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.
翻译:在本文中,我们关注的是一系列单向随机变量的序列,这些变量有零碎的多元度手段和独立的亚加西噪音。 基础的多元度允许有任意度,但有固定度。 所有其他模型参数允许根据样本大小而有所不同。 我们提议了一个基于 $\ ell_ 0$ 的处罚的两步估计程序, 并提供本地化错误的上限。 我们通过得出一个全球信息- 理论下限来补充这些结果, 该下限显示我们的两步估计值几乎是迷你式速率最佳的。 我们还显示, 我们的估算值通过在基本信号的单个变化点的平滑度达到个别的本地化差错而享有接近最佳的适应性表现。 此外, 在特殊的平滑度限制下, 我们提供了一种小负值在本地化错误上的最小值下限。 这个下限是独立于多元顺序的, 并且比全球小微缩度下限要快。