The silhouette, concentration functions, and ML-density estimation under order restrictions

Based on empirical Levy-type concentration functions a new graphical representation of the ML-density estimator under order restrictions is given. This representation generalizes the well-known representation of the Grenander estimator of a monotone density as the slope of the least concave majorant of the empirical distribution function. From the given representation it follows that a density estimator called silhouette which arises naturally out of the excess mass approach is the ML-density estimator under order restrictions. This fact brings in several new aspects to ML-density estimation under order restrictions. Especially, it provides new methods for deriving asymptotic results for ML-density estimators under order restrictions based on empirical process theory.