent variables and fitting the unknown multivariate link functions with free-knots splines.To accelerate the estimation procedure,we make use of the marginalization and parameter expansion and reparameterization techniques,and design a generalized Metropolis sampling step in our algorithm.In addition,the performance of the proposed model and estimation method are checked through a simulation and applied to a real dataset.Finally,the other type of data studied in this paper is about martingale difference sampling data,which is a broad class of non-independent random data.In practical applications,the assumption that sample sampling is independent and identical dis-tributed cannot always be satisfied.Therefore,the study of such data is even more universal.In the fourth chapter of this paper,based on nonparametric regression model with martingale difference sample errors,we consider a non-parametric lin-ear smoothing estimator.Some large sample properties of estimator consistency are investigated and obtained.As an application,with heteroscedasticity of errors,we do some simulation for the consistency of the nearest neighbor estimator.