Non-iterative Sampling Algorithm For Robust Regression Models And Statistical Inferences For Time Series Models Of Counts

In order to deal with the heavy-tailed phenomenon in continuous data and the over-dispersancy in time series data of counts,in this thesis,we discuss some related models and the parameter estimation problems.The models considered include the scale mix-ture Normal regression model,censored Student-t regression model,mixture negative binomial integral ARCH model and the Neyman type-A GARCH models,involving sev-eral statistical algorithms,such as non-iterative sampling algorithm and expectation maximization algorithm.The necessary and sufficient condition of stationary proper-ty of related model are obtained and some model selection procedures are proposed.Compared with traditional models and algorithms,the procedures we considered have satisfying performance in both simulations and real data analysis,and have better ap-plications in such field as Economy and Medicine.1.A Non-iterative Bayesian Sampling Algorithm for Linear Regression Models with Scale Mixture of Normal Distributions Scale mixtures of Normal(SMN)distributions,are a class of heavy-tailed distributions,which are often used asa robust alternative to the the routine use of normal distributions when outliers are present in the data.The maximum likelihood estimation(MLE)via the expectation maximization(EM)algorithm and Bayesian method based on monte carlo markov chain(MCMC)have been developed in the documents to estimate the parameters in SMN regression and related linear models.For example,Andrews(1974),Dempster(1980)and Lange(1993)developed the EM algorithm for SMN distributions and discussed their applications in robust regression.Fernndez and Steel(2000)discussed MCMC method for SMN linear regression model from a Bayesian perspective.Abanto-Valle et al.(2010)applied the SMN distributions to stochastic volatility models for robust Bayesian analysis.Rosa et al.(2003,2004)extended the SMN linear regression model to SMN linear mixed effect model or longitudinal data and adopted a Bayesian framework to carry out posterior analysis for robust inferences.Garay et al.(2015,2017)extended the SMN regression model to censored situation,and discussed the EM algorithm and MCMC method for robust inferences.Although the Gibbs sampler is widely used for statistical inferences for its general applicability and ease of implementation,there are,however,two vital issues regarding such iterative sampler,which are too easily overlooked by users.First,the variables gen-erated in the same iterative process in the Gibbs sampling are hardly ever independent,second,it is harder to check convincingly whether the stage of convergence has been reached upon termination of iteration.Tan et al.(2003)developed a non-iterative sam-pling algorithm based on inverse Bayes formula(IBF)in missing data structure,which can approximatively generate independent and identically distributed(i.i.d.)samples exactly or approximately from the observed posterior distribution,which can be used for statistical inferences immediately,thus eliminates the two problems in Gibbs samling.Inspired by Tan et al.(2003),in the first chapter,the idea of IBF is applied to the SMN regression model and a non-iterative Bayesian posterior sampling algorithm is developed.Our procedure combines the robustness of the SMN regression with the computational effectiveness of the non-iterative sampler,which can generate indepen-dently and identically distributed posterior samples.Simulation studies are conducted to assess the finite sample performance of the procedure,in addition,model selection criteria and influential statistic based on the sampled posterior samples are calculated in order to choose the most fitted model and to detect possible outliers.Finally,the SMN regression model with IBF algorithm is applied to the U.S.Treasury bond prices data.Compared with the Normal regression and the iterative Gibbs sampler,our procedure has better performance both in simulation studies and real data analysis.2.A Non-iterative Bayesian Sampling Algorithm for Censored Student-t Linear Regression Models Censored Student-t linear regression models(CTR)are more robust than censored Student-t linear regression models(CTR)when dealing with data with outliers.In the second chapter,we develop a non-iterative sampling algorithm based on to analyze censored Student-t linear regression model.The core of the algorithm lies in the hierarchical representation of the Student-t distribution and the censored data structure,which make the CTR model naturally has a MCEM structure.First,we augment the observed data with two type latent data,one is the mixture variables to represent the Student-t distribution,the other is the missing data of censoring,and obtain the structure of augmented conditional predictive distributions as in the MCEM structure.Then,we conduct the EM algorithm to get the posterior mode,which then is used to get the best importance sampling density,so that the overlap area under the target density and the importance sampling density(ISD)is large.Finally,we use the IBF and sampling/important resampling(SIR)twice to generate i.i.d.samples approximately from the observed posterior distribution.The samples can be used in model selection and influence diagnosis to choose the best degree of freedom and detect latent outliers.We conduct simulation studies to investigate the performance of the CTR model with IBF algorithm,and apply the procedure to two real censored data sets,one is the wage rates data with left censoring and the other is the insulation life data with right censoring,finding that the proposed procedure is more effective than the usual censored Normal linear regression model.3.Statistical Inferences for a Mixture Integer-valued GARCH model Based on Negative Binomial Distribution In the third chapter,a mixture integer-valued ARCH model based on negative binomial distribution is proposed for modeling time series of counts with overdispersion and multimodality.This model consists of K stationary or non-stationary integer-valued ARCH components,and each component has a marginal distribution of negative binomial distribution conditionally.Compared with the single-component model,the advantages of the mixture model include the ability to handle not only overdispersion but also multimodality and non-stationary components.The necessary and sufficient first-order and second-order stationary conditions are giv-en.Then some autocovariance and autocorrelation functions are derived.Because of the mixture distribution,the maximum likelihood estimations of parameters are done through an EM algorithm.The performances are studied via simulations.Finally,themodel is applied to a real data of counts,and the model is selected by three information criterions.4.Statistical Inferences for the Neyman type-A Integer-valued GARCH Model In the forth chapter,we discuss a particular compound Poisson integer-valued GARCH model,which is known as the Neyman type-A integer-valued GARCH mod-el.For Neyman type-A integer-valued GARCH(p,q)model,we derive also some first-order and second-order stationary condition.Then autocovariance and autocorrelation functions are given,which can be used for Yule-Walker estimation.As for estimation of parameters,we present three approaches:Yule-Walker estimation(YW),conditional least square estimation(CLS)and maximum likelihood estimation(MLE).For complex-ity of probability distribution law of Neyman distribution,EM algorithm is employed in maximum likelihood approach.From results of numerical simulation studies,we can find that the performances of three approaches are more perfect,especially when sample size increases.At last,campylobacter data is modeled by Neyman type-A integer-valued ARCH model with the help of AIC and BIC,and data fitting and residual test are done.