How biased is the sample maximum?

Question

Serial numbers are uniform between 0 and an unknown maximum $\theta$ . From $n = 9$ captured numbers you use the largest, $M$ , to estimate $\theta$ . Take $\theta = 20$ .

What is the bias of $M$ as an estimator of $\theta$ ?

Accepted Answer

The idea. Every observation is at most 	heta, so their maximum is at most 	heta too, and on average strictly below it. The estimator systematically undershoots, which is a pure bias. Compute the expected maximum. For n uniform draws on (0,	heta), \mathbb{E}[M] = \frac{n}{n+1}	heta. Intuitively, the n points cut the interval into n+1 gaps of equal average size 	heta/(n+1), and the top gap, from M up to 	heta, is the piece the maximum misses. The bias. 	ext{Bias} = \mathbb{E}[M] - 	heta = -\frac{	heta}{n+1} = -\frac{20}{10} = -2. The maximum is, on average, 2 units below the true ceili…

How biased is the sample maximum?

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