Localising change points in piecewise polynomials of general degrees

Abstract

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 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.