Large and moderate deviations principles for kernel estimation of a multivariate density and its partial derivatives

This paper studies the large deviations behaviour of the kernel estimator of a probability density f, by considering the case when the kernel takes negative values. It establishes large and moderate deviations principles for the kernel estimators of the partial derivatives of f. The estimators of the derivatives exhibit a quadratic behaviour for both the large and the moderate deviations scales, whereas for the density estimator there is a classical gap between the large deviations and the moderate deviations asymptotics.