UniformL1-distance large deviations in nonparametric density estimation

In this paper, we are concerned with uniform large deviations probabilities for theL1-error in kernel density estimation. Several results are stated taking into account uniformity over classes of density functions as well as over families of kernels and intervals of smoothing parameters. Some limits are well-identified and are universal in the sense that they do not depend neither on the distribution of the observations nor on the estimation kernel. Our results allow to compare performances of the test based on theL1-deviation of the kernel density estimator to that of the Kolmogorov-Smirnov test when testing goodness-of-fit of composite hypotheses.