Statistical Analysis On Several Integer-valued Time Series Models With Random Coefficient

Time series is an important.tool for the study of dependent data.Time series analysis is a main branch of statistics.the related theoretical methods and practical applications have always been the international frontier and hot issues.Traditional time series models usually deal with continuous data.and has poor performance for integer-valued data.However,integer-valued data widely exists in our daily life.for example:the number of newly diagnosed cases of some infectious disease per day,the claim number per month of a insurance company,and so on.Therefore,the modeling and statistical inference of integer-valued time series emerge as the times require,and have become a research area and investigation field that more and more people pay attention to.This thesis mainly insists of three sections.Firstly,we consider a class of first-order generalized random coefficient integer-valued time series model.The empirical likeli-hood method is used to diseuss the point estimation,confidence region and hypothesis testing of the unknown parameters.Secondly,we study a kind of first-order mixed integer-valued autoregressive models with random coefficients.The unknown parame-ters are estimated by two-step least square method,and the large sample properties of the estimators are given.furthermore,the hypothesis whether the model has stochastic thinning parameter is tested.Finally,we investigate a class of second-order random coefficient integer-valued autoregressive models.The estimation method of parameters and the hypothesis testing for the order of the model are presented.Next.we introduce the statistical models and the main results in this thesis.1.Empirical likelihood for a generalized integer-valued random coefficient INAR（1）modelIn order to make the integer-valued time series model more flexible and widely used.Gomes et al.（2009）proposes a class of generalized random coefficient INAR（1）model.Definition 1 The GRCINAR（1）model is a sequence of random variables {Xt,t∈N} defined hy the following recursive equation:Xt=φt（?）G Xt-1+εt,t ∈ N,in which（ⅰ）{φt,t≥1} is a positive i.i.d.random variable sequence with φ=E（φt）andσ12=Var（φt）.The distribution function of φt is denoted by Pφ1.（ⅱ）{εt,t≥1} is a non-negative integer-valued i.i.d.random variable sequence with λ=E（εt）and σ22=Var（εt）.The probability mass function of φt is denoted by fε.（ⅲ）The generalized operator φt（?）G Xt-1 is defined as:（?）t（?）G Xt-1|φt Xt-1～G（φtXt-1.δtXt-1）.where G（φtXt-1,δtXt-1）denote the distribution of a discrete random vacriable with mean φtXt-1 and variance δtXt-1.（ⅳ）X0、{φt≥ 1} and {εt,t ≥ 1} are independent.Gomes et al.（2009）gives the conditional least squares and conditional maximum likelihood estimation methods for the model parameters.Here,We mainly discuss their empirical likelihood inference method.Let θ=（φ,λ）T,it is easy to get the log ELR function#12 where#12 b（θ）satisfies#12 We can obtain the asymptotic distribution of l（θ）as follow.Theorem 1 If{Xt,t∈N]is a strict stationary and ergodic process,and E（Xt4）<+∞,then we have l（θ）（?）x2（2）,n→+∞.Based on the above theorem,we can construct the EL confidence region of the parameters.Theorem 2 If{Xt,t∈N} is a strict stationary and ergodic process,and E（Xt4）<+∞,then the 100（1-α）%EL confidence region of θ is:CαnEL={θ|l（θ）≤cα},in which,0<α<1,and cα satisfies P（x2（2）≥cα）=α.On the other hand,the MEL estimator of θ can be obtained by minimizing lE（θ）,that is#12 in which#12 The following theorem gives the asymptotic property of θMEL.Theorem 3 If {Xt,t∈N} is a strict stationary and ergodic process,and E（Xt4）<+∞,then the MEL estimator θMEL is consistent,and（?）（θMEL-θ0）（?）N（0,V-1W（θ0）V-1）,n→+∞,where θ0 is the true value of 0,#12#12 In practice,we could also consider the following hypothesis about θ:H0:θ=θ0 v.s.H1:θ≠θ0.Based on EL method,we can get the ELR statistic:Q（θ）=l（θ）-l（gMEL）.which has the following limit,distribution:Theorem 4 If {Xt,t∈N} is a strict stationary and ergodic process,and E（Xt4）<under the null hypothesis H0.we have Q（θ0）（?）λ2（2）,n→+∞.Therefore,the rejection region for significance level α is given by Wn,α={Q（θ0）≥ cα},where ca satisfies P（X2（2）≥cα）=α.2.Statistical inference on a mixed INAR（1）model with random coefficientIn order to overcome the limitation that the thinning parameters of the mixed INAR（1）model are constant,we propose a class of mixed INAR（1）models with random coefficients.Definition 2 The RCMINAR（1）model is a sequence of random variables {Xt,t∈N} defined by the following recursive equation:#12 in which "（?）" and "*" denote the Binomial thinning parameter and Negative Binomial thinning parameter,respectively.Furthermore,we assume that（ⅰ）{φt,t≥1} is a sequence of i.i.d.random variables with cumulative distribution function Pφ1 on（0,1）.Denote φ=E（φt）and σ12=Var（φt）;（ⅱ）{εt,t ≥ 1} is a sequence of non-negative integer-valued i.i.d.random variables with probability mass function fε.Denote λ=E（εt）and σ=Var（εt）;（ⅲ）X0、{φt,t≥ 1} and {εt,t≥1} are independent.（ⅳ）For any given t and s（t≠s）,εt is independent of {Bi（t-l）,i≥1} and{Wi（t-l）,i≥1}（l≥0）,{Bi（t）,i≥1} and {Bj（s）,j≥1},as well as {Wi（t）,i≥1}and {Wj（s）,j≥1} are also independent.（ⅴ）For any given t,s≥1,{Bi（t）,i≥1} and {Wi（t）,i≥1} are independent condi-tional on φt.First,the following theorem establishes the strict stationarity and ergodicity of the proposed model.Theorem 5 If 0<σ12+φ2<1,there exists a unique strict stationary integer-valued random series satisfying Definition 2.Furthermore,the process is ergodic.Next,we discuss the estimation of the model parameters.For η=（φ,λ）T,we have#12 In which Yt=（Xt-1,1）T.In order to estimate θ=（σ12,p,σ22）T,we use two-step least square method.Denoteδ=(σ12,（1-2p）（σ12+φ2）+φ,σ22T,Zt=（Xt2-1,Xt-1,1）T,we can obtain its estimator of δ:#12Note that δ is a function of η、we rewrite it as δ（η）.Substituting the least squares estimator of η into the above equation leads to S（η）,then we can get the estimator ofδ asθ1＝σ12=δ1（η）,θ2=p=1/2-δ2（η）-η1/2（δ1（η）+η12,θ3=σ22=δ3（η）,in which θ1,θ2 and θ3 are the three components of θ,δ1（η）,δ2（η）and δ3（η）are the three components of δ（η）,and η1 is the first component of η.We give the large sample properties of the estimators if the following theorem.Theorem 6 If {Xt,t∈N）is a strict stationary and ergodic process with E（Xt8）<+∞,then（?）,n→+∞.the covariance matrix is#12 where V=E（YtYtT）,Φ=E[（Xt-YtTη0）2YtYtT],U=E（ZtZtT）,Δ=E[（VZ-trδ0）2ZtZtT],Π=E[（Vt-ZtTδ0）（Xt-YtTη0）ZtYtT].Theorem 7 If {Xt,t∈N} is a strict stationary and ergodic process with E（Xt8）<+∞.then（?）,n→+∞in which（?）At last,we study the following hypothesis problem:H0:σ12=0 v.s.H1:σ12>0.Let e=（0,0,1,0,0）T,then the testing statistic and its asymptotic distribution can be obtained:（?）,n→+∞Based on this,we give the rejection region for significance level a as#12 in which uα/2 is the upper α/2 quantile of N（0,1）.3.Statistical inference on a random coefficient INAR（2）model Based on the INAR（2）model with constant thinning parameter,we propose a.kind of random coefficient INAR（2）model.Definition 3 The BRCINAR（2）model is a sequence of random variables {Xt,t∈N} defined by the following recursive equation:#12 in which the thinning parameters α1,α2∈（0,1）are constants,{εt} is a sequence of non-negative integercvalued i.i.d.random variables with Gean λ and variance σε2.Furthermore,{εt} and {Xt-1} are supposed to be independent.The strict stationarity and ergodicity of the proposed model are given in the following theorem.Theorem 8 If α1,α2∈（0,1）and p1,p2∈（0,1）,then there exists a unique strict stationary and ergodic integer-valued random series satisfying Definition 3.Next,we estimate the unknown parameters by two-step least square method.Letη=（β1,β2,λ）T.where β1=p1α2,β2=p2α2,thenη=M-1b,in which#12#12 Denoteθ=（α1β1-β12,α2β2-β22,β1-α1β1,β2-α2β2,2β1β2,σε2）T,then#12 in which Vt=Xt-β1Xt₁-β2Xt₁-λ,Let θ（η）be the estimator of θ（η）with 77 replaced by η.Define η1.η1 as com-ponents of η,and θ1（η）,η2（η）as components of θ（η）.We obtain the estimators ofα1,α2,p1,p2 as follows:α1=θ1（η）+η12/η1,α2=θ2（η）+η22/η2,p1=η12/θ1（η）+η12,p2=η22/θ2（η）The following theorem gives the asymptotic properties of the estimators.Theorem 9 If {Xt,t∈N} is a strict stationary and ergodic process with.E|Xt]8<+∞,then（?）,n→+∞,the covariance matrix is#12 in which（?）.∑=E（Xt-ηTDt）2DtDtT,Γ=EZtZtT,W=E（（Vt-ZtTθ）2ZtZtT）,Π=E（（Vt-ZtTθ）（Xt-ηTDt）ZtDtT）.Dt=（Xt-1,Xt-2,1）T.Theorem 10 If {Xt,t ∈ N} is a strict stationary and ergodic process with E|Xt|8<+∞,then α1,α2,p1 and p2 are consistent estimators of α1,α2,p1 and p2,respectively.Furthermore,we have（?）（α1-α1,α2-α2,β1-β1,β2-β2）T（?）N（0,ΦΩΦT）,in which#12The following hypothesis problem is considered:H0:α2==0 vs.H1:α2>0.We construct testing statistics as（?）（α2-α2）/γ.in whichγ=β22ω22+（β22-θ2）2ω88+2β2（β22-θ2）ω28/β24.where ω22,ω28 and ω88 are consistent estimators of the corresponding elements.We can prove that the asymptotic distribution of the testing statistics is standard normal distribution.Based on this result,it is easy to get the rejection region of the null hypothesis.