Overlapping Batch Confidence Intervals on Statistical Functionals Constructed from Time Series: Application to Quantiles, Optimization, and Estimation
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时间序列统计泛函的重叠分批置信区间：分位数、优化与估计应用
- 研究问题
- 对于平稳时间序列构造的统计泛函，如何利用重叠数据批次刻画依赖并构造置信区间，改善覆盖率等有限样本表现？
- 方法与思路
- 作者以重叠批次估计方差参数，借助点估计量的中心极限定理推导学生化统计量的 OB 极限，并用维纳过程泛函确定临界值。
- 材料陈述的贡献
- 摘要称方法适用于分位数、M 估计及 CVaR 等泛函；数值实验显示，大重叠批次配合相应临界值常能改善区间质量。
- 阅读关注点
- 应核对中心及函数型极限定理条件、批次大小和重叠比例的渐近设定，以及各 OB 极限对应的统计量与临界值计算。
- 阅读边界
- 这是 arXiv 预印本，同行评审及期刊接收未获核实；摘要未列出完整依赖条件，分布无关极限和数值优势不构成任意样本保证。
原文摘要
We propose a general purpose confidence interval procedure (CIP) for statistical functionals constructed using data from a stationary time series. The procedures we propose are based on derived distribution-free analogues of the $χ^2$ and Student's $t$ random variables for the statistical functional context, and hence apply in a wide variety of settings including quantile estimation, gradient estimation, M-estimation, CVAR-estimation, and arrival process rate estimation, apart from more traditional statistical settings. Like the method of subsampling, we use overlapping batches of time series data to estimate the underlying variance parameter; unlike subsampling and the bootstrap, however, we assume that the implied point estimator of the statistical functional obeys a central limit theorem (CLT) to help identify the weak asymptotics (called OB-x limits, x=I,II,III) of batched Studentized statistics. The OB-x limits, certain functionals of the Wiener process parameterized by the size of the batches and the extent of their overlap, form the essential machinery for characterizing dependence, and consequently the correctness of the proposed CIPs. The message from extensive numerical experimentation is that in settings where a functional CLT on the point estimator is in effect, using \emph{large overlapping batches} alongside OB-x critical values yields confidence intervals that are often of significantly higher quality than those obtained from more generic methods like subsampling or the bootstrap. We illustrate using examples from CVaR estimation, ARMA parameter estimation, and NHPP rate estimation; R and MATLAB code for OB-x critical values is available at~\texttt{web.ics.purdue.edu/~pasupath/}.