Polygonal smoothing of the empirical distribution function

We present two families of polygonal estimators of the distribution function: the first family is based on the knowledge of the support while the second addresses the case of an unknown support. Polygonal smoothing is a simple and natural method for regularizing the empirical distribution function (F_n) but its properties have not been studied deeply. First, consistency and exponential type inequalities are derived from well-known convergence properties of (F_n). Then, we study their mean integrated squared error (MISE) and we establish that polygonal estimators may improve the MISE of (F_n). We conclude by some numerical results to compare these estimators globally, and also together with the integrated kernel distribution estimator.