  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.
