On Convergence of Densities of Translation and Scale Statistics

ABSTRAC1 Convergence of densities leads directly to convergence of their distribution functions via Scheffe's theorem. This paper is concerned with the converse: what are sufficient conditions to obtain convergence of densities from convergence of distribution functions? A general lemma is given and results are obtained for translation and scale statistics with applications to bootstrap estimates of densities and convergence of Bayes posteriors.