An Alternative Unimodal Density Estimator with a Consistent Estimate of the Mode

The traditional maximum likelihood unimodal density estimator (Grenander (1956)) pieces together two isotonic density estimators at a known mode. It is discontinuous at the mode, and does not directly adapt to the case of unknown mode. This paper presents an alternative unimodal density estimator in the form of a generalized isotonic regression on a partial order which is continuous at the mode, and moves easily to the case of unknown mode. A penalized version is introduced to control the spiking at the mode, and is proved to be consistent everywhere. It is shown that the penalized estimator also provides a consistent estimate of the mode. Simulation results compare the penalized estimator to other nonparametric estimators in the literature in terms of Hellinger distance, the squared error loss of the estimate of the mode, and the height at the mode. Two important advantages of the new estimator are that it provides the density estimate and the mode estimate simultaneously, and that it is “fully automatic,” that is, no pre-grouping or bounds on density height are necessary to prevent spiking.