Statistical Inference For Some Integer-valued Time Series Based On Negative Binomial Thinning Operator

Integer-valued time series exist in many fields,such as meteorology,epidemiology,insurance,transportation,criminology,etc.In recent years,the modeling of integer-valued time series has attracted the attention of statisticians.Bivariate time series of counts are commonly encountered,and the bivariate INAR(1)models based on thin-ning operator are commonly used to fit bivariate count time series.Generally,the desired bivariate INAR(1)model can be constructed in two different approaches.In one approach,the marginal distributions of the model are prespecified,then the dis-tribution of the innovations is identified in the required form to hold the stationarity of the process.The other approach to construct the desired bivariate INAR(1)model is the choice of an appropriate distribution for the innovations to lead to the speci-fication of the underlying marginal distributions.However,to our knowledge,based on the negative binomial thinning operator,there seems no bivariate INAR(1)model constructed by prespecifying the distribution for the innovations,the details are stat-ed in the Remark 2.1.1 in Chapter 2.For this,we give an extension of the negative binomial thinning operator.We propose a bivariate INAR(1)process based on the extended negative binomial thinning operator by prespecifying the distribution for the innovations and study the probabilistic properties and the parameter estimation of the process.In addition,we propose an INAR(1)process based on the extended negative binomial thinning operator,which is a special case of the bivariate INAR(1)process proposed above in univariate case.And the empirical likelihood inference of the process is studied.Finally,based on the binomial thinning operator and the extended negative binomial thinning operator,the mixed INAR(1)process is proposed and the statistical inference of the process is considered.In the following part,we will introduce our main results of this thesis briefly.1.The modelling and statistical inference of the bivariate INAR(1)process based on the extended negative binomial thinning operator.In order to better characterize two correlated integer-valued time series with a productive data generating scheme,we propose an extended negative binomial thinning operator.The definition is as follows.Definition 1 Assume that X is a non-negative integer-valued random variable and let ??(0,1).Then an extended negative binomial thinning operator denoted by"*" is defined as follows(?)(1)where {Wj} is a sequence of i.i.d.geometric random variables with probability mass function P(Wj=k)=?k/(1+?)1|k,k?0.Based on the extended negative binomial thinning operator,we propose a bivariate INAR(1)process by prespecifying the distribution for the innovations.The definition is following.Definition 2 The bivariate non-negative integer-valued series {Xt}t?N is said to be a BNBINAR(1)process if it satisfies the following recursion(?) where(?)"*" is the extended negative binomial thinning operator defined by(1);(?)A*is a matrical thinning operation,which acts as the usual matrix multiplication keeping the properties of the extended negative binomial thinning operation,and the thinning operations ?1*X1t and ?*X2t are mutually independent for all t ? N;(?){Rt} is a sequence of i.i.d.nonnegative integer-valued random vectors.For each time t,Rt is independent of A*Xt-1 and Xs for s<t.It can be seen from the definition of the model,the BNBINAR(1)process is a Markov process defined on N02 with the following one-step transition probabilities(?)where f(k,s)=P(R1t=k,R2t=s),k,s ? N.In order to derive the properties of the model,let mean and variance of {Rit},and covariance between {Rit} and{R2t} be E(Rit)=?i,Var(Rit)=vi?i,vi>0,?i>0,i=1,2,and Cov(R1t,R2t)=?,??R.The following theorem states that the BNBINAR(1)process is strictly stationary and ergodic.Theorem 1 If{Xt}t?N satisfies BNBINAR(1)process,then the process is strictly stationary and ergodic.The probabilistic properties of BNBINAR(1)process are given in the following proposition.Proposition 1 Suppose {Xt}t?N is a BNBINAR(1)process,for i,j=1,2,h?0,then(?)The first-order and second-order marginal conditional moments of {Xt}t?N are E(Xi,t+h|Xi,t-1)=?ih+1Xi,t-1+(?i+?i)1-?ih+1/1-?i,(?)+?i(?i+?i)/)1+?i)2(1-?ih)(1-?ih+1)+?i(1+?i)/1-?i2(1-?i2(h+1)),Cov(Xi,t+h,Xit|Xi,t-1)=?ih[?i(1+?i)(Xi,t-1+1)+vi?i],Cov(Xi,t+h,Xjt|Xi,t-1,Xj,t-1)=?ihCov(Rit,Rjt)=?ih?,j?i.(?)The first-order and second-order marginal moments of {Xt}t?N are E(Xit)=?i+?i/1-?i,Var(Xit)=?i(?i+1)/(1-?i)2+vi??/1-?i2,Cov(X1t,X2t)=cov(R1t,R2t)/1-?1?2=?/1-?1?2,Cov(Xi,t+h,Xjt)=?ihCov(Xit,Xjt),Cov(Xit,Rjt)=Cov(Rit,Rjt)=?,j?i,Further,we have(?)(?)Proposition 2 Suppose {Xt}t?N satisfy BNBINAR(1)process,then the dispersion Index of {Xit} isIi=Var(Xit)/E(Xit)=?i(1+?i)(1+?i)+vi?i(1-?i)/(1-?i2(?i+?i),furthermore,the marginal distribution of {Xit} is overdispersed,equdispersed or under-dispersed when vi+?i2(-+?i+2?i)/(1-?i)?i is greater than 1,equal to 1 or less than 1,respectively,i=1,2.Next,we consider the parameter estimation of BNBINAR(1)process by the meth-ods of conditional least squares and conditional maximum likelihood.Moreover,we obtain the asymptotic normality of the conditional least squares estimator.Theorem 2 Let ?CLS be the conditional least squares estimators of BNBINA-R(1)process,then ?CLS is consistent and has the following asymptotic distribution(?) where(?),gt=(g1,g2,g?)T,?t=(?1,?2,??)T,gi= E(Xit | Xi,t-1)=?i(Xi,t-1+1)+?i,g?=Cov(X1t,X2t | Xl,t-1,X2,t-1)=?,?i=Xit-gi???=?1?2-g?,i=1,2.Next,we consider two specific BNBINAR(1)processes by prespecifying the dis-tributions for the innovations with equdispersion and overdispersion.One BNBINAR(1)process,the innovations of which follow jointly a bivariate Poisson distribution,is denoted as BP-BNBINAR(1)process.The other BNBINAR(1)process,the in-novations of which follow a bivariate negative binomial distribution,is denoted as BVNB-BNBINAR(1)process.For these two processes,we not only obtain the estima-tors of unknown parameters by the methods proposed above but also discuss the cases of dispersion for them.Corollary 1 Let {Xt}t?N be from BP-BNBINAR(1)process,then the distribu-tion of {Xit} is overdispered,i=1,2.Corollary 2 Let {Xt}t?N be from BVNB-BNBINAR(1)process,then the distri-bution of {Xit} is overdispered,i=1,2.We discuss the performances of two estimators of these two specific BNBINAR(1)processes by simulation study.The results show that the CML is superior to the CLS estimator.At last,we apply our new model to monthly counts of drug offense reported in two police departments in Rochester in USA.By comparing them with the other candidate models,the BVNB-BNBINAR(1)process is most appropriate for this data set.Moreover,we analyze the Pearson residual to check the adequacy of the BVNB-BNBINAR(1)process.2.Empirical likelihood inference of INAR(1)process based on the extended neg-ative binomial thinning operatorIn order to model integer-valued time series data with a productive data generat-ing scheme,an INAR(1)process is proposed by prespecifying the distribution for the innovation based on the extended negative binomial thinning operator,which is the special case of BNBINAR(1)process in univariate case.The definition is as follows.Definition 3 A non-negative integer-valued process {Xt}t?N given by Xt=?*Xt-1+?t,t?N,(2)is said to be an INAR(1)process based on the extended negative binomial thinning operator,denoted as GNBINAR(1)process,where(?)"*" is the extended negative binomial thinning operator defined by(1);(?){?t} is a sequence of i.i.d.non-negative integer-valued random variables,whose probability mass function is denoted as f?>0,and the mean and variance of the {?t}are E(?t)=?,Var(?t)=v?,v>0,?>0.For fixed t,?t is independent of ?*Xs and Xs,when s<t.The following theorem states that the GNBINAR(1)process is strictly stationary and ergodic.Theorem 3 Suppose {Xt}t?N satisfy GNBINAR(1)process defined by(2),then{Xt}t?N is strictly stationary and ergodic.The following theorem states some probabilistic properties of GNBINAR(1)pro-cess.Proposition 3 If {Xt}t?N satisfies GNBINAR(1)process defined by(2),then(?)for h?0,first-order and second-order conditional moments of GNBINAR(1)process areE(Xt+h|Xt-1)=?h+1Xt-1+(?+?)1-?h+1/1-?,(?)Cov(Xt+h,Xt |Xt-1)=?h[?(1+?)(Xt-1+1)+v?].(?)First-order and second-order marginal moments of GNBINAR(1)process are E(Xt)=?+?/1-?,Var(Xt)=?(?+1)/(-?)2+v?/1-?2.Furthermore,we have(?)(?) It is easy to obtain the following remark by Proposition 2.Remark 1 If {Xt}t?N is a GNBINAR(1)process,then the dispersion Index of{Xt} is I=Var(Xt)/E(Xt)=?(1+?(1+?)+/(1-?)+v?(1-?)/(1-?2(?+?),and the marginal distribution of {Xt} is overdispersed,equdispersed and underdis-persed,when v+?2(1+?-2?)/(1-?)? is greater than 1,equal to 1 or less than 1,respectively.Next,we consider the estimation of GNBINAR(1)process by empirical likelihood(EL)method.In order to obtain the empirical likelihood ratio statistic and its asymp-totically distribution,we make some assumptions as follows.(C1){Xt}t=1n is a stationary process;(C2)E(|Xt|6)<?.Denote the unknown parameter vector as ?=(?,?)T,and the true value as ?0.Let mt1(?)={Xt-?Xt-1-?-?)(Xt-1+1),mt2(?)=Xt-?Xt-1-?-?,according to dual likelihood given by Mykland(1995),an empirical log-likelihood ratio statistic can be obtained based on mt(?)=(mt1,mt2),(?) The following theorem states one empirical likelihood confidence region for the param-eters ?0.Theorem 4 Under the assumptions(C1)and(C2),we have(?)Theorem 5 Under the assumptions(C1)and(C2),the EL confidence regions with the confidence level(1-?)of ? can be constructed by FEL(?)={??R2|l(?)???2(2)},and FEL(?)is a convex set,where 0<?<1 and ??2(2)is the upper ? quantile of ?2 distribution with degrees of freedom 2.According to the log EL function proposed by Zhang et al.(2011)(?) the maximum empirical likelihood estimators of ? can be defined as follows ?MEL=arg inf? LE(?).The asymptotic normal distribution of ?MEL is provided by the follow-ing theorem.Theorem 6 Under the assumptions(C1)and(C2),the maximum empirical likelihood estimators ?MEL are consistent and asymptotically normal,i.e.(?) where D(?0)=E(mt(?0)mtT(?0))and(?).The following theorem states the asymptotic properties of CLS estimator.Theorem 7 The conditional least squares estimators ?CLS of GNBINAR(1)process are consistent and have asymptotical distribution(?) where(?)?t=Xt-gt.In simulation study,we compare the performances of these three estimation meth-ods on coverage probability of confidence regions and estimation of parameters.The results show that the maximum empirical likelihood estimation has much better advan-tages in the coverage probability of confidence regions than the other two estimators,and has less advantages on estimation than CML estimators but better than CLS es-timators.Finally,we apply the model to monthly counts of drug offense reported in a police department in Rochester.By comparing it with the other competitive models,our model is more appropriate for this data set.Then we check the adequacy of the model by analyzing the residual of the model.3.Modelling and statistical inference for mixed INAR(1)process based on the binomial thinning operator and the extended negative binomial thinning operator.Some random events may persist or disappear during observation time,or they may become more active and produce more random events over time.For example,patients with a disease that can be transmitted during the incubation period may recover or die,or they may spread the disease to other people during the incubation period to increase the number of patients.To characterize this kind of phenomenons,we propose a mixed INAR(1)process by prespecifying the distribution for the innovation based on the binomial thinning operator and the extended negative binomial thinning operator.Before giving the definition of mixed INAR(1)process,we first review the defini-tion of binomial thinning operator(Steutel and van Harn,1979).Assume that X is a non-negative integer-valued random variable and let ??(0,1).Then binomial thinning operator denoted by is defined as follows(?)(3)where {[Bi} is a sequence of i.i.d.Bernoulli random variables with probability mass function P(Bi=1)=1-P(Bi=0)= ? and is independent of X.In the following,we give the definition of mixed INAR(1)process.Definition 4 Suppose {Xt}t?N be a non-negative integer-valued series.If {Xt}t?N satisfies the following recursion(?)(4)then {Xt} is said to be a mixed INAR(1)process based on the binomial thinning operator and the extended negative binomial thinning operator,which is denoted as GNBBMINAR(1)process,where(?)"(?)" is the binomial thinning operator defined by(3),and "*" is the extended negative thinning operator defined by(1).For every t ? N,thinning operators?(?)Xt and ?*Xt are independent;(ii){?t} is a sequence of i.i.d.non-negative integer-valued random variables,whose probability mass function f?>0,and the mean and variance of ?t are denoted as E(?t)=?i,Var(?t)=v?,v>0,?>0.For fixed t,suppose that ?t is independent of ?(?)Xt,?*Xt and Xs,s<t.Proposition 4 GNBBMINAR(1)process is a Markov process defined on N0 and its one-step transition probabilities is given as follows(?) where s=max{0,xt-xt-1}.The following theorem states that GNBBMINAR(1)process is strictly stationary and ergodic.Theorem 8 If {Xt}t?N satisfies GNBBMINAR(1)process defined by(4),then{Xt}t?N is strictly stationary and ergodic.The following proposition states some probabilistic properties of GNBBMINAR(1)process.Proposition 5 Suppose {Xt}t?N satisfy GNBBMINAR(1)process defined by(4),then(?)for h?0,first-order and second-order marginal conditional moments of {Xt}t?N are given as followsE(Xt)=?+(1-p)?/1-a1,E(Xt2)=(2a1?+a3+a4+2(1-p)?2)((1-p)?+?)/(1-a1)(1-a2)+a4+(1-p)?(?+2?)/1-a2+?2+v?/1-a2,where a1=p?+(1-p)?,a2=p?2+(1-p)?2,a3=p?(1-?)and a4=(1-p)?(1+?);(ii)first-order and second-order moments of {Xt}t?N are E(Xt+h|Xt-1)=a1h+1Xt-1+[?+(1-p)?]1-a1h+1/1-a1,(?)Furthermore,the relationships between the conditional moments and the mo-ments of {Xt}t?N are(?)(?)Remark 2 If {Xt}t?N is a GNBBMINAR(1)process,then the dispersion Index of {Xt} is I=Var(Xt)/E(Xt)=a3+a4/1-a2+(1-a1)(a4+x?)/(1-a2[?+(1-p)?],and the marginal distribution of {Xt} is overdispersed,equdispersed or underdispersed,when v+[1-p?+(1-p)?+2?](1-p)?2/[1-p?-(1-p?]? is greater than 1,equtal o 1 or less than 1,respectively.Two specific GNBBMINAR(1)processes are considered by prespecifying the marginal distributions of the innovation with equdispersion and overdispersion,respectively One GNBBMINAR(1)process is assumed that the innovation follows poisson distri-bution,denoted as Poi-GNBBMINAR(1)process,the other GNBBMINAR(1)pro-cess is assumed that the innovation has Geometric distribution,denoted as Geo-GNBBMINAR(1)process.According to Remark 2,the dispersion of these two GNBB-MINAR(1)processes can be obtained and stated by the following two propositionsProposition 6 If {Xt} satisfies Poi-GNBBMINAR(1)process,then the marginal distribution of {Xt} is overdispersed.Proposition 7 If {Xt} satisfies Geo-GNBBMINAR(1)process,then the marginal distribution of {Xt} is overdispersed.We evaluate the performances of CML estimator on the estimation of two spe-cific GNBBMINAR(1)process by numerical simulation.The results show that CML estimators have asymptotic property and can provide reliable estimates.Then the GNBBMINAR(1)process is used to fit a crime data and is compared with other I-NAR(1)models.The results show that the Poi-GNBBMINAR(1)process is more suitable for fitting this set of data sets and the adequacy of the model is verified by residual analysis.