Consistency of penalized maximum likelihood estimation for multidimensional change-in-velocity detection
Abstract
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 ρk(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→∞. The maximum changepoint-location error is Oℙ{(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.
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BibTeX
@misc{do2026consistency,
title = {{Consistency of penalized maximum likelihood estimation for multidimensional change-in-velocity detection}},
author = {Linh Do and Dat Do and Scott A. McKinley},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04191},
eprint = {2610.04191},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04191},
}
RIS
TY - UNPB
TI - Consistency of penalized maximum likelihood estimation for multidimensional change-in-velocity detection
AU - Linh Do
AU - Dat Do
AU - Scott A. McKinley
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04191
UR - https://arxiv.org/abs/2610.04191
AB - 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→∞. The maximum changepoint-location error is O_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.
ER -