Consistency of the frequency polygon estimators of density function and density mode under m -END samples

The frequency polygon estimation, which is based on the histogram technique, has a similar convergence rate as those of modern non negative kernel estimators and the advantage of computational simplicity. This article will study the uniformly complete consistency and the rate of uniformly complete consistency of frequency polygon estimation for m-END samples under some general conditions, which improve and extend the corresponding ones in the literature. As an application, the uniformly complete consistency and the corresponding rate of density mode estimator are also obtained. Some numerical simulations are also carried out to support the theoretical results.