Bayesian Analysis For Single-index Varying-coefficient Models

The single-index varying-coefficient model was first proposed by Xia and Li in the 1999. Due to its good properties that it does not surfer from the curse of dimensionality which is often encountered in multivariate nonparametrics settings, and it has a strong interpretability. Recently, the single-index varying-coefficient model is widely applied to various fields including psychology, medicine, econometrics and education. The single-index varying-coefficient model combines many advantages of parametric models and nonparametric models. Also this model includes a class of statistical models, such as the single index model, the varying-coefficient model as its special cases. Hence, many authors have studied the model. However, to our knowledge, there is little work done on Bayesian inference of single-index varying-coefficient model. At the same time, with the development of statistical computing tools, bayesian statistics has received a lot of attention in recent years due to its simple operation and consider the history information. This paper has mainly studied Bayesian inference on the single-index varying-coefficient model, the main studies are as follows:(1) For the smooth functions in single-index varying-coefficient model, we use the B-spline to approximation to them and then obtain their corresponding Bayesian estimators.(2) Based on uninformative prior of single index, the conjugate normal-inverse gamma prior for the model error variance and components of B spline coefficients, we derive their corresponding posterior distributions.(3) Based on the Gibbs sampling within Metropolis-Hastings algorithm, we develop a hybrid algorithm to simultaneously obtain Bayesian estimates of parameters and smooth functions. At the same time, we present a formula to evaluate the standard errors of Bayesian estimators.(4) Through simulation studies, we verify the efficiency of the Bayes method proposed in this article and show that this Bayes method produces small biases, then we use a real example to illustrate the application of our method.