Optimal Adaptive Estimation on R or R^+of the Derivatives of a Density

In this paper, we consider the problem of estimating the d -th order derivative f⁽ᵈ⁾ of a density f , relying on a sample of n i.i.d. observations X₁,,Xₙ with density f supported on R or R⁺ . We propose projection estimators defined in the orthonormal Hermite or Laguerre bases and study their integrated L² -risk. For the density f belonging to regularity spaces and for a projection space chosen with adequate dimension, we obtain rates of convergence for our estimators, which are optimal in the minimax sense. The optimal choice of the projection space depends on unknown parameters, so a general data-driven procedure is proposed to reach the bias-variance compromise automatically. We discuss the assumptions and the estimator is compared to the one obtained by simply differentiating the density estimator. Simulations are finally performed. They illustrate the good performances of the procedure and provide numerical comparison of projection and kernel estimators