Robust Estimation of Sparse, High Dimensional Time Series with Polynomial Tails
多项式尾部下稀疏高维时间序列的稳健估计AI 解读 ↓
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多项式尾部下稀疏高维时间序列的稳健估计
- 研究问题
- 在高维、时间依赖与重尾噪声同时存在时，如何一致估计稀疏 VAR 参数，并量化时间依赖带来的估计代价？
- 方法与思路
- 作者采用稳健惩罚估计方法，以重尾依赖过程的集中不等式为技术工具，处理同方差或异方差噪声驱动的稀疏模型。
- 材料陈述的贡献
- 摘要称，推导了参数估计的最优一致性速率及有限样本界，速率与独立样本情形对应，但包含时间依赖造成的额外因子。
- 阅读关注点
- 应核对尾部与矩条件、稳定性和依赖限制、稳健损失与惩罚的具体选择，以及最优速率所依据的比较标准或下界。
- 阅读边界
- 所给版本为 arXiv 预印本，同行评审或期刊接收未核实；摘要未列具体矩条件、误差界及调参规则，不能据此判断普适性。
原文摘要
High dimensional Vector Autoregressions (VAR) have received a lot of interest recently due to novel applications in health, engineering, finance and the social sciences. Three issues arise when analyzing VAR's: (a) The high dimensional nature of the model in the presence of many time series that poses challenges for consistent estimation of its parameters; (b) the presence of temporal dependence introduces additional challenges for theoretical analysis of various estimation procedures; (b) the presence of heavy tails in a number of applications. Recent work, e.g. [Basu and Michailidis, 2015],[Kock and Callot,2015], has addressed consistent estimation of sparse high dimensional, stable Gaussian VAR models based on an $\ell_1$ LASSO procedure. Further, the rates obtained are optimal, in the sense that they match those for iid data, plus a multiplicative factor (which is the "price" paid) for temporal dependence. However, the third issue remains unaddressed in extant literature. This paper extends existing results in the following important direction: it considers consistent estimation of the parameters of sparse high dimensional VAR models driven by heavy tailed homoscedastic or heteroskedastic noise processes (that do not possess all moments). A robust penalized approach (e.g., LASSO) is adopted for which optimal consistency rates and corresponding finite sample bounds for the underlying model parameters are obtain that match those for iid data, albeit paying a price for temporal dependence. The theoretical results are illustrated on VAR models and also on other popular time series models. Notably, the key technical tool used, is a single concentration bound for heavy tailed dependent processes.