On the Strong Approximation of the Distributions of Estimators in Linear Stochastic Models, I and II: Stationary and Explosive AR Models
线性随机模型估计量分布的强近似：平稳与爆炸性 AR 模型AI 解读 ↓
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线性随机模型估计量分布的强近似：平稳与爆炸性 AR 模型
- 研究问题
- 在平稳与爆炸性自回归模型中，估计量分布何时能够以强于弱收敛的方式逼近极限，并获得一定的一致性？
- 方法与思路
- 作者引入强渐近平移等变性，结合似然光滑性建立分布强收敛，再为相关模型推导更易检验的充分条件。
- 材料陈述的贡献
- 摘要称平稳情形下若干估计量可在 Borel 集及适当参数邻域上一致逼近高斯分布，并将强收敛方法扩展至爆炸情形。
- 阅读关注点
- 应核对强收敛与等变性的定义、似然光滑性要求，以及平稳和爆炸情形各自的归一化与极限分布。
- 阅读边界
- 摘要未给出具体正则条件，不能把平稳情形的高斯极限直接套用于爆炸模型；索引摘要稿件版本未注明。
原文摘要
It is shown that in the stationary autoregressive case (Part I [Sections 1-5)] the distributions of least squares estimators and their close relatives such as sample autocovariances and the recently introduced general $M$-estimators converge to suitable Gaussian distributions uniformly over all Borel sets and uniformly over suitable neighborhoods of the parameter. Specifically, the notion of strongly asymptotically shift equivariance is introduced and it is shown that the distributions of any estimators satisfying this asymptotic equivariance condition converge in the preceding strong sense whenever they converge weakly (in law), provided the likelihood function of the sample is appropriately smooth. This smoothness of the likelihood is verified under mild conditions. Then, restricted to a broad class of models which include autoregressive models, a more easily verifiable condition implying the aforementioned asymptotic equivariance is derived and is shown to be satisfied by the estimators mentioned earlier. The methods used in the present paper are different from the usual method of characteristic functions; some indications of their possible wider scope are given. In Part II (Sections 6-8) the explosive autoregressive model is considered and a simple extension of the preceding result is applied to show that the least squares estimators converge in the preceding strong sense under suitable random or nonrandom normalization.