Statistical Inference For Some Classes Of Integer-valued Threshold And Change Point Models

Nonlinear time series observations are common in real life and cover a broad range of fields such as economics,actuarial science,social science,etc.The threshold model is one of the most important models in nonlinear time series model.This paper stud-ies statistical inference for some classes of integer-valued threshold and change point models.The main content is divided into three parts.In the first part,we introduce a first-order self-exciting threshold generalized Poisson integer-valued autoregressive process.Based on conditional least squares estimation method and conditional maxi-mum likelihood estimation method,the estimation problems for the model parameters are discussed.The forecast problem is considered,too.The effect of the estimates are studied through numerical simulations.The model is used to fit a set of real da-ta at last.In the second part,we redefine NBTINAR(1)under a weaker condition.Some probabilistic and statistical properties,parameter estimation problems are discussed.Parameters' point estimation and interval estimation problems are consid-ered.A method to test the nonlinearity of the data is provided.At last,The effect of the estimates are studied through numerical simulations.The model is used to fit a set of crime data.In the third part,we further extend the NBTINAR(1)model to interesting cases,then we obtain a generalized d-order delay NBTINAR(1)model and the CP1-NBINAR(1)model.A brief discussion on statistical properties and parameter estimation problems of the new models.Using new models to fit a set of actual data.In what follows,we introduce the main results of this thesis.1.Statistical inference of the SETGPINAR(1)processTo capture the piecewise phenomenon of integer-valued time-series observation-s,we introduce a self-excited integer-valued threshold autoregressive process,called SETGPINAR(1)process,which is defined as follows:Definition 1 The SETGPINAR(1)process is defined by the following recursive equation:Xt=?st+1oXt-1+?t,t?z,(1)where(i){st} is a sequence of Befrnoulli random variables taking value 1 when Xt-1?r and 0 otherwise,r is the unknown threshold value;(ii)"o" is the binomial thinning operator defined in(1.1.1);(iii){?t} is a sequence of i.i.d.GP(?,?)random variables and its probability mass function(p.m.f)is P(X=x)= ?(?+?x)x-1e-(?+?x)/x!,x=0,1,2…where ?>0,-1<?<1 and P(X=x)=0,for x?m if ? + ?m?0.(iv)For fixed t and i(i=1,2),?t is assumed to be independent of {?ioXt-1}{Xt-l} for all l?1.Firstly we discuss some properties of the SETGPINAR(1)process.Proposition 1 Let {Xt}t?z be the process defined in(1).Then(i){Xt}t?z is an irreducible,aperiodic and positive recurrent Markov chain(hence ergodic);(ii)There exists a strictly stationary process satisfying(1).The next proposition ensures the first four moments exist.Proposition 2 Let {Xt} be the process defined by(1).Then E(Xtk)<? for k=1,2,3,4.Next,we study the parameter estimation of this process.The following proposition gives the moments and conditional moments of the SETGPINAR(1)process(1).Proposition 3 Suppose {Xt} is a stationary process defined by(1),then for t?1,(i)E(Xt|Xt-1)=?iXt-1I1,t+?2Xt-1I2,t+?/1-k;(ii)E(Xt)=?1p1u1+?2p2u2+?/1-k;(iii)Var(Xt|Xt-1)=?1(1-?1)Xt-1I1,t+?2(1-?2)Xt-1/2,t+?/(1-k)3;(iv)Var(Xt)=?i=12[?2pi?i2+?i(1-?i)piui]+p1p2(?1u1-?2u2)2+?/(1-k)3;(v)?(k)=?jk=1?2j-1(?1-?2)?k-j(1)+?2k?(0).The following result states the strong consistency and asymptotic normality of the CLS-estimators.Theorem 1 Let {Xt} be a SETGPINAR(1)process.Then the CLS estimators?CLS are strongly consistent and asymptotically normal,(?)where ?0=(?1,0,?2,0,?0)T denotes the true value of ?,V:=E((?)/(?)?g(?0,X0)(?)/(?)?Tg(?0,X0)),W=E(q(?0)(?)/(?)?g(?0,X0)(?)/(?)?Tg(?0,X0))with q(?):=(Xt-g(?,Xt-1))2.The following theorem states the asymptotic behavior of(?MTCLS,?MTCLS)T.Theorem 2 Let{Xt} be a SETGPINAR(1)process.Then the MTCLS estimators(?MTCLS,?MTCLS)T are strongly consistent and asymptotically normal,where ?=diag(ITV-1WV-1I,E[(X1-E(X1|X0))2-Var(X1|X0)]2),D0=D(?0,?02).the CML-estimators ?CML:=(?1,CML,?2,CML,?CML)T of ? are obtained by maximizing the conditional log-likelihood functionThe following results establish the strong consistency and the asymptotic normal-ity of the CML-estimators.Theorem 3 Let {Xt} be a SETGPINAR(1)process.Then the CML-estimators?CML are stro'gly consistent and asymptotically normal,i.e.,where I(?)is the Fisher information matrix.We use the SMLS algorithm proposed by Yang et al(2018a)to estimate the thresh-old r.The SMLS algorithm for the SETGPINAR(1)model are given below.Step 1.For each r ?[r,r]?N0,find ?CLS(r)such that ?CLS(r)=arg min? Q(?),where Q(?)is defined in(4).Step 2.The threshold is estimated by searching over all candidates,i.e.,Remark In practice,r and r can be selected as the minimum and maximum values of the samples,or the lower and upper ?-quantiles of the samples with some ?(e.g.,?=0.1).We compare the estimates by simulation and simulate the forecast distributions of the SETGPINAR(1)model,then apply this model to fit a set of actual data and compare it under different integer-valued models.The SETGPINAR(1)model is more suitable for fitting this set of data.2.Empirical likelihood inference of the NBTINAR(1)processTo capture the piecewise phenomenon of integer-valued time-series observation-s,Yang et al.(2018c)introduced an integer-valued threshold autoregressive(NBTI-NAR(1))process.To better describe the characteristics of time series of counts such as overdispersion or structural change,we redefine the NBTINAR(1)model under a weaker condition,which is defined as follows:Definition 2 The NBTINAR(1)process is defined by the following recursive equation:where(i)I1,t(r)=I{Xt-1÷r},I2,t(r)= 1-I1,t(r)=I{Xt-1?r=I{Xt-?r},where r is the unknown threshold value;(ii)"*" is the negative-binomial thinning operator(Ristic et al.,2009),defined as where ?i?(0,1),{Wk(i)} represents a sequence of independent and identically distributed(i.i.d.)geometric random variables with parameter ?i/1+?i;(iii)For fixed i(i = 1,2),{Zi,t} is an i.i.d non-negative integer-valued sequence with a probability mass function(p.m.f.)fz(·)>0,and E(Zi,t)=?i<?.Further,Zi,t is assumed to be independent of Xt-l and ?i*Xt-l for all l?1.(iv){Z1,t} and {Z2,t} are mutual independent.We discuss some basic properties of the new NBTINAR(1)process.Proposition 4 Suppose {Xt} is a stationary process defined by(2),then for t?1,In what follows,we prove the the ergodicity of {Xt}.Theorem 4 The process {Xt,t?1} defined in(2)is an ergodic Markov chain.Next,we discuss the parameter estimation problem of NBTINAR(1)process(2),give the conditional least squares(CLS)estimation and empirical likelihood estimation(EL)of parameter.The CLS-estimators(CLSE)?CLSE for the NBTINAR(1)process(2)can be obtained by minimizing the following CLS criterion function:where g(?,Xt-1)=E(Xt|Xt-1)=?i=12(?iX-1+?i)Ii,t.We make the following assumptions to study the asymptotic properties of ?CLSEAssumptions:(C1){Xt} is a stationary process;(C2)E|Xt|4<?;We state the strong consistency and asymptotic normality of ?CLSE in the follow-ing theorem.Theorem 5 Under assumptions(C1)-(C2),the CLSE ?CLSE are strongly con-sistent and asymptotically normal,i.e.,(?)(?CLSE-?0)(?)N(0,G),as n??,where G = V-1WV-1,V and W are square matrices of order 4.The elements of V and W take the forms:Vi,j:= E[(?)/(?)?ig(?,Xt-1)(?)/(?)?jg(?,Xt-1)]and Wi,j:=E[ut2(?)(?)/(?)?ig(?,Xt-1)(?)/(?)?jg(?,Xt-1)],with ut(?)=Xt-g(?,Xt-1).In what follows,we discuss Empirical likelihood inference of the new NBTINAR(1)process.Let p1,…,pn,be non-negative numbers adding to unity.Then,the log empirical likelihood ratio(ELR)evaluated at ?,a candidate value of ?0,is(?)where mt(?)=(m1,t(?),m2,t(?),m3,t(?),m4,t(?))T,with mi,t(?)=(Xt-g(?,Xt-1))Xt-1Ii,t(i=1,2),and mi,t(?)=(Xt-g(?,Xt-1))Ii-2,t(i=3,4).By introducing Lagrange multipliers ??R and b ? R4,standard derivations in the empirical likelihood lead to l(?)=2 ?t=1n log(1+b(?)Tmt(?)),where b(?)satisfies 1/n ?t=1n mt(?)/1+b(?)Tmt(?)=0.The following theorem shows that l(?0)converges to the chi-squared distribution with degree 4.Theorem 6 Under assumptions(C1)-(C2).If ?0 is the true value of ?,then l(?0)?x42,as n??,where ?42 is a chi-squared distribution with 4 degrees of freedom.In what follows,we will consider the maximum empirical likelihood estimator(MELE)for the parameter ?.Following Qin and Lawless(1994),the log EL function is defined as and the MELE can be obtained by minimizing the above equation,i.e.,To study the asymptotic properties of the MELE ?MELE,we make a further assumption:(C3)E|Xt|6<?.Then,we can get the asymptotic normality of ?MELE in the following theorem.Theorem 7 Under assumptions(C1)and(C3),the MELE ?MELE is asymptot-ically normal,where I(?0)=E[mt(?0)mt(?0)T]and J=E((?)mt(?0)/(?)?).We use the SMLS algorithm proposed by Yang et al.(2018a)to estimate r for the new NBTINAR(1)process in this thesis,then we obtain threshold parameter r:Next,we discuss the issue of testing nonlinearity for the threshold model.By definition(2)of the NBTINAR(1)processes,the null hypothesis and the al-ternative hypothesis take the form:Assumme that ? is asymptotically normally distributed around the true value ?0,i.e.,for some covariance matrix ?=(?i,j)4×4.Theorem 8 Let ? be some consistent estimators of ? satisfying(4).Then the statistic for test problem(3)is where I =(1,1),A =(1,-1,1,-1).Furthermore,u?N(0,1),n??,when H0 is true.We study the performances of the estimations and the effectiveness of nonlinear test.3.Statistical Inference of CP1-NBINAR(1)modelMotivated by Chen and Lee(2016),we further extend the NBTINAR(1)model defined in(2)to a more general version.The NBTINAR(1)model can be extended to the following interesting cases.(i)An immediate extension of model(2)can be achieved by letting the threshold vari-able Wt be chosen as Xt-d with Xt-d denotes a lag variable of the observed time series,then we obtain a generalized d-order delay NBTINAR(1)model.(ii)If the threshold variable Wt takes the form of a time index,i.e.,Wt=t,then the NBTINAR(1)model becomes a change point model with r denotes the change point.We call the resulting model a NBINAR(1)model with an unknown structural break,and denoted it by CP1-NBINAR(1).For the CP1-NBINAR(1)model,we use a meaningful notation T instead of r in(2)to denote the change point in this paper.Specially,when T = 0,the CP1-NBINAR(1)model reduces to the ordinary NBINA-R(1)proposed by Ristic et al.(2012).(iii)The threshold variable Wt also can be chosen as the linear or nonlinear combination of lag or explanatory variables,see,e.g.,Chen(1995),Chen et al.(2006).A common conditional expectation of the above extensive models takes in the following form:E(Xt|Xt-1)=(?1Xt-1+?1)I1,t+(?2Xt-1+?2)I2,t.Proposition 5 Suppose {Xt} is a process generated by the above extensive models(4.1.1)is a ergodicity Markov chain with state sapce N0.The estimation method and test method discussed in the above are also applicable to the extended models,at last we use models to fit a group of actual data.