Statistical Inference For Some Classes Of Random Coefficient Binomial Autoregressive Models

In real life applications,finite range count time series are widely used in many fields,such as the weekly number of rainy days,the number of occupied rooms in a hotel,etc.To model this kind of data,we propose some classes of random coefficient binomial autoregressive models in this paper.The construction of the coefficients are established from different aspects and we consider the statistical inference for the models.The main content is divided into four parts.In the first part,we consider the influence of the explanatory variables and propose the binomial AR(1)process with explanatory variables.In the second part,we consider the influence of the observations for the coefficients.In the third part,we propose a new class of flexible models using copulas.Finally,we relax the parametric model assumptions for the coefficients,consider a more widely class of models and use Bayesian empirical likelihood inference.In the following part,we will introduce the main results of our studies.1.Statistical inference for the binomial AR(1)model with explanatory variables(BAR(1)-X)In real life applications,time series data often exhibit changing dynamic behaviors because of the influence of the environmental factors.In order to capture the influence of the explanatory variables,we use logit transformation and consider the possibility of a multivariate autoregressive structure for the random coefficients.Thus,the explanatory variables can be included into the corresponding models.Using this approach,we propose a new class of binomial autoregressive models with explanatory variables,denoted by BAR(1)-X.The conditional least square(CLS)and conditional maximum likelihood(CML)estimators are discussed and the related asymptotic properties are derived.We also investigate the influence of the types and dimensions of the explanatory variables for the proposed methods in simulation studies.Finally,a real data example in meteorology is provided to illustrate the model.Using weekly maximum temperature as explanatory variables,we apply the BAR(1)-X model to the data set of rainy days per week at Hamburg-Neuwiedenthal in Germany.2.Statistical inference for the binomial autoregressive model with observation driven coefficients(LLBAR(1))For the binomial autoregressive processes,the thinning probabilities at a time are often depending on the processes up to that time.To model this kind of updating mechanism,we propose a new class of observation driven binomial autoregressive(LLBAR(1))models,where the coefficients are specified based on the generalized linear model(GLM)theory.We consider the CML estimation method and derive the asymptotic properties of the estimators.Furthermore,we also give the consistent estimator of the conditional probability mass function.Finally,a real data example of Yersiniosis infections in Germany is analyzed to illustrate our model.3.Statistical inference for the flexible binomial AR(1)model with copula(FBAR(1))In order to accurately and flexibly capture the correlation structure between the random coefficients in the binomial AR(1)process,we propose a new class of models with copula,which we call the FBAR(1)model.We derive the properties of the process,such as stationarity,ergodicity,expectation and variance.We use the two-step CLS and CML methods to estimate the parameters.In simulation studies,we investigate the accuracy of the estimators and the robustness of our model with the contaminating data.Finally,we apply our new model to two real data sets in the fields of meteorology and finance.By comparing the fit quality performances of all the candidate models,the FBAR(1)model shows the best performance in these cases.4.Bayesian empirical likelihood inference for the random coefficient binomial AR(1)modelWe make no parametric assumption for the autoregressive coefficients and propose a new class of binomial autoregressive models,which we only assume that the coefficients are i.i.d.bivariate random variables.The related model is called the random coefficient binomial AR(1)(RCBAR(1))model.We use the Bayesian empirical likelihood inference for the model.First,we establish a nonparametric likelihood using the empirical likelihood(EL)approach.Then,an efficient Markov chain Monte Carlo(MCMC)procedure is described for the required computation of the posterior distribution.In the simulation study,we analyze the efficiency of the MCMC algorithm.Furthermore,we consider the simulation under contaminated data sets.The results show that the BEL method is more robust than the CLS method.Finally,a real data example of transactions at the Korea stock market is analyzed to illustrate our method.