Generalization And Statistical Inference Of Random Coefficient Integer-valued Autoregressive Process

Integer-valued time series data is ubiquitous in real life,for example,the monthly number of people suffering from an infectious disease in an area,the annual number of reproductions of a rare species,the monthly number of traffic accidents in a certain area,the monthly claims of insurance companies and so on.The most common integer-valued time series model is the integer-valued autoregressive(INAR)model based on thinning operator.The proposed model and its continuous improvement have great-ly promoted the development of integer-valued time series modeling.Generally,the thinning parameters in INAR model are assumed to be fixed constants.Although this assumption brings great convenience to the research,it is obviously unreasonable.Because the observed data are influenced by various environmental factors,and the environment changes with time,so the thinning parameters of the model should not be fixed,but should be random.Therefore,in practical application,it is rough to describe the observed data based on the constant coefficient INAR model,which often can not effectively describe the inherent law of data change.At this time,it is imperative to establish the random coefficient INAR model.In this paper,we generalize the random coefficient integer-valued autoregressive process introduced by Zheng et al.(2007),and study the statistical inference of some generalized models.The main contents are divided into three parts.In the first part,we relax the assumption that the thinning parameters are independent and identically distributed in the random coefficients integer-valued autoregressive process.By allow?ing the thinning parameters of the model to be state-dependent,we propose a class of observation-driven random coefficient integer-valued autoregressive processes based on the negative binomial thinning operator and some basic probability and statistics properties are derived.Based on conditional least squares and empirical likelihood methods,the parameter estimation and hypothesis testing of the model are studied.The effects of parameter estimation,the coverage probability of the confidence region and the power of the test are studied by numerical simulation,and a set of crime data is fitted with the proposed model.In the second part,the univariate random coefficient integer-valued autoregressive process is extended to the multivariate case,we propose a new bivariate first-order random coefficient integer-valued autoregressive process.The strict stationarity,ergodicity and moment properties of the process are discussed.The parameter estimation and coherent prediction of the model are studied.Finally,the proposed model is applied to a set of real data,and the fitting results are compared with other bivariate integer-valued time series models.In the third part,we further generalize the bivariate first-order random coefficient integer-valued autoregressive pro-cess proposed in the second part,and introduced a new bivariate generalized first-order random coefficient integer-valued autoregressive process.The probabilistic statistical properties and parameter estimation of the new model are studied.The effect of the estimation is discussed by numerical simulation,and the proposed model is used to fit a set of real data.In what follows,we introduce the main results of this thesis.1.Statistical inference of the observation-driven NBRCINAR(1)processesIn order to dynamically describe the time-varying characteristics of the thinning parameters for INAR model,by assuming that the thinning parameters of the model are state-dependent,we propose a class of observation-driven random coefficient integer-valued autoregressive processes based on the negative binomial thinning operator "*",called observation-driven NBRCINAR(1)process,which is defined as follows:Definition 1 {Xt} is called an observation-driven NBRCINAR(1)process if it satisfies the following regression equation xt=?t*Xt-1+?t,log ?t/1-?t=v(Xt-1;?),t?1,(1)where(i)"*" is the negative binomial thinning operator introduced by Ristic and Bakouch(2009).?t*Xt-1=(?)?i(t),given ?t,{?i(t)}is a sequence of i.i.d.geometric random variables with probability mass function P(?i(t)=x)=?tx/(1+?t)1+x,x=0,1….(?)? is a l-dimensional parameter vector;the function u(·;·)belongs to a specific parametric family of functions g(?)= {v(Xt-1;?);??(?)},(?)is an open subset of Rl and v(x;?)is three times continuously differentiable with respect to ?;(?){?t} is a sequence of i.i.d.non-negative integer-valued random variables with probability mass function f?>0 and ?t is independent of {Xs}s<t.Obviously,the observation-driven NBRCINAR(1)process {Xt} is a Markov chain on N0 with the following transition probabilities:(?)Let E(?t)= ?,Var(?t)= ??2.Assume that they are all finite.The following proposition give some basic properties of the observation-driven NBRCINAR(1)pro-cess.Proposition 1 Let {Xt} be the process defined by(1).Then for t?1,(?)Proposition 2 If sup v(x;?)<+?,??(?),then the process {Xt} defined by x?N0(1)is an ergodic Markov chain.For observation-driven NBRCINAR(1)process,we discuss the estimation prob-lems of the parameter ? =(?T,?)T based on conditional least squares estimation and empirical likelihood estimation.Let ?0 is the true value of ?.First,we make some assumptions about the function u(x;?)as follows:Assume that there exists a neighborhood ? of ?0 and a real positive function N(x),such that:(A1)1?i,j?l,|(?)v(x;?)(?)?i|and|(?)2v(x;?)/(?)?i(?)?j|are continuous in ? and bounded by N(x)for ? in ?;(A2)1?i,j,k?l,|(?)3v(x;?)/(?)?i(?)?j(?)?k|are continuous in ? and bounded by N(x)for ? in ?;(A3)E|Xt|6+?<? and E|N(Xt)|6+?<? hold for some ?>0;(A4)E((?)u(Xt;?0)/(?)?·(?)v(Xt;?0)/(?)?T)is of full rank;(A5)the parameters of v(Xt;?)are identifiable,i.e.,if ???0,then Pv(xt,?)?Pv(Xt;?0),where Pv(Xt;?)denotes the marginal distribution law of v(Xt;?).Let then the conditional least squares estimators ?CLS are obtained by?CLS= arg ? minS(?).Theorem 1 Under the assumptions(A1)-(A4),the conditional least squares estimators ?CLS are consistent and have the following asymptotic distribution:(?)Then a conditional least squares confidence region for ?0 can be construct as follows:where 0<?<1,c? satisfies P(xl+12?c?)=?,Gn(?)=1/n(?)MtT(?),According to the dual likelihood introduced by Mykland(1995),we can derive an empirical log-likelihood ratio statistic where satisfies(?)Theorem 2 Ufnder the assumptions(A1),(A3)and(A4),as n??,By Theorem 2,we can construct an empirical likelihood confidence region for the parameters ?0 as follows:IEL={?|2L(?)?C?},where c? satisfies P(xl+12?c?)= ?,for 0<?<1.In addition,by minimizing the log empirical likelihood ratio function L(?)or equivalently maximizing the empirical likelihood ratio function R(?),we can get the maximum empirical likelihood estimators?MEEL.Theorem 3 Under the assumptions(A1)-(A4),the maximum empirical likelihood estifmators ?MEL are consistent and have the following asymptotic distribution:where W(?0)= E(Mt(?0)MtT(?0))and U(?0)= E((?)Mt(?0)/(?)?T).Now we consider the hypothesis testing problem H0:?= ?0??H1:???0.As in the independent data case,we can use the empirical likelihood ratio statistic defined by Q(?0)= 2L(?0)-2L(?MEL).By Theorems 2 and 3,it's easy to derive the following theorem.Theorem 4 Under the assumptions(A1)-(A4),(?)(?0)?x2(l+1)when H0 is true.The conditional maximum likelihood estimation is used as a benchmark,and we compare the performance of the two estimation methods by numerical simulation.The simulation results show that although the maximum empirical likelihood estimation is similar to the conditional least squares estimation and has no obvious advantages,the empirical likelihood method shows great advantages in the coverage probability of confidence regions,and its coverage probability is obviously higher than that of conditional least squares estimation.Finally,we apply the model to the data of a count of CAD shots fired calls per month in Pittsburgh,Pennsylvania,USA,and compare it with other common integer-valued models.2.Modeling and statistical inference of BRCINAR(1)process.In order to describe the two interrelated integer-valued time series better,we propose a new bivariate first-order random coefficient integer-valued autoregressive process,namely the BRCINAR(1)process,which is defined as follows:Definition 2 The bivariate process {Xt}is called BRCINAR(1)process if it satisfies the followi'ng regression equation(?)where(?){?1,t}and {a2,t} are two mutually independent sequences of i.i.d.random vari-ables with cumulative distribution function(CDF)P?i(ui)(i = 1,2)on[0,1);(?)is a random matrix,the operation Ato is a random matrical thinning operation which acts as the usual matrix multiplication and keeps the properties of the random coefficient thinning operation.For all t ?Z,given?i,t(i = 1,2),the thinning operation ?1,t o X1,t-1 and a2,t o X2,t-1 are performed independently each other;(?){Zt} is a sequence of i.i.d.bivariate non-negative integer-valued random vec-tors and follow some bivariate distribution with joint probability mass function fz(x,y)>0.For fixed t,Zt is independent of At o Xt-1 and Xs for s<t.Obviously,the BRCINAR(1)process {Xt}t?Z is a Markov process on N02 with the following transition probabilities:where l1(t)= min(x1,t,x1,t-1)and l2(t)=min(x2,t,x2,t-1).Let E(?i,t)= ?i,Var(?i,t)= ??i2,E(Zi,t)= ?i,Var(Zi,t)= ?zi2 and cov(Z1,t,Z2,t)=?,where i = 1.2.Assume that they are all finite.The following propositions give some basic probabilistic and statistical properties of BRCINAR(1)process.Proposition 3 If0<?i2 +??i2<1,i = 1,2,then there exists a unique strictly stationary bivariate integer-valued random series {Xt}t?z satisfying(2).Furthermore,the process is an ergodic process.Proposition 4 Suppose {Xt}t?z is a strictly stationary process given by(2),then fort?1,k?0,i,j=1,2 and i?j,In what follows,we discuss the parameter estimation problems of the BRCINAR(1)process based on Yule-Walker estimation,conditional least squares estimation and conditional maximum likelihood estimation.Our primary interest lies in estimating the parameters ?=(?1,?2,?1,?2,?)'.Moreover,the estimation of corresponding variances is also considered.Firstly,let ?i(0)Var(Xi,t),?i(1)= Cov(Xi,t,Xi,t-1),?ij(0)= Cov(X1,t,X2,t),where i,j =1,2 and i?j.Then their sample equivalents have the form as follows:where From Proposition 4,we can derive ?i =?i(1)/?i(0)and ?=(1-?1?2)?ij(0).Thus we can get the Yule-Walker estimators as follows:Moreover,we define,then the parameters ?i can be estimated asTheorem 5 If|E|Xi,t|4<?,i=1,2,then?YW are strongly consistent estima-tors for ?.Secondly,we use the two-step conditional least squares method introduced by Karisen and Tj(?)stheim(1988)to estimate the parameters ?.In the first step,we derive the conditional least squares estimators of the parameters ?1=(?1,?2,?1,?2)'.By minimizing the following function we can obtain the conditional least squares estimators ?1CLS.Furthermore,by solving the equations(?)Q(?1)/(?)?1=0,we can derive the conditional least squares estimators of ?1 as follows:?1CLS=Bn-1b,where and The following theorem states the strong consistency and asymptotic normality of the conditional least squares estimators ?1CLS.Theorem 6 If E|Xi,t|4<?,i= 1,2,then the conditional least squares estima-tors ?CLS are strong consistent and have the following asymptotic distribution:In the second step,we consider conditional least squares estimation of ?.Define a new random variable Yt =(X1,t-E(X1,t|Xt-1))(X2,t-E(X2,t|Xt-1)).Therefore,we can construct the criterion function The conditional least squares estimator of ? is given by By solving the equation(?)S(?)/(?)?=0,we can obtainTheorem 7 If E|Xi,t|4<?,i=1,2,for the conditional least squares estimator?CLS,we haveFinally,by the Markov property of BRCINAR(1)process,it's easy to derive the conditional likelihood function where the transition probability P(Xt = xt|Xt-1= xt-1)are given by(3),? is the parameter vector from the joint probability mass function fz and the cumulative distribution function P?i.The conditional maximum likelihood estimators ?CML are obtained by maximizing the above conditional likelihood function.Besides,we use two nonparametric methods to construct consistent estimators of the variance ?.The first method is based on conditional least squares.The second method is based on Schick(1996).We compare the performance of the three estimation methods by numerical sim-ulation.The simulation results show that as a parameter method,the result of con-ditional maximum likelihood estimation is obviously better than that of Yule-Walker estimation and conditional least squares estimation.Finally,we apply the model to to fit a bivariate count time series of offence data which represent monthly counts of aggravated assaults(AGGASS)and robbery(ROBBERY)in Pittsburgh,and compare it with other common bivariate integer-valued models.3.Modeling and statistical inference of BGRCINAR(1)processTo make the BRCINAR(1)process more general,we further generalize it and pro-pose a new bivariate generalized random coefficient INAR(1)process called BGRCI-NAR(1)process,which is defined as follows:Definition 3 An bivariate integer-valued stochastic process {Xt}t?z is said to be a BGRCINAR(1)process if it satisfies the following recursive equation:where(?)For i = 1,2,{(?i1,t,?i2,t)}t?z is a sequence of i.i.d.bivariate random variables with joint cumulative distribution function(CDF)Pi(?i1,?i2)on[0,+?)2.For fixed t,(?11,t,?12,t)and(?21,t,?22,t)are mutually independent;(?)The operation At·Xt-1 is defined by where"?"is the generalized random coefficient thinning operator defined as Given ?ij,t(i,j=1,2),all counting series{?k(i,j,t)}of the thinning operation are mutually independent non-negative integer-valued random variables with(?){Zt} is a sequence of i.i.d.bivariate non-negative integer-valued rafndom vec-tofrs and follow some bivariate distribution with joint probability mass function fz(x,y)>0.For fixed t and any s<t,Zt is independent of At?Xt-1 and Xs.From(4),we can see that the BGRCINAR(1)process ?Xt}t?Z is a Markov process on N02 with the following transition probability:where For i,j= 1,2,the function ?(x;?ij,xj,t-1)and the sum' upper and lower limit Ui(t),Vi(t),ki(t),li(t)are specified according to the distribution of the counting series{?k(i,j,t)}.Moreover,the integral area(?)ionly depends on the joint distribution function pi(?i1,?i2).Let E(?i,j,t)=??ij,E(?ij,t)=??ij,E(Zi,t)=?i,Var(?ij,t),Cov(?i1,t,?i2,t)=?i,Var(Zi,t)=?zi2 and cov(Z1,t,Z2,t)=?,where i,j=1,2.Assume that they are all finite.The following proposition gives a sufficielt condition for the BGRCINAR(1)process to be strictly stationary and ergodic.Proposition 5 If p(A+B)<1,where ?(A+B)is the spectral radius of A+B,then BGRCINAR(1)process has a unique strictly stationary and ergodic solution.For BGRCINAR(1)process,we have the following conclusions about moments and conditional moments.(?)Conditional expectation at lag k(?)Conditional variance at lag 1 where(?)Conditional covariance at lag 1(?)Expectation(?)Covariance functionIn what follows,we discuss the parameter estimation problems of the BGRCI-NAR(1)process based on conditional least squares estimation,Yule-walker estimation and conditional maximum likelihood estimation.Our primary interest lies in estimating the parameters ?=(??11,??12,??21,??22,?1,?2,?)'.In addition,we also consider the estimation of parameters ? =(??11,??12,??21,??22,??112,??22,?1,?2,?z12,?z22)'Firstly,we use the two-steps conditional least squares method introduced by Karlsen and Tj(?)stheim(1988)to estimate the parameters ?.Let ? =(?1',?)'.In the first step,we derive conditional least squares estimators of the parameters ?1.The criterion function is constructed as The conditional least squares estimators of ?1 are given by Setting(?)S(?1)(?)?1= 0,we can obtain whereTheorem 8 If E |Xi,t}4<?,i=1,2,then the conditional least squares estima-tors ?1CLS are strong consistent and have the following asymptotic distribution:In the second step,we give the conditional least squares estimation for the param-eter ?.Define a new random variable Since ?1CLS is strong consistent,we can construct the criterion function The conditional least squares estimator of ? is obtained by minimizing S(?).By solving(?)S(?)/(?)??0,we can obtainTheorem 9 If E |Xi,t|4<?,i= 1.,2,for the conditional least squares estimator?CLS,we have(?)(?CLS-?)(?)N(0,?2),where?2-E[(X1,t-??11X1,t-1-??12X2,t-1-?1)(X2,t-??21X1,t-1-??22X2,t-1-?2)-?]2.Secondly,let?(h)=Cov(Xt+h,Xt)=(?).Then the sample equivalents of ?(0)and ?(1)have the following elements:?ii(0)=1/n(?)(Xi,t-Xi)2,?ij(0)=1/n(?)(Xi,t-Xi)(Xj,t-Xj),?ii(1)=1/n-1(?)(Xi,t+1-Xi)(Xi,t-Xi),?ij(1)=1/n-1(?)(Xi,t+1-Xi)(Xj,t-Xj),whereXi-1/n(?)Xi,t,i,j=1,2 and i?j.Since?(1)=A?(0),?=(1-??11??22-??12??21)?12(0)-??11??21?11(0)-??12??22?22(0).Suppose that the covariance matrix ?(0)is non-singular,then we can get the Yule-Walker estimators as follows:AYW=?(1)?-1(0),Moreover,we define Zi,t=Xi,t-??i1YW X1,t-1-??i2YW X2,t-1,then the parameters ?i can be estimated as?iYW=1/n(?)Zi,t,i=1,2.Theorem 10 If E|Xi,t|4<?,i = 1,2,then the Yule-Walker estimators ?YW are strongly consistent.Finally,by using the Markov property of BGRCINAR(1)process,we can derive the conditional log-likelihood function where T are the parameters from the distribution function?(x;?ij,xj,t-1),Pi(?i1,ai2)and fz(x,y),the transition probability P(Xt=xt|Xt-1= xt-1)are given by(5).The conditional maximum likelihood estimators TCML are obtained by maximizing the conditional log-likelihood function.We do some simulations to compare the performance of the the above three es-timation methods.The simulation results show that when the sample size is large,conditional least squares estimators and Yule-Walker estimators are equivalent.The conditional maximum likelihood estimation performs better than Yule-Walker estima-tion and conditional least squares estimation,although it takes a long time to calculate.Finally,the proposed BGRCINAR(1)model is used to fit a set of daytime and night-time road accidents in Schiphol area,in the Netherlands for the year 2001,and compare it with the bivariate integer autoregressive model with constant coefficients.