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Author Jankowski, Hanna K. ♦ Wellner, Jon A.
Source arXiv.org
Content type Text
File Format PDF
Date of Submission 2009-10-16
Language English
Subject Domain (in DDC) Natural sciences & mathematics ♦ Mathematics ♦ Probabilities & applied mathematics
Subject Keyword Mathematics - Statistics Theory ♦ Mathematics - Probability ♦ 62G05 ♦ 62G07 (primary) ♦ 62G20 ♦ 62G30 (secondary) ♦ math ♦ stat
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.
Educational Use Research
Learning Resource Type Article
Page Count 39


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