Unbiased but inefficient, the uniform MoM

Question

For a Uniform(0, 	heta) sample, the method of moments gives \hat	heta = 2\bar X. Show that \hat	heta = 2\bar X is unbiased, explain how it can nonetheless produce a value below the largest observation, and compare how fast its error shrinks to that of the maximum-based estimator.

Accepted Answer

Unbiased. Each observation has mean 	heta/2, so the sample average has mean 	heta/2, and doubling gives \mathbb{E}[\hat	heta] = 2\,\mathbb{E}[\bar X] = 2 \cdot \frac{	heta}{2} = 	heta. On average the estimate lands exactly on the truth. That is a genuine virtue of the method of moments here. Yet it can be embarrassing. The estimate uses only the average, so it can fall below a value you actually observed. Suppose you draw 1, 2, 9: the average is 4, giving \hat	heta = 8, but you already saw a 9, which is impossible under 	heta = 8. Being right on average does not stop a single estimate …

Unbiased but inefficient, the uniform MoM

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