### Estimation of a discrete monotone distributionEstimation of a discrete monotone distribution

Access Restriction
Open

 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