Tests for normality based on density estimators of convolutions

Recent results show that densities of convolutions can be estimated by local U-statistics at the root-nn rate in various norms. Motivated by this and the fact that convolutions of normal densities are normal, we introduce new tests for normality which use as test statistics weighted L1L1-distances between the standard normal density and local U-statistics based on standardized observations. We show that such test statistics converge at the root-nn rate and determine their limit distributions as functionals of Gaussian processes. We also address a choice of bandwidth. Simulations show that our tests are competitive with other tests of normality.