Electronic Journal of Statistics

Estimation of a discrete monotone distribution

Hanna K. Jankowski and Jon A. Wellner

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We study and compare three estimators of a discrete monotone distribution: (a) the (raw) empirical estimator; (b) the “method of rearrangements” estimator; and (c) the maximum likelihood estimator. We show that the maximum likelihood estimator strictly dominates both the rearrangement and empirical estimators in cases when the distribution has intervals of constancy. For example, when the distribution is uniform on {0,,y}, the asymptotic risk of the method of rearrangements estimator (in squared 2 norm) is y/(y+1), while the asymptotic risk of the MLE is of order (logy)/(y+1). For strictly decreasing distributions, the estimators are asymptotically equivalent.

Article information

Electron. J. Statist., Volume 3 (2009), 1567-1605.

First available in Project Euclid: 7 January 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E20: Asymptotic distribution theory 62F12: Asymptotic properties of estimators
Secondary: 62G07: Density estimation 62G30: Order statistics; empirical distribution functions 62C15: Admissibility 62F20

Maximum likelihood monotone mass function rearrangement rate of convergence limit distributions nonparametric estimation shape restriction Grenander estimator


Jankowski, Hanna K.; Wellner, Jon A. Estimation of a discrete monotone distribution. Electron. J. Statist. 3 (2009), 1567--1605. doi:10.1214/09-EJS526. https://projecteuclid.org/euclid.ejs/1262876706

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