Estimation For Random Coefficient Bifurcating Autoregressive Model

The meticulous experimentation and analysis conducted by E.O.Powell in the 1950s and 1960s provide the motivation for this current paper. The statistical methodology can be enhanced by better models of dependence, by advanced computational methods. Methods of statistical analysis of cell lineage data, which explicitly considered the dependence between the cells,began with the bifurcating autoregressive BAR(l) model Cowan and Staudte (1986).{(ε_2t ,ε_2t+1)} is a sequence of independent identically distributed (i.i.d.) bivariate normally random variables with common meanμ, common varianceσ²,and correlation coefficientρ.This model regards each line of descent as an AR(1) process ,and account for similarities between sister cells due to environmental effects it allows correlation between the residuals of sister cells.Huggins and Basawa(1999)proposed bifurcating ARMA(p,q)models to accommodate this extended dependence in the family tree. Huggins and Basawa(2000) discussed maximum likelihood estimation for a Gaussian bifurcating AR(p) and established the consistency and asymptotic normality of the maximum likelihood estimators of the estimators of the model parameters. Basawa ahd Zhou(2004) introduced non-Gaussian bifurcating autoregressive models and studied some preliminary estimation problems. Zhou and Basawa (2004) have discussed maximum likelihood estimation for an exponential bifurcating AR(1) process.Zhou and Basawa(2005) considered the asymptotic properties of the least-squares (LS) estimators of parameters in a bifurcating BAR(p) process.Integer-valued bifurcating autoregressive model were introduced by Zhou and Ba-sawa(2005).In this model,the parameter plays the role. In some situations,the parameter may vary and it may be random. In this paper, we extend the above model to a random coefficient model:random coefficient bifurcating autoregressive model. Our main objective is to investigate basic probabilistic and statistical properties of this model and inferential methods for relevant parameters associated with model.The paper is organized as follows. In Section l,a general class of bifurcating models for cell lineage data is presented. In Section 2, we reexamination the model bifurcating poisson model and we get the Markov property. As following, the random coefficient model is defined and described in detail , some statistical properties are established. we propose estimation methods for the model parameters. we also derive the limit distributions of parameters.Proposition2.1. The limiting distribution of the process X_t is poisson with mean ,and .Proposition2.2. One single cell lineage is an irreducible and aperiodic Markov chain.Just as X₁, X₂, X₄, X₈,...The random coefficient bifurcating autoregressive model is defined by equationThe mean ofφ₁â—‹X_t is defined byX_t,ε_2t,ε_2t+1 are integer-valued variables, {(ε_2t,ε_2t+1)}, i = 1,2,…are i.i.d., Eε_2t = Eε_2t+1 =μ, var(ε_2t) = var(ε_2t+1) =σ_ε², corr(ε_2t ,ε_2t+1) =ρ.φ₁ is a variable with distribution function P_φ,(2) is given byφ₁～U[0,1], P_φ= x, Theorem2.1.Theorem2.2.(2) One single cell lineage is an irreducible Markov chain. Just as X₁, X₂, X₄, X₈,…Consume(1) EX_t^k→m_k, t→∞.(2) X_t is ergodic, and ,n→∞, m_k <∞, k = 1,2,3,4.(3) X_t andε_t have finite moment.Under the conditions, the following version of martingale central limit theorem will be used in the derivation of the limits distribution of the CEL estimator.Lemma 2.3. Let Y_t, t = 1,2,…,be a sequence of zero-mean vector martingale differences satisfying the following condition.(a) E(Y_tY'_t) =Ω_t,Ω_t is a positive definite matrix, and, n→∞,Ωis apositive definite matrix. (b) E(Y_itY_jtY_ltY_mt) <∞, t, i, j, l, m = 1,2,∞,Y_rt denotes the rth element of the vectorY_t.thenTheorem2.3.ELS estimator of (μ,θ)' is seen to beLimit distribution of (μ|^,θ|^)' is seen to beB and A are defined byConsidering Quasi-likelihood estimators(QLE),we get the quasi-score functions...