论文标题
一般程度的分段多项式中的本地变化点
Localising change points in piecewise polynomials of general degrees
论文作者
论文摘要
在本文中，我们关注的是一系列具有分段多项式均值和独立的高斯噪声的单变量随机变量。允许基础多项式为任意但固定的度。所有其他模型参数都可以根据样本量而变化。 我们提出了一个基于$ \ ell_0 $ - 二元化的两步估计过程，并在本地化错误上提供上限。我们通过得出全球信息理论下限来补充这些结果，这表明我们的两步估计器几乎是最小的速率 - 最佳速率。我们还表明，我们的估计器通过在基础信号的各个变更点处的平稳性水平达到平稳性，通过达到单个定位错误，享有几乎最佳的自适应性能。另外，在特殊的平滑度约束下，我们在本地化误差上提供了最小的下限。该下限独立于多项式订单，并且比全局最小值下限更加清晰。
In this paper we are concerned with a sequence of univariate random variables with piecewise polynomial means and independent sub-Gaussian noise. The underlying polynomials are allowed to be of arbitrary but fixed degrees. All the other model parameters are allowed to vary depending on the sample size. We propose a two-step estimation procedure based on the $\ell_0$-penalisation and provide upper bounds on the localisation error. We complement these results by deriving a global information-theoretic lower bounds, which show that our two-step estimators are nearly minimax rate-optimal. We also show that our estimator enjoys near optimally adaptive performance by attaining individual localisation errors depending on the level of smoothness at individual change points of the underlying signal. In addition, under a special smoothness constraint, we provide a minimax lower bound on the localisation errors. This lower bound is independent of the polynomial orders and is sharper than the global minimax lower bound.
