Estimation of a discrete monotone distribution

TL;DR

This paper compares three estimators for discrete monotone distributions, showing the maximum likelihood estimator outperforms others when the distribution has constant intervals, especially in uniform cases.

Contribution

It provides a theoretical comparison of empirical, rearrangement, and MLE estimators, highlighting the superiority of MLE in certain monotone distribution scenarios.

Findings

MLE strictly dominates rearrangement and empirical estimators with constant intervals.

Asymptotic risk of rearrangement estimator is y/(y+1) for uniform distributions.

MLE has asymptotic risk of order (log y)/(y+1) for uniform distributions.

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.