Study On Markov-switching Integer-valued Time Series Models

In real life,integer-valued time series can be seen everywhere,such as the number unemployed people in a certain area,the monthly claims of insurance companies,the number of hospitalizations,the number of daily traded in a stock,and so on.In recent years,the analysis of integer-valued time series has been paid more and more attention by statistics.At present,there are mainly two types of modeling integer-valued data.? the thinning operator model(with thinning operator).Steutel and Van Harn(1979)proposed the thinning operator "?",which lays a foundation for the analysis of integer-valued data.Based on the thinning operator,Al-Osh and Alzaid(1987)defined the first-order integer-valued autoregressive process(INAR(1)).A process {Xt} follows the recursion is said to be an INAR(1)process,where {Zt} be an i.i.d.process with range N0,denote E(Zt)=?,??(0,1).The proposition of this process promotes the development of integer-valued time series analysis.In the INAR(1)family,it is very important and popular that the first-order integer-values Poisson autoregression(PINAR(1)),that is,the innovation {Zt} is i.i.d.Poisson process.While the condition least squares estimations of the model's parameters exist biases and are not effective for small sample sizes,we proposed the maximum empirical likelihood estimate(MELE),based on the empirical likelihood method and making effective use of auxiliary information.Through the study and analysis of numerical simulation,the proposed estimation method has a good result.Note that,the thinning parameter ? may vary with time.Zheng et al.(2007)proposed the first-order random coefficient integer-valued autoregressive model(RCINAR(1)),which may be defined as follows:where {?t} is a sequence of i.i.d.random variables defined on[0,1)with E[?t]=? and Var(?t)=??2;{Zt} is a sequence of i.i.d.non-negative integer-valued random variables with E(Zt4)<?;X0,{?t} and {Zt} are independent.There is an interesting problem that testing the thinning parameter ?t is constancy or not.This is essentially same as testingBased on the empirical likelihood,a testing statistic is given,where we assume that the innovation {Zt} is an i.i.d Poisson process.We studied the feasibility of the pro-posed method through numerical simulation.The results supported our theory and the proposed method was applied to the analysis of a set of data on the burn claims counts.Meanwhile,the assumption that the thinning parameters are independent of each other has some limitations.We proposed a first-order integer-valued autoregressive process with Markov coefficient(MSINAR(1))under relaxing the assumption.? State space model(with the latten process).Ferland et al.(2006)studied INGARCH(p,q)process.Fokianos and Tj?stheim(2011)emphasized that the INGARCH process,although de-fined using the hidden conditional means,is observation-driven in the sense of Cox(1981);that is,the serial dependence can be explained by past observations.In con-trast,a very popular type of parameter-driven model for count processes is the hidden-Markov model(HMM),which is the first proposed by Baum and Petrie(1966),who referred to them as "probabilistic functions of Markov chains".Combining these two class models,we proposed a Markov-switching integer-valued generalized autoregres-sive conditional heteroscedastic(MSINGARCH)process.The necessary and sufficient conditions for the first-order stationary and the existence of the second order moment and some basic properties of the process are derived.The maximum likelihood esti-mates of model parameters are computed by using EM algorithm,and the consistency and asymptotic normality of the estimation are verified by numerical simulation.To illustrate the usefulness of the model,we apply it to a set of real data on the earthquake counts.In addition,based on the empirical likelihood method,the confidence region-s of unknown parameters in the autoregressive model with explanatory variables is given.The simulation results show that the confidence region based on empirical like-lihood has some advantages over that based on the conditional least squares estimation.