| Integer-valued time series data is widely used in everyday life. The value of such data are non-negative integers, and typically exhibit dependencies, heteroscedasticity and asymmetric characteristics. Therefore, it is more difficult to study comparing with the traditional continuous time series. Although the models based on the thinning operators can make a reasonable interpretation of the background of data in some cases, but can not capture the heteroscedasticity of the data. The classical Poisson linear autoregressive model solve the problem of modeling the heteroscedasticity of the data successfully. However, such models can not describe the asymmetric of data. For count data with heteroscedasticity and asymmetry, the existing models can not reactive the objective laws of the observed data. Bankruptcy is the core theory of risk theory. The classical risk model is too idealistic to reflect the relationship between the various factors. In fact, there are some dependence structures between premiums and claims. However, the existing risk model ignores the dependence structure. Therefore, it is necessary to study the dependencies between premiums and claims.This paper studies the modeling and statistical inference of two kinds of integer-valued time series. The main content is divided into three parts. First, we focus on the integer-valued time series data with heteroscedasticity and asymmetric, propose a class threshold Poisson log-linear autoregressive model, study the stationary and ergodicity of the model, give the maximum likelihood estimator of the parameters and the asymptotic distribution of the estimator, study the effect of estimation via the simulations. Also, we fit a set of real data using the proposed model. Then, we give the empirical likelihood inference of the threshold Poisson log-linear autoregressive model. Give the confidence regions for the model parameters based on the normal approximation and empirical likelihood methods. Compare the coverage accuracy of the two confidence regions via simulations. Finally, we propose a class of risk model which the premiums and claims have dependent structure, study some properties of the model, including the moments and the probability generating function. Given the normal approximation of ruin probability, when the number of claims follows a Poisson process,we propose a further risk model,give the upper bound of the ultimate ruin probability.In what follows we introduce the main results of this paper.1.The threshold Poisson log- linear autoregressive model.To capture the heteroscedasticity and asymmetric of integer- valued time series data,we propose a class of threshold Poisson log- linear autoregressive model:Definition 1 Let{Xt,t∈Z)be a time series,Assume that the conditional distribution of Xt based on past information is given as follows: where vt=log λt,I1,t=I{Xt≤r},I2,t=1-I1,t=I{Xt>r},r∈N and r<r*<∞, r*is a constant,Ft=σ{xs,s≤t)is the σ-filed generated by{Xt,xT-1,Xt-2,…}, θi=(di,ai,bi)T(i=1,2)be the parameter of model(1),θ1≠θ2.This model is called threshold Poisson log- linear autoregressive model,denoted by TPLAR(1).In what follows we discuss some basic properties of the TPLAR(1)model. Let dt=d1I1,1,t+d2I2,t,at=a1I1,t+a2I2,t,bt=b1,I1,t+b2I2,t,ct=dt+btlog(Xt+1),then the stationary solution of vt takes the following form: We assume that when k=1,the formula Ⅱ0j=1 at-j=1,so that the above equation is always reasonable.To study the properties of TPLAR(1)model and the parameter estimation prob-lems,we give two assumptions:(C1)Assume the parameters ai,bi,di satisfy:(C2)Let D be the compact set ofIR6,θ=(θT1丁,θT2)T∈D is the parameter to be estimated with true value 90.In the following theorem and corollary,we give the stationary and ergodicity of sequences{vt,t>0)and{(vt,Xt),t>0}.Theorem 1 Consider the TPLAR(1) model defined by definition 1. Assume that condition (C1) holds.Then the Markov chain{vt} is as stationar and ergodicity process.Corollary 1 Assume that the conditions in theorem all hold,then the Markov chain{(vt,Xt)}is a joint stationary and ergodicity process.Assume thatv1=log λ1 and X1 are both known,{Xt)nt=1 is a sample generate by model(1)based on the initial value X1 and λ1.And{vt)nt=2 is a sequence calculate by the initial value v1(vt=log(λt)).Thus,we have the likelihood function the log- likelihood function where and the score functionThe root of Sn(θ)=0,if exist,gives the maximun likelihood estimator of θ, denoted by θ.For the case that r is unknown.Let l(r,θ)be the log- likelihood function when r is unknown.Then,we can give the estimation of unknown parameters r and 9 by the following two steps:Step 1 For each rj∈[0,r*]∩ N,use the above method t0 calculate the maximum likelihood estimator θrj,i.e.Step 2 For the [r*]+1 results obtained by Step 1,{(rj,θrjT)T,j=0,1,…,[r*])find the group which can maxinmum l(r,θ),i.e,We introduce the following notations:let (?)t(θ)= xt · vt(θ) — exp{vt(θ)}, where vt(0) is calculated by (2), (?)(θ)= Σnt=1 (?)t(θ).LetThe following theorem gives the strong consistency and asymptotic normality of 0.Theorem 2 Assume that{Xt}nt=1 is a sample satisfy the TPLAR(1) process. As-sume that the conditions of(C1)- (C2) hold, then 0 is strong consistent. Furthermore, 0 has the following distribution, where G can be consistently estimated byWe studied the performance of the maximum likelihood estimator via simulations. We also fit a set of real data using the proposed model, which obtain a good fitting result.2. Empirical likelihood inference for threshold Poisson log-linear autoregressive model.By theorem 2,0 follows a normal distribution with mean 0 and variance G-1. Then we have the confidence regions based on the normal approximation method as follows: where 0< δ< 1, X26(δ) denotes the up δ-quantile of X2 distribution with degree of freedom 6, G is the consistent estimator of G.In what follows, we consider the empirical likelihood (Owen,1988) inference for TPLAR(1) model. Let p= (p1,^…,pn)T be a probability vector satisfies the log empirical likelihood ratio function can be constructed asThe following theorem gives the limit distribution of l(θ0).Theorem 3 Assume that (C1)- (C2) hold, then where x26 is the X2 distribution with degree of freedom 6.By theorem 3, for 0< δ< 1, the confidence region of θ with confidence level 100(1 —δ )% takes the following form: where X26(δ) denotes the up δ-quantile of X2 distribution with degree of freedom 6.Based on the simulations, we compare the coverage accuracy of the normal approx-imation confidence region and the empirical likelihood confidence region. The results show that under the same confidence level, the empirical likelihood confidence region resulting higher coverage accuracy.3. A class of ruin probability model with dependent structureOn the basis of classical risk model, we consider random premium income and the dependence structure between the premium income and claims, and propose a new risk model: where Un is the total surplus during a period of time, u is the initial reserve of an insurance company, Xi, (i= 1,2, …,n) are independent and identically distributed random variables, random variable h(Xi) is depend on Xi,Ii (i= 1,2,…,n) is the claim times of the ith insurance policy, n is the total amount of policies in the time interval (0,t].We introduce some notations:E(Xi)= μx, E(h(Xi))= μh, E(Xih(Xi))= μxh, Var(Xi)= σ2x, Var(h(Xi))= σ2h. Then, we have the expectation and variance of Un and the probability generating function of Un,By the central limit theorem, we can normality approximate the total surplus in model (3) as follows:Assume that the insurance policies follows a Poisson process with parameters λ, then process (1) can be extend to a long-term collective risk model: where u(u≥0)is the initial reserve of a insurance company,N(t)is a Poisson precess with parameter λ,which denotes the number of insurance policies,∑N(t)i=1 Xi is the premium income until the time l,∑N(t)i=1 h(Xi)Ii is the total claims until the time l.We can see form(4)that,the amount of claims and premium income has a dependency structure,which both depend on a Poisson process{N(t)}.Let S(t)=∑N(t)i=Xi-∑N(t)i=1 h(Xi)Ii be the acumulated earnings of a insurance company until time t.In order to run properly,we always require the expectation value for S(t)to be positive,that is 1.e,We give the some definitions.Definition 2 Let E(Xi)=μx,E(h(Xi))=μh,q=P(Ii=1),where Ii follows a Bernoulli distribution.Assume that μX—qμh>0,we define safe load factorDefinition 3 Let U(t) be the total surplus during[0,t].The ruin time is defined byDefinition 4 Let U(t)be the total surplus during[0,t],T is the ruin time.The ultimate ruin probability is defined byLet F(y)be the cumulative distribution function of.Xi-h(Xi)Ii,i=1,2,3,… Assume that the second order moment of Xi-h(Xi)Ii,i=1,2,3,…exist,then we have: where g(r)=λ(∫+∞-∞e rydF(y)-1).We have the following results.Lemma 1 The equation g(r)=λ(∫|∞-∞e-rydF(y)-1)=0 has a unique positive solution,and we call it the adjustment coefficient,denoted by R.Lemma 2 For the surplus process{S(t),t>0),define Fs={Fst,t≥0},then {Mu(t),Fst,t≥0)is a martingale,whereLemma 3 Under the above notations,T is the stopping time of Fs.Theorem 4 For the Poisson risk model{Y(t),t≥0),where premium and claim income are dependent,the ultimate ruin probability satisfy the following inequality where R is the adjustment coefficient.Theorem 5 For the Poisson risk model{U(t),t≥0),where prenium and claim income are dependent,the ultimate ruin probability takes the following form... |
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