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=?t???G Xt-1+?t,t ? N,in which???{?t,t?1} is a positive i.i.d.random variable sequence with ?=E??t?and?12=Var??t?.The distribution function of ?t is denoted by P?1.???{?t,t?1} is a non-negative integer-valued i.i.d.random variable sequence with ?=E??t?and ?22=Var??t?.The probability mass function of ?t is denoted by f?.???The generalized operator ?t???G Xt-1 is defined as:???t???G Xt-1|?t Xt-1?G??tXt-1.?tXt-1?.where G??tXt-1,?tXt-1?denote the distribution of a discrete random vacriable with mean ?tXt-1 and variance ?tXt-1.???X0?{?t? 1} and {?t,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 ?=??,??T,it is easy to get the log ELR function#12 where#12 b???satisfies#12 We can obtain the asymptotic distribution of l???as follow.Theorem 1 If{Xt,t?N]is a strict stationary and ergodic process,and E?Xt4?<+?,then we have l??????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-??%EL confidence region of ? is:C?nEL={?|l????c?},in which,0<?<1,and c? satisfies P?x2?2??c??=?.On the other hand,the MEL estimator of ? can be obtained by minimizing lE???,that is#12 in which#12 The following theorem gives the asymptotic property of ?MEL.Theorem 3 If {Xt,t?N} is a strict stationary and ergodic process,and E?Xt4?<+?,then the MEL estimator ?MEL is consistent,and?????MEL-?0????N?0,V-1W??0?V-1?,n?+?,where ?0 is the true value of 0,#12#12 In practice,we could also consider the following hypothesis about ?:H0:?=?0 v.s.H1:???0.Based on EL method,we can get the ELR statistic:Q???=l???-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??0?????2?2?,n?+?.Therefore,the rejection region for significance level ? is given by Wn,?={Q??0?? c?},where ca satisfies P?X2?2??c??=?.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???{?t,t?1} is a sequence of i.i.d.random variables with cumulative distribution function P?1 on?0,1?.Denote ?=E??t?and ?12=Var??t?;???{?t,t ? 1} is a sequence of non-negative integer-valued i.i.d.random variables with probability mass function f?.Denote ?=E??t?and ?=Var??t?;???X0?{?t,t? 1} and {?t,t?1} are independent.???For any given t and s?t?s?,?t 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.???For any given t,s?1,{Bi?t?,i?1} and {Wi?t?,i?1} are independent condi-tional on ?t.First,the following theorem establishes the strict stationarity and ergodicity of the proposed model.Theorem 5 If 0<?12+?2<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 ?=??,??T,we have#12 In which Yt=?Xt-1,1?T.In order to estimate ?=??12,p,?22?T,we use two-step least square method.Denote?=(?12,?1-2p???12+?2?+?,?22T,Zt=?Xt2-1,Xt-1,1?T,we can obtain its estimator of ?:#12Note that ? is a function of ??we rewrite it as ????.Substituting the least squares estimator of ? into the above equation leads to S???,then we can get the estimator of? as?1??12=?1???,?2=p=1/2-?2???-?1/2??1???+?12,?3=?22=?3???,in which ?1,?2 and ?3 are the three components of ?,?1???,?2???and ?3???are the three components of ????,and ?1 is the first component of ?.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?,?=E[?Xt-YtT?0?2YtYtT],U=E?ZtZtT?,?=E[?VZ-tr?0?2ZtZtT],?=E[?Vt-ZtT?0??Xt-YtT?0?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:?12=0 v.s.H1:?12>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 u?/2 is the upper ?/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 ?1,?2??0,1?are constants,{?t} is a sequence of non-negative integercvalued i.i.d.random variables with Gean ? and variance ??2.Furthermore,{?t} 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 ?1,?2??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.Let?=??1,?2,??T.where ?1=p1?2,?2=p2?2,then?=M-1b,in which#12#12 Denote?=??1?1-?12,?2?2-?22,?1-?1?1,?2-?2?2,2?1?2,??2?T,then#12 in which Vt=Xt-?1Xt₁-?2Xt₁-?,Let ????be the estimator of ????with 77 replaced by ?.Define ?1.?1 as com-ponents of ?,and ?1???,?2???as components of ????.We obtain the estimators of?1,?2,p1,p2 as follows:?1=?1???+?12/?1,?2=?2???+?22/?2,p1=?12/?1???+?12,p2=?22/?2???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-?TDt?2DtDtT,?=EZtZtT,W=E??Vt-ZtT??2ZtZtT?,?=E??Vt-ZtT???Xt-?TDt?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 ?1,?2,p1 and p2 are consistent estimators of ?1,?2,p1 and p2,respectively.Furthermore,we have?????1-?1,?2-?2,?1-?1,?2-?2?T???N?0,???T?,in which#12The following hypothesis problem is considered:H0:?2==0 vs.H1:?2>0.We construct testing statistics as?????2-?2?/?.in which?=?22?22+??22-?2?2?88+2?2??22-?2??28/?24.where ?22,?28 and ?88 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.