# UW STAT 561 - Estimation of a Discrete monotone Distribution (39 pages)

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## Estimation of a Discrete monotone Distribution

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## Estimation of a Discrete monotone Distribution

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Lecture Notes

Pages:
39
School:
University of Washington
Course:
Stat 561 - Special Topics in Applied Statistics

Unformatted text preview:

Estimation of a Discrete Monotone Distribution Hanna K Jankowski and Jon A Wellner October 19 2009 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 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 1 Introduction This paper is motivated in large part by the recent surge of acitivity concerning method of rearrangement estimators for nonparametric estimation of monotone functions see for example Fouge res 1997 Dette and Pilz 2006 Dette et al 2006 Chernozhukov et al 2009 and Anevski and Fouge res 2007 Most of these authors study continuous settings and often start with a kernel type estimator of the density which involves choices of a kernel and of a bandwidth Our goal here is to investigate method of rearrangement estimators and compare them to natural alternatives including the maximum likelihood estimators with and without the assumption of monotonicity in a setting in which there is less ambiguity in the choice of an initial or basic estimator namely the setting of estimation of a monotone decreasing mass function on the non negative integers N 0 1 2 Suppose that p px x N is a probability mass function i e px 0 for all x N and x N px 1 Our primary interest here is in the situation in which p is monotone decreasing px px 1 for all x N The three estimators of p we study are a the raw empirical estimator b the method of rearrangement estimator c the maximum likelihood estimator 1 Notice that the empirical estimator is also the

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