Error of the maximum with five uniform draws

Question

Reaction times are uniform between 0 and an unknown maximum 	heta. From n = 5 independent readings you estimate 	heta by the largest reading M. Take 	heta = 12. What is the mean squared error of M?

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The idea. The maximum is always below the true ceiling, so it is biased low; its MSE combines that bias with its variance. There is a closed form. Use the MSE formula. For the maximum of n uniform draws on (0,	heta), variance plus squared bias simplify to 	ext{MSE}(M) = \frac{2	heta^2}{(n+1)(n+2)}. Plug in n = 5, 	heta = 12: 	ext{MSE}(M) = \frac{2 \cdot 144}{6 \cdot 7} = \frac{288}{42} = \frac{48}{7} \approx 6.86. More data helps only slowly here: going from 4 to 5 draws barely moved the error, because the maximum's accuracy improves like 1/n, not like the 1/\sqrt{n} you might expect fro…

Error of the maximum with five uniform draws

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