This paper concerns about the limiting distributions of change point estimators, in a high-dimensional linear regression time series context, where a regression object $(y_t, X_t) \in \mathbb{R} \times \mathbb{R}^p$ is observed at every time point $t \in \{1, \ldots, n\}$. At unknown time points, called change points, the regression coefficients change, with the jump sizes measured in $\ell_2$-norm. We provide limiting distributions of the change point estimators in the regimes where the minimal jump size vanishes and where it remains a constant. We allow for both the covariate and noise sequences to be temporally dependent, in the functional dependence framework, which is the first time seen in the change point inference literature. We show that a block-type long-run variance estimator is consistent under the functional dependence, which facilitates the practical implementation of our derived limiting distributions. We also present a few important byproducts of our analysis, which are of their own interest. These include a novel variant of the dynamic programming algorithm to boost the computational efficiency, consistent change point localisation rates under temporal dependence and a new Bernstein inequality for data possessing functional dependence. Extensive numerical results are provided to support our theoretical results. The proposed methods are implemented in the R package \texttt{changepoints} \citep{changepoints_R}.
翻译:本文关注在高维线性回归时间序列背景下变化点估计值的有限分布。 在高维线性回归时间序列背景下, 每个时间点都会观测回归对象$(y_t, X_t)\ in\mathbb{R}\ times\ mathb{R}\ times\ times $t\in\\ _1,\ldb{r\\\\\\\\\\\美元。 在未知的时间点, 被称为变化点, 回归系数会变化, 以 $\ell_2$-norm 测量跳点大小。 在最小跳跃尺寸消失且保持恒定的系统中, 我们提供的变化点估计值的分布。 在功能依赖框架内, 我们允许共变和噪声序列在时间上都取决于时间。 我们显示, 区型长期差异估计值在功能依赖下是一致的, 这有利于实际执行我们限制分布的源值。 我们还展示了少数重要的分析产品, 在最小跳跃度大小的系统性调整中, 这是他们自己在运行的理论性分析结果的模型分析中, 包括了我们不断增长的模型分析结果的模型分析的模型分析结果。 这些变变数 。 包括了我们自我的模型的模型的模型的模型的模型的模型的模型的模型的模型的计算。 这些变变变式 。 这些是 。