Adaptive smoothing for a penalized NPMLE of a non-increasing density

A penalized version of the well-known non-parametric maximum likelihood estimator of a non-increasing density f has been developed recently. The penalized version depends on a smoothing parameter, as well as the data. Here some adaptive choices of the smoothing parameter are considered. The asymptotically optimal smoothing parameter depends on f through beta = -1/2F(0)f'(0). In the adaptive procedures, estimates of beta are used to determine the smoothing parameter. Two such procedures are shown to be theoretically correct and practically viable.