Speed of convergence for estimators of finite translation mixture distributions

A suitably tapered Fourier transform is used to obtain a simple and explici t construction of estimators for the c.d.f.'s of finite translation mixture s. An a.s. speed of L-1-convergence is derived for these c.d.f.'s which cor roborates the lower bound for such rates recently derived in Chen (1995) fo r arbitrary mixtures. In the translation case the rates are determined by t he tail behavior of the Fourier transform of the kernel of the given locati on family, i.e., by the degree of ill-posedness of the mixture problem.