Estimation of a discrete monotone distribution

Abstract

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 ${\ell }_2$ norm) is $y/(y+1)$, while the asymptotic risk of the MLE is of order $(\log y)/(y+1)$. For strictly decreasing distributions, the estimators are asymptotically equivalent.

39 pages. See also http://www.stat.washington.edu/www/research/reports/2009/ http://www.stat.washington.edu/jaw/RESEARCH/PAPERS/available.html

References in corpus (4)

• Improving Point and Interval Estimates of Monotone Functions by Rearrangement
• Nonparametric inference for competing risks current status data with continuous, discrete or grouped observation times
• Limit properties of the monotone rearrangement for density and regression function estimation
• On the Grenander estimator at zero

Cited by in corpus (1)

• Binomial Mixture Model With U-shape Constraint