Estimating both ends of a uniform

Question

You observe X1, \dots, Xn i.i.d. Uniform(a, b) with both endpoints unknown. Find the MLE of (a, b), show the endpoints are biased, and correct them.

Accepted Answer

The MLE. The density is \frac{1}{b-a} on [a, b], so the likelihood is L(a, b) = \frac{1}{(b - a)^n}\,\mathbf{1}\{a \le \mini Xi\}\,\mathbf{1}\{b \ge \maxi Xi\}. To maximize, make the width b - a as small as the data allows: push a up to the sample minimum and b down to the sample maximum. \hat{a}{	ext{MLE}} = X{(1)}, \qquad \hat{b}{	ext{MLE}} = X{(n)}. The bias. Both estimators sit inside the true interval. Using the order-statistic means for a uniform, \mathbb{E}[X{(1)}] = a + \frac{b - a}{n + 1}, \qquad \mathbb{E}[X{(n)}] = b - \frac{b - a}{n + 1}. So the observed range R = X{(n)} - X{(1)…

Estimating both ends of a uniform

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