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On change-point estimation under Sobolev sparsity. (English) Zbl 1439.62152

Summary: In this paper, we consider the estimation of a change-point for possibly high-dimensional data in a Gaussian model, using a maximum likelihood method. We are interested in how dimension reduction can affect the performance of the method. We provide an estimator of the change-point that has a minimax rate of convergence, up to a logarithmic factor. The minimax rate is in fact composed of a fast rate – dimension-invariant – and a slow rate – increasing with the dimension. Moreover, it is proved that considering the case of sparse data, with a Sobolev regularity, there is a bound on the separation of the regimes above which there exists an optimal choice of dimension reduction, leading to the fast rate of estimation. We propose an adaptive dimension reduction procedure based on Lepski’s method and show that the resulting estimator attains the fast rate of convergence. Our results are then illustrated by a simulation study. In particular, practical strategies are suggested to perform dimension reduction.

MSC:

62H30 Classification and discrimination; cluster analysis (statistical aspects)
62G05 Nonparametric estimation
62G10 Nonparametric hypothesis testing
62C20 Minimax procedures in statistical decision theory

Software:

wbs; EBayesThresh
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References:

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