Robust Estimation Methods For Extended T-process Regression Models

As an important tool for functional data analysis,Gaussian process regression mod-els can model the nonlinear relationship between a response and a set of large dimen-sional covariates and have many good properties,such as efficient modeling and es-timation procedure,closed conditional and predictive distribution and some statistical asymptotic properties.The extended t-process regression model inherits these proper-ties from Gaussian process regression models and has been proved more robust against outliers,due to its heavy-tailed extended multivariate t distribution.However,the over-estimated problem of the degree of freedom by maximum likelihood methods leads to an undesirable result that the extended t-process regression model loses its robustness.Therefore,robust estimation by using Bayesian methods are developed for the extended t-process regression model in this paper.First,by applying penalty priors,we can obtain smaller and more stable estimates of the degree of freedom.For data with high dimensional covariates,to reduce the com-plexity of the model,we carry out a variable selection procedure for parameters in the covariate kernel function by using Spike-and-slab priors.Besides,statistical properties such as consistency of maximum a posterior estimators and information consistency are also obtained.Numerical studies show that the proposed method has smaller prediction errors against outliers than original extended t-process regression models.Moreover,the proposed methods are applied to the extended t-process regression models with in-dependent errors,which assumes the independence between regression functions and error functions and has more flexible structures.Because of the existence of two latent scare variables,the likelihood function and predictive procedure of this model has no close forms,a MCEM algotithm is implemented to attach their stochastic approxima-tion.Simulation studies and real examples show that this method performs more robust and applicable to general situations than the proposed method before.