含参数随机环境中(2,1)随机游动M估计量的渐近正态性

Asymptotic Normality of M-estimator for a(2,1)Random Walk in a Parametric Random Environment

考虑随机环境中(2,1)随机游动,其中环境独立同分布,包含特定参数.[Stoch.Models,2019,35(3):338-356]基于对游动轨道的一次观测,对环境参数构造了M-估计量,并得到其弱相合性.本文进一步证明M-估计量的渐近正态性.研究工具是构造(2,1)随机游动的内蕴分支结构.虽然M-估计量没有解析表达式,但可借助内蕴分支结构,通过建立准则函数梯度向量的中心极限定理,得到M-估计量的渐近正态性,其极限协方差矩阵正是Fisher信息矩阵的逆矩阵.本文还给出了Fisher信息矩阵的估计方法.最后进行数值模拟,验证了所提出的估计量的弱相合性;根据理论得到的渐近正态分布结论,进一步构造经验置信区间,验证了结果的合理有效性.

We consider a(2,1)random walk in an i.i.d.random environment,whose environment involves certain parameter.Based on a single observation of the path till the time it reaches a distant site,[Stock.Models,2019,35(3):338-356]constructed an M-estimator for the environment parameter and obtained its weak consistency.In this paper,we further prove the asymptotic normality of the M-estimator.Although there is no analytical expression for the value of the M-estimator,the intrinsic branching structure within(2,1)RWRE can help us establish the central limit theorem for gradient vector of the criterion function,and then obtain the asymptotic normality of the M-estimator.The limiting covariance matrix is exactly the inverse of Fisher information matrix.We also explain how to estimate the Fisher information matrix.Furthermore,the paper explores in a simulation setting the numerical behavior of weak consistency and asymptotic confidence regions for the parameter value.