| Non-linear integer-valued time series observations are common in production and life,such as the number of epidemics,the number of outpatients in a hospital at a certain time,etc.This paper studies the modeling and statistical inference of several nonlinear time series.The main content is divided into three parts.In the first part,we consider the inference of a self-excited integer-valued threshold autoregressive process.Based on the empirical likelihood(EL)method,the estimation and testing problems for the model parameters are considered.The effect of the estimates and the power of the test are studied through numerical simulations.The model is used to fit a set of real data at last.In the second part,we propose a class of random coefficients integer-valued threshold autoregressive process.Some basic properties of the model and parameter estimation problems are discussed.The forecast problem is considered,too.In the third part,we consider the empirical likelihood inference of a class of threshold binomial autoregressive models.Parameters' empirical likelihood estimation,as well as the asymptotic properties of the estimators are given.In what follows,we introduce the main results of this thesis.1.Empirical likelihood inference of SETINAR(2,1)processTo capture the threshold characteristics of integer-valued time-series observations,we consider a self-excited integer-valued threshold autoregressive process,called SETI-NAR(2,1)process,which is defined as follows:Definition 1 The SETINAR(2,1)process is a sequence of random variables{Xt}t?Z defined by the following recursive equation,Xt =?t(?)Xt-1+Zt,t?Z,where(?)?t:=?1It1,1(r)+?2It-1,2(r),It1,1(r)= I{Xt-1?r},It 1,2(r)=1-It 1,1(r)=I{Xt-1>r},r is the unknown threshold variable;(?)"(?)" is the Binomial thinning operator defined in(1.1.1);(?){Zt} is a sequence of i.i.d.non-negative integer-valued random variables with probability mass function(p.m.f.)fz>0 afnd E(Zt)= ?<?.For fixed t,Zt is assumed to be independent of Xt-l,?t and ?t(?)Xt-l(l ? 1).In the following,we will omit(r)in It-1,i(r)(i = 1,2)to make the notation easy without ambiguity.Theorem 1 The process {Xt,t?1} defined ifn(1)is an ergodic Markov chain.In what follows,we introduce the EL inference for the SETINAR(2,1)models.Let ?=(?1,?2,?)T.By Mykland(1995),the log empirical likelihood ratio(ELR)statistic takes the form:(?)where mt(?)=(mt1(?),mt2(?),mt3(?))T,mti(?)=(Xt-?1Xt-1It-1,1-?2Xt-1It-1,2-?)Xt-1It-1,i,i=1,2,mts(?)= Xt-?1Xt-1It-1,1-?2Xt-1It-1,2-?.To study the asymptotic behaviour of the ELR statistic,we make the following assumptions:(C1){Xt} is a stationary process;(C2)E|Xt|6<?;Theorem 2 Under assumptions(C1)-(C2),we have that l(?0)(?)?(3)2.Theorem 3 Under the assumplions(C1)-(C2),for 0<?<1,the 100(1-?)%EL confidence region of ? can be written as:C?(?} ={??R3:l(?)??32(?)},where c?{?} is a convex set.In what follows,we will consider the maximum empirical likelihood estimator(MELE)for the parameter ?.Following Qin and Lawless(1994),the MELE is defined by minimizing the log empirical likelihood function,i.e.,(?)Theorem 4 Under assumptions(C1)-(C2),the MELE ?MELE is asymptotically normal,(?)where I(?0)= E[mt(?0)mt(?0)T]and V=E((?)?/(?)mt(?0)).We use the NeSS algorithm(Li and Tong,2016)to estimate the threshold r.Let(?)Let us now tackle the issue of testing nonlinearity for the threshold model with the null hypothesis and the alternative hypothesis take the form:H0:?1=?2 VS H1:?1??2.Assumming that ? is the consistent estimators of ?=(?1,?2)T,and is asymptotically normally distributed around the true value ?0,i.e.,(?)We have the following theorem.Theorem 5 Let ?=(?1,?2)T be some cofnsistent estifmators of ? satisfying(3).Then the statistic for test problem(2)is u=(?).Furthermore,as n??,u(?)N(0,1)when H0 is true.2.Modeling and statistical inference of the RCTINAR(1)processTo overcome the limitation that the autoregressive coefficients of the SETINAR(2,1)model are constants.we propose a class of random coefficients integer-valued threshold autoregressive processes,namely the RCTINAR(1)process,which is defined as:Definition 2 The RCTINAR(1)process is a sequence of random variables {Xt}t?Z defined by the following recursive equation,Xt=(?1,t(?)Xt1)It-1,1 +(?2,t(?)Xt1}It1,2+Zt,t?Z,(4)where(?)It-1,1 = I{Xt-i ? r},It-1,2= 1-It-1,1 = I{Xt-1>r},where r?[r,r]is theunknown threshold variable,r and r are some upper and lower bounds of r;(?){?i,t}is an i.i.d.Beta rafndofm variable sequence,?i,t?Beta(qi,k-qi),k is some known constant,0<qi<k,i=1,2,{?1,t}and{?2,t} are independent of each other;(?)"(?)" is the Binomial thinning operator definded by(1.1.1);(?){Zt} is a sequence of i.i.d.Poisson random variables with mean ?;(?)For fixed t and i(i = 1,2),Zt is assumed to be independent of ?i,t(?)Xt-1 and Xt-l(l?1).First,we discuss some basic properties of the RCTINAR(1)process.Proposition 1 Let {Xt}t?Z be the process defined in(4).Then {Xt}t?Z is an irreducible,aperiodic,and positive recurrent(and hence ergodic)Markov chain.Thus,there exists a strictly stationary process satisfying(4).Proposition 2 Let {Xt} be the process defined by(4).Then E(Xtk)<? for k = 1,2,3,4.Proposition 3 Let {Xt} be the process defined by(4).Then(?)E(Xt|Xt-1)=?1Xt-1It-1,1+?2Xt-1It-1,2+?;(?)E(Xt)= p1?1u1+p2?2u2+?;(?)(?)(?)(?)(?)Cov(Xt,Xt-h)=(?1p1 + ?2p2_h Var(Xt-h);(?)p(h):= Corr(Xt,Xt-h)=(?1p1+?2p2)h.In what follows,we discuss the parameter estimation problems of the RCTI-NAR(1)process.The conditional least squares estimators ?CLS of parameter ??(q1,q2,?)T are ob-tained by minimizing the residual square sum function:Q(?):=?t=1n(Xt-g(?,Xt-1))2=?t=1nUt2(?),,where Ut(?)= Xt-k/q1Xt-1It-1,1-k/q2Xt-1It-1,2-?.Theorem 6 Let {Xt} be a RCTINAR(1)process.Then the CLS-estimators?CLS:=?(q1,CLS,q2,CLS,?CLS)T are strongly consistent and asymptotically normal,(?)where V and W are square matrices of order 3,with elements Vij= E((?)?i/(?)g(?,Xt-1)(?)?j/(?)g(?,Xt-1),and Wij= E(Ut2(?)(?)?i/(?)(?,Xt-1)(?)?j/(?)(?,Xt-1),respectively,with ?1=q1,?2 = q2 and ?3 = ?.The conditional maximum likelihood estimators ?CML of parameter ? can be ob-tained by maximizing the conditional log-likelihood function:l(?)=(?)log(p(xt-1,xt,q1It-1,1 + q2It-1,2,?)).Theorem 7 Let {Xt} be a RCTINAR(l)process satisfying(C1)-(C6).Then the conditional maximum likelihood estimators ?CML=(q1,CML,q2,CML,?CML)T of ?is strongly consistent.Theorem 8 Under the assumptions of Theorem 7,the CML-estimzators ?CML is asymptotically normal,i.e.,(?)(?CML-?)(?)N(0,I(?)-1).We have improved the NeSS algorithm,and estimate the threshold parameter r of RCTINAR(1)model based on the new algorithm.Finally,the model is applied to the German measles data.3.Empirical likelihood inference of SET-BAR(1)processThe SETBAR(1)process is proposed by Moller et al.(2016a),and is defined by:Definition 3 The SET-BAR(1)process is a sequence of random variables {Xt}t?Z defined by the following recursive equation,Xt = ?t(?)Xt-1+?t(?)(N-Xt-1),t?Z(5)where(?)N ?N is predetermined upper limit of the range,R ?[0,N)is the threshold variable;(?)?t:= ?1It-1,1+ ?2It-1,2,?t:?1It-1,1+ ?2It-1,2,where It-1,1=I{Xt-1?R},It-1,2 =1-It-1,1=I{Xt-1>R};(?)"(?)" is the Binomial thinning operator definded by(1.1.1);(?)For i ? {1,2},?i= ?i+ ri?(0,1),?i=?i(1-ri)?(0,1),where ?i?(0,1),ri?(max{-1-?i/?i,-?i/?i},1).Let ?=(r1r2,?1,?2)T,mt(?)=(mt1(?),mt2(?),mt3(?),mt4(?))T with mti(?)=(Xt-g(?,Xt-1))(Xt-1-?iN)It-1,i(i= 1,2),mui(?)=(Xt-g(?,Xt-1))(1-ri-2)NIt-1,i-2(i= 3,4).Thus,the log ELR statistic has the form:(?)To study the asymptotic behaviour of the ELR statistic,we make the following assumptions:(C1){Xt} is a stationary process;(C2)E|Xt|6<?;Theorem 9 Under assumptions(C1)-(C2),we conclude that l(?0)(?)?(4)2.Theorem 10 Under assumptions(C1)-(C2),for 0<?<0,the 100(1-?)%EL confidence region of ? can be written as:C?{?} = {??R4:1(0)??42(?)},where c0{?} is a convex set.The maximum empirical likelihood estimator ?MELE of the parameter ? can be obtained by:(?)Theorem 11 Under assumptions(C1)-(C2),the maximum empirical likelihood estifmators ?MELE is asymptotically normal,(?)where I(?0)= E[mt(?0)mt(?0)T]and J = E((?)?/(?)mt(?0)).We propose a new algorithm for estimating the threshold parameters,called SMLS algorithm.The steps are as follows:Step 1.For each R?[0,N)?N0,find ?CLS(R)such that ?CLS(R)= arg min ? Q(?).Step 2.The threshold is estimated by searching over all candidates,i.e.,(?)... |
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