Convergence in the Hausdorff metric of estimators of irregular densities, using Fourier-Cesàro approximation

The problem of estimating a density which is allowed to have discontinuities of the first kind is considered. The usual Fourier-type estimator is based on the Dirichlet or sine kernel and is not suitable to eliminate the Gibbs phenomenon. Fourier-Cesàro approximation yields the Fejér kernel which is the square of the sine function. Density estimators based on the Fejér kernel do control the Gibbs phenomenon. Integral metrics are not sufficiently sensitive to properly assess the performance of estimators of irregular signals. Therefore, we use the Hausdorff distance between the extended, closed, graphs of estimator and estimand, and derive an a.s. speed of convergence of this distance.