When maximum likelihood and method of moments disagree

Question

Suppose your data are n independent draws from a Uniform(0, 	heta) distribution, and you want to estimate the upper bound 	heta. Derive both the method-of-moments and the maximum likelihood estimators, and explain how they can disagree, sometimes dramatically.

Accepted Answer

Method of moments (MoM). The mean of Uniform(0,	heta) is 	heta/2. Setting the sample mean equal to the theoretical mean, \bar x = 	heta/2, gives \hat	heta{	ext{MoM}} = 2\bar x. Maximum likelihood (MLE). The density is 1/	heta on [0,	heta] and zero above it, so the likelihood 	heta^{-n} is only valid when 	heta \ge \maxi xi; it decreases in 	heta, so it is largest at the smallest admissible value: \hat	heta{	ext{MLE}} = \maxi xi. How they disagree. Both are consistent, but they trade off differently. The MLE is biased low with \mathbb{E}[\max] = 	frac{n}{n+1}	heta, yet has small …

When maximum likelihood and method of moments disagree

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