Localising change points in piecewise polynomials of general degrees

07/20/2020

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by   Yi Yu, et al.

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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. We propose a two-step estimation procedure based on the ℓ_0-penalisation and provide upper bounds on the localisation error. We complement these results by deriving 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.