A hand-sized uniform ceiling estimate

Question

You have four draws from a Uniform(0, 	heta) distribution: 3, 7, 5, 9. The mean of this distribution is 	heta/2. Use the method of moments to estimate 	heta.

Accepted Answer

Step 1: the sample average. \bar X = \frac{3 + 7 + 5 + 9}{4} = \frac{24}{4} = 6. Step 2: match the mean. The mean is 	heta/2, so set 	heta/2 = \bar X = 6: \hat	heta = 2\bar X = 12. A small surprise. The largest value you actually saw is 9, yet the estimate is 12. That is fine: the method only uses the average, and on average the true ceiling is twice the mean. (A sharper estimator would lean on that maximum, but that is a story for a harder problem.) Takeaway. For a uniform starting at zero, double the average: \hat	heta = 2\bar X = 12.

A hand-sized uniform ceiling estimate

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