一般次数分段多项式中变点的局部化
Localising change points in piecewise polynomials of general degrees
摘要：
本文研究了具有独立次高斯噪声和分段多项式均值的一元随机变量序列。基础多项式允许是任意的但固定的次数。允许所有其他模型参数根据样本量而变化。我们提出了一个两步估计过程的基础上的$\ell_0$ -惩罚，并提供上界的定位误差。我们补充这些结果，推导出一个全球信息理论的下界，这表明我们的两步估计是近极大极小率最优的。我们还表明，我们的估计享有接近最佳的自适应性能，通过实现个人的定位误差取决于在个别的变化点的基础信号的平滑程度。此外，在一个特殊的平滑约束下，我们提供了一个极小极大下界的本地化错误。这个下界与多项式阶数无关，并且比全局极小极大下界更尖锐。
英文摘要：
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.
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