Theoretical Research And Application Of First-order Random Coefficient Mixed Thinning Integer Valued Autoregressive Model

The paper develops a first-order random coefficient mixed-thinning integer-value autoregressive time series model（RCMTINAR（1））,which improve the first-order mixed-thinning integer-valued autoregressive model（MTINAR（1））and the constant coefficient is replaced by the random coefficient,is more convenient to solve the counting data related to the random variable and more suitable for fitting the actual data.First of all,we introduce a random coefficient mixed-thinning.We assume X is a non-negative random variable,and W_i is i.i.d.random variable.The random coeffi-cient mixed-thinning is defined by the following equation,（?）,whereas α∈（0,1）,and W has mixed distrbution:here Bi and G_i are random variable with Bernoulli（α_t）distribution and geometric（α_t/（1+α_t））distribution.We introduce the main results of this paper as below.1.Definition of RCMTINAR（1）modelDefinition 1 A non-negative integer-valued process given by X_t=α_t·_pX_t-1+ε_t,t≥2,（1）If the following conditions as satisfied:（1）{α_t} is an i.i.d.sequence with cumulative distribution function（CDF）P_α on[0,1);（2）{ε_t} is an i.i.d.non-negative integer valued sequence with a probablity mass function（PMF）f_ε>0,E（ε_t⁴）<∞,E（X₁⁸）<∞;（3）X₁,{α_t} and {ε_t} are independent for each t;（4）random variable X_t-i and ε_t are independent for all i≥1;（5）ε_t is independent of the counting series {W_i};（6）Let α=E（α_t）,σ_α²=Var（α_t）,μ_ε=E（ε_t）,σ_ε²=Var（ε_t）,τ²=α²+σ_α²,and note that they are all assume finite.{X_t} will be said to be RCMTINAR（1）model.And some moments,autocovariance functions,k-step ahead conditional mean and k-step ahead conditional variance for this model are studied discuss as the distribution of the innovation sequence is unknown,the conditional marginal distribution is also analyzed with given random coefficient.Then,the conditional least squares（CLS）and modified quasi-likelihood（MQL）are adopted to estimate the model parameters.We give the strong consistency and asymptotically normally distributed of CLS estimates by the following two theorems.Theorem 1 Let {X_t}be a strictly stationary ergodic RCMTINAR（1）process andθ=(α,μ_ε}^T.Then the CLS estimator （?）_CLS will be strongly consistent and bivari-ate asymptotically normally distributed.Theorem 2 Let {X_t} be a strictly stationary ergodic RCMTINAR（1）process and β=（σ_α²,p,σ_ε²）^T.Then the CLS estimator （?）_CLS will be strongly consistent and tr-ivariate asymptotically normally distributed.Next,the asymptotically normally of MQL estimator given by,Theorem 3 Let {X_t} be a strictly stationary ergodic RCMTINAR（1）process andθ=（α,μ_ε）^T.Then the j oint limit distribution of MQL estimator （?）_MQL given by:（?）where （?） where T₁（J₁）=E[V_J1^-1（X₂|X₁）X₁²],T₂（J₁）=E[V_J1^-1（X₂|X₁）],T₃（J₁）=E[V_J1^-1（X₂|X₁）X₁].Afterwards,Using the mean square error and mean absolute deviation error as criteria,and the performances of these two estimaton methods are investigated via simulation,and compared with false modified quasi likelihood estimation.We obtain that MQL is better than CLS.Finally,the practical relevance of the model is illustrated by using two applications to a SIMpass data set and a burglary data set of the Crime data with a comparison,and residual test of the model are discussed.We get RCMTINAR（1）model is more suitable for analysis real data.