Asymptotic Inference For Discrete-time Risk Models Based On Integer-valued Time Series

In the classical discrete time risk model, the claims made in different time periods are assumed to be independent (and identically distributed). The assumption on in-dependence is often unrealistic in application and was made merely for mathematical convenience. A recent trend of the study is to consider various dependent structures, such as dependence among the claims, dependence between the inter-arrival-time and the claim, and the dependence of the discrete variate time series. Under the depen-dency structure, most of existing methods fail in obtaining precise estimate. A recent breakthrough came from the asymptotic inference in risk models with heavy-tailed claim sizes. Cossette et al.(2010) propose a type of discrete-time risk models that use dependent integer-valued time series as counting random variables. A substantial part of the work by Cossette et al.(2010) is the co-relation among the discrete variate time series brought by thinning operations which serve as a natural extension of the scalar multiplications. Due to this progress, the asymptotic inference in risk models with heavy-tailed claim sizes and dependent structure became a hot topic recently.Partially inspired by the work of Cossette et al.(2010), we consider the risk models with dependent for counting random variables. Our goal is the precise large deviations for the aggregate claims and the uniform asymptotic formula for the finite-time ruin probability in the setting when the claim follows a heavy tail distribution.Let the random variable (r.v.) Wk represent the aggregate claim amount in period k for an insurance portfolio. In general, the r.v. Wk follows a compounded distribution, which means one can write Wk as where Nk and{Xk,j,j≥1} represent, respectively, the number of claims and the individual claim amounts made during the period k. Throughout, we assume that {Xk.j, k≥1, j≥1} forms an independent and identically distributed (i.i.d.) array and that{Xk,j,k≥1,j≥1} and{Nk} are independent. We adopt the notations X for a generic copy of{Xk} and Fx for the common distribution function (d.f.) of the i.i.d. array{X,Xk,j,k≥1,j≥1}. DefineThen the r.v. Sk represents the accumulated aggregate claim amount over the first k periods.The surplus process is given bywhere c is a positive constant corresponding to the premium income rate and Uq u>0is the initial reserve of the company.The time of ruin for the risk model(1) is described by T=inf{t, U(t)<0}. The finite-time ruin probability for some time t>0isFor the sake of application, we show that the accumulated aggregate claim Sk has the same distribution as a stopped random walk.theorem1In the risk model(1), let the i.i.d. array{X,Xk,j,k≥1,j≥1} be the claim amounts with d.f. Fx, and{Y, Yj,j≥1} be a sequence of i.i.d. non-negative r.v.’s such that Y and X are identically distributed,then where "d=" denotes the identically distribution.Consequently, to establish the asymptotic for Sk,all we need is to work with whose behavior is determined by the distributions of X and Ni+…+Nk.It is worthy of mentioning that under dependency, N1+…+Nk is not necessarily Poisson random variable, even if N1,…,Nk are Poissoning. As a consequence,the classic treatment for independent setting is no longer working here. In this work we establish the asymptotic inference in different settings classified according to the way in which the counting random variables N1,…, Nk are generated.1. Discrete-time risk models based on INMA(1) for counting random variables. Let (?)={Nk,k∈N+} be a Poisson MA(1) process defined bywhere (?)={εk,k∈N} is a sequence of i.i.d. r.v.’s with the Poisson distribution of the mean (?).The thinning procedure "o" is defined by,where{(?)k-1,j,j≥1} is a sequence of i.i.d. Bernoulli r.v.’s with the mean α∈[0,1], and the sequences{(?)k-1,j,j≥1} generated by different k are independent. The risk model(1) is said to be Discrete-time risk model based on INMA(1) if the counting random variables satisfy (3).In the following theorems we establish the precise large deviations for the sums of heavy-tailed claims, and derive the uniform asymptotic formula for the finite-time ruin probability.theorem2In the discrete-time risk model based on INMA(1),let the i.i.d.array {X,Xk,j,k≥1,j≥1} be the claim amounts with the d.f. Fx(x)∈C and the finite expectation μ. Then, for any fixed γ>0,Theorem2indicates that the INMA(1) dependence structure of counting random variables does not affect the asymptotic behavior of the large deviations of Sk.theorem3In Discrete-time risk model based on INMA(1), let the i.i.d. array {X,Xk,j,k≥1,j≥1} be the claim amounts with the d.f. Fx(x)∈C and the finite expectation μ.Then, for any fixed γ>0, uniformly over x>γk1+δ for some δ>0as k�'∞. Note that, the zones of uniform convergence in theorem2and theorem3are different. The relation in theorem3with respect to P(Sk>χ) is more natural for studying the asymptotic of the finite-time ruin probabilities, where x represents the initial capital of an insurance company. In addition, we restrict ourselves to the case x∈[γk1+δ,+∞] in Theorems3. One implication of this constraint is that the initial capital χ satisfies k=o(x).With the help of Theorem3, we can estimate the finite-time ruin probability defined in (2) as follows.theorem4Suppose that the conditions of Theorem3hold. Then, for any fixed γ>0and t>0, uniformly over u>γt1+δ for some δ>0as u�'∞Theorem4indicates that the uniform asymptotic formula for the finite-time ruin probability be valid if t=o(u).Furthermore, we generate above results to the risk model with a Poisson MA(q) dependent structure.2. Discrete-time risk models based on INMA(q) for the counting random variables.Let Nk is a Poisson MA(q) process defined bywhere (?)={εk,k∈Z} is a sequence of i.i.d. r.v.’s with the Poisson distribution of the mean (?) as its common distribution, where a=Σq αi and α0=1. The thinningwhere{(?)i,j(-i)} for j=1,2,…, and for every i is a sequence of i.i.d. Bernoulli r.v.’s with mean(∈[0,1]). The risk model (1) is called Discrete-time risk model based on INMA(q) if the counting random variables N1,…, Nk satisfy (4).theorem5In the discrete-time risk model based on INMA(q), let the i.i.d. array {X,Xk,j,k≥1,j≥1} be the claim amounts with the d.f. Fx(x)∈C and the finite expectation μ. Then, for any fixed γ>0, uniformly for x>γk as k�'∞.theorem6In the discrete-time risk model based on INMA(q), let the i.i.d. array {X,Xk,j,k≥1,j≥1} be the claim amounts with the d.f. Fx(x) E C and the finite expectation μ. Then, for any fixed γ>0,theorem7Suppose that the conditions of Theorem6hold.Then,for any fixed γ>0and t>0, uniformly over u>γt1+δ for some δ>0as u�'∞.3. Discrete-time risk models based on INAR(1) for counting random variables.The pattern we consider here has strong background in the financial practice. A classic application is on the unemployment insurance, which can be extended to a larger class of insurance models where the large insurance claims takes significant portion. In the health insurance, for example, an insured patient has been financially supported by his/her insurance company until full recovery (or death). The cost in some cases can be extremely high and so it counts a significant portion out of the total claims, despite the low frequency of such extreme cases. These phenomena are captured in mathematics by heavy tail distribution. Another factor comes from the newly increasing number of the hospitalized patients in the same claim period. Nk counts two parts of the claims. One is contributed by the patients existing since the previous period, and another is by the patients newly added in the current period. Together, the model is mathematically formulated as following:The claim numbers (?)={Nk,k∈N+} form a Poisson AR(1) process defined by where (?)={εk,kN+} is a sequence of i.i.d. r.v.’s with the Poisson distribution of the mean (1—α)γ,α∈[0,1]. In addition, N0is assumed to be deterministic. The risk model (1) is called Discrete-time risk model based on INAR(1) if the counting random variables satisfy (5). Note that if α≠1, then model given in (5) yields a stationary sequence of Poisson r.v.’s of the mean γ.Similarly, we have theorem8In the discrete-time risk model based on INAR(1), and α∈[0,1), let the i.i.d. array{X,Xk,j,k≥1,j≥1} be the claim amounts with the d.f. Fx(χ) E C and the finite expectation μ. Then, for any fixed γ>0, holds uniformly for χ>γk.theorem9In discrete-time risk model based on INAR(1), and α∈[0,1), let the i.i.d. array{X,Xk,j,k≥1,j≥1} be the claim amounts with the d.f. Fx(χ)E C and the finite expectation μ. Then, for any fixed γ>0, holds uniformly for χ>γk1+δ for some δ>0as k�'∞.In addition, we establish the tail asymptotic Sk in the case when F E S.theorem10In discrete-time risk model based on INAR(1), and α∈[0,1), let the i.i.d. array{X,Xk,j, k≥1,j≥1} be the claim amounts with the d.f. Fx(χ) E S. Then, for any fixed γ>0, holds uniformly for x>γk as k�'∞.In comparison to Theorem9, Theorem10extends the class of the d.f. F from C to S with a relaxed asymptotic co-relation "χ> rk " versus "χ> rk1+δ" imposed in Theorem9. We derive lower bound for the sums of S claims. As the consequence of Theorem9, we establish the asymptotic form (given in Theorem11below) for the ruin probability.theorem11Suppose that the conditions of Theorem9hold. Then, for any fixed γ>0and t>0, holds uniformly foru>γt1+δ for some δ>0as u�'∞.4. Discrete-time risk models based on INARCH(1) for the counting random vari-ables.Here the claim-number process (?)={Nk,k∈N+} is described by the Poisson ARCH(1) process, which is defined as where a0>0, a1≥0and (?)k-1denotes the information up to time k-1. In addition, N0is assumed to be deterministic.The risk model(1) is called Discrete-time risk model based on INARCH(1) if the counting random variables satisfy (6).The Poisson ARCH was proposed by Rydberg-Shepherd (2000) and by Streett(2000) independently. The basic hypothesis is the linear co-relation between the previous claim number the conditional (on the past data) expected average of the current claim number. Note that if0≤a1<1, then model given in (6) yields a stationary sequence.theorem12In the discrete-time risk model based on INARCH(1),0≤a1<1, let the i.i.d. array{X,Xk,j,k≥1,j≥1}be the claim amounts with the d.f. Fx(χ)∈C and the finite expectation μ. Then, for any fixed γ>0, holds uniformly for x>γk.theorem13In the discrete-time risk model based on INARCH(1),0≤a1<1, let the i.i.d. array{X,Xk,j,k≥1,j≥1}be the claim amounts with d.f. Fx(x)∈C and the finite expectation μ. Then, for any fixed γ>0, holds uniformly for x>γk1+δ for some δ>0as k�'∞.theorem14In the discrete-time risk model based on INARCH(1),0≤a1<1,let the i.i.d. array{X, Xk，j, k≥1,j≥1}be the claim amounts with the d.f. Fx(x)∈S. Then,for any fixed γ>0, holds uniformly for x>γk as k�'∞.As before, Theorem13lead to the asymptotic for ruin probabilities.theorem15Suppose that the conditions of Theorem13hold. Then, for any fixed γ>0and t>0, holds uniformly for u>γt1+δ for some δ>0as u�'∞.Finally, in this paper, we consider the empirical likelihood for the autoregres-sive error-in-explanatory variable models. With the help of validation, we develop an empirical likelihood ratio test statistic for the parameters of interest. 5. First-order autoregressive error-in-explanatory variable modelsIn many research settings, the exact measurement of some important variables is difficult, time consuming, or expensive, and can only be performed for a few items in a large scale study. Thus they are usually replaced by surrogate observations, which are available by some relatively simple measuring methods. Generally, the relation-ship between the surrogate variables and the true variables can be rather complicated compared to the classical additive error model usually assumed (Fuller,1987). In many practical settings, it is difficult to even specify the relationship between true variables and their surrogates. Perhaps, the most realistic treatment does not include error structure or distribution assumption of true variables by given surrogate variables. To address this problem, some statisticians developed statistical inference techniques based on the surrogate data and validated observations without specifying any error structure and the distribution assumption of the true variable given the surrogate vari-able. They include Pepe and Fleming (1991), Pepe (1992), Wang (1999),and Wang and Rao(2002).We now try to apply this idea to the following first-order, univariate, autoregressive model with p explanatory variables:Here β∈R, α=(α1,…, αp), ZtT=(Z1t,…, Zpt); α and α are unknown parameters of interest, the Zt’s are independent identically distributed explanatory variables, and {εt} are independent identically distributed random forcing sequence.Many economic relationships are dynamic in nature. They are characterized by the presence of a lagged dependent variable among the regressors and therefore can be represented by model (7). However, in many applications, the exact measurement of Zt may be difficult, so a surrogate Zt is observed instead of Zt. In this paper, without specifying any structure equation and the distributed assumption of Zt given Zt, the estimator of parameters is defined to an empirical likelihood ratio test statistic based on validation data.We first calibrate the model (7) and obtain an equivalent calibration regression model. In what follows, we assume that in addition to the primary data{(Yt, Zt)tN=1}, n independent and identically distributed observations{(Zi,Zi)i=N+1N+n} are available. Let u(Zt)=E(Zt｜Zt).Then(7) can be rewritten as where ηt=εt+αZt—αu(Zt). Under the surrogate assumption, E(εt｜Zt)=0and E(ηt｜Zt)=0. Then we can obtain the conditional least squares (CLS) estimator of (β,α) defined That is,the CLS estimator (β,α) is the solution to the following equationMotivated by (9) and Owen (1991), we can obtain the following empirical loglike-lihood functionNeverthless,the estimate of (β,α)can not be derived from the above function because (?)(β,α) contains the unknown terms u(Zt). To solve this problem, one might replace u(·) in (?)(β,α) by the following estimatorwhere K(·) is a kernel function and hn is a bandwidth that decreases to zero as n increases to infinity. To avoid technical difficulties caused by small values in the de-nominator of un(z), we consider replacing u(·) in (?)(β,α) by a truncated version of un(z). In Wang and Rao (2002), the truncated version of un(z) is then given by where fn(z)=-kr]T K{K^), and fbn(z)=max{fn(z), bn} for some positive con-stant sequence bn, p is dimensionality of variable Zt in fn(z). Here we let htn(β,α) and Jtn denote ht(β,α)and Jt with u((?)) replaced by (?)n((?)).Then we define an estimated empirical loglikelihood asBy the Lagrange multiplier nethod,(?)(β,α)Can be represented as where λ∈Rp+1satisfiesIn some regularity conditions(see Subsection4.2),we prove (?)(β,α)its asymptotic distribution is that of a weighted Sum of independent standard X2random variables with unknown weights.theorem16Under some regularity conditionswhenever(β,α)is true value,where(?)denotes convergence in distribution, x12,i (1≤i≤p+1)are independent x12variables,and ω1,ω2…ωp+1are the eigenval-ues of D(β,α)=∧0-1.(β,α)∧(β,α).Here∧(β,α)=∧0(β,α)+rAl(β,α),∧0(β,α)=E(JtJtTηt2),ηt=Yt—βYt-1—αu((?)t),To apply Theorem16to construct a confidence region for(β,α),we must estimate the unknown weights ωi consistently.Wang and Rao(2002)develop an adjusted empir-ical likelihood method for error-in-covariables linear models with the help of validation data.We here extend this method to model (7).We now propose an adjusted empirical log-likelihood whose asymptotic distribu-tion is a standard chi-square.By examining the asymptotic expansion of (?)(β,α),we define an adjusted empirical log-likelihood by which can be proved to be asymptotically xp+12,where theorem17Under the regularity conditions given in Theorem16,if(β,α)is the true parameter,we haveAsymptotically,the confidence region for(β,α)with the significance level1—δ can be given aswith cδbe decided by the equation P{xp+12≤cδ)=1—δ.