发表机构
Duke University School of Medicine; University of Chicago; Tulane University（杜克大学医学院; 芝加哥大学; 杜兰大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了CPLASS方法中惩罚最大似然估计量在高频数据下对分段线性轨迹变点检测的一致性，给出了变点位置误差的收敛速率。
AI 中文摘要
我们建立了CPLASS（一种用于检测$d$维时间序列数据中速度变化的方法）所基于的惩罚最大似然估计量的统计一致性。信号被建模为连续分段线性轨迹，并观测到独立的高斯噪声。与经典的均值变化模型不同，相邻分段之间的连续性耦合了它们的参数，从而阻碍了标准分割论证的直接应用。在紧致参数空间、最小段长和速度跳跃条件以及增强的Schwarz信息准则惩罚$\rho_k(\log n)^\gamma$（其中$\gamma>1$）下，我们证明了在高频极限$n\to\infty$下，估计的段数和所有变点位置具有联合一致性。最大变点位置误差为$O_{\mathbb{P}}\{(\log n/n)^{1/2}\}$。证明首先利用经验过程理论建立每个固定模型大小下拟合信号、方差和似然的收敛性。然后，它将过拟合和欠拟合论证与两阶段几何定位分析相结合，得出初始的$(\log n/n)^{1/3}$速率，并通过利用局部两段连续分段线性结构加以改进。
英文摘要
We establish statistical consistency for the penalized maximum-likelihood estimator underlying CPLASS, a method for detecting changes in velocity in $d$-dimensional time-series data. The signal is modeled as a continuous piecewise-linear trajectory observed with independent Gaussian noise. Unlike classical change-in-mean models, continuity across adjacent segments couples their parameters and prevents direct application of standard segmentation arguments. Under compact parameter spaces, minimum segment-length and velocity-jump conditions, and a strengthened Schwarz information criterion penalty $ρ_k(\log n)^γ$ with $γ>1$, we prove joint consistency of the estimated number of segments and all changepoint locations in the high-frequency regime $n\to\infty$. The maximum changepoint-location error is $O_{\mathbb{P}}\{(\log n/n)^{1/2}\}$. The proof first uses empirical-process theory to establish convergence of the fitted signal, variance, and likelihood for each fixed model size. It then combines overfitting and underfitting arguments with a two-stage geometric localization analysis, yielding an initial $(\log n/n)^{1/3}$ rate that is sharpened by exploiting the local two-segment continuous piecewise-linear structure.
Comments30 pages, 1 figure
