The Annals of Statistics

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

Wolfgang Polonik

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Based on empirical Lévy-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 to higher dimensions and arbitrary order restrictions. 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 provides a new point of view to ML-density estimation from which one gains additional insight to this problem, as demonstrated in the present paper.

Article information

Ann. Statist., Volume 26, Number 5 (1998), 1857-1877.

First available in Project Euclid: 21 June 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07
Secondary: 62A10 62G30 62G20

Empirical processes excess mass Grenander density estimator level set estimation least concave majorant minimum volume sets nonparametric maximum likelihood estimation


Polonik, Wolfgang. The silhouette, concentration functions and ML-density estimation under order restrictions. Ann. Statist. 26 (1998), no. 5, 1857--1877. doi:10.1214/aos/1024691360.

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