Minimax wavelet estimation for multisample heteroscedastic nonparametric regression

Abstract

The problem of estimating the baseline signal from multisample noisy curves is investigated. We consider the functional mixed effects model, and we suppose that the functional fixed effect belongs to the Besov class. This framework allows us to model curves that can exhibit strong irregularities, such as peaks or jumps for instance. The lower bound for the L 2 minimax risk is provided, as well as the upper bound of the minimax rate, that is derived by constructing a wavelet estimator for the functional fixed effect. Our work constitutes the first theoretical functional results in multisample non parametric regression. Our approach is illustrated on realistic simulated datasets as well as on experimental data.

Nonparametric Pointwise Estimation for a Regression Model with Multiplicative Noise

“…holds almost everywhere on ℝ d . Besov spaces are important in theory and applications, which contain Hölder and L 2 lebesgue spaces as special examples [9][10][11]. The next lemma provides equivalent definitions of Besov space [12].…”