Rao-Blackwellizing a crude guess for a uniform ceiling

Question

You have i.i.d. observations X1, \dots, Xn from Uniform(0, 	heta) with unknown ceiling 	heta. A crude unbiased estimator uses only the first observation: T = 2X1, which works because \mathbb{E}[2X1] = 2\cdot(	heta/2) = 	heta. Rao-Blackwellize T by conditioning on the sufficient statistic S = X{(n)} = \maxi Xi, and identify the improved estimator.

Accepted Answer

Setup. The sample maximum X{(n)} is sufficient for 	heta. Rao-Blackwell tells us T^\star = \mathbb{E}[2X1 \mid X{(n)}] is unbiased with variance no larger than that of 2X1. Condition on the maximum. Fix X{(n)} = m. The first observation X1 is the maximum with probability 1/n (in which case X1 = m), and otherwise, with probability (n-1)/n, it is one of the non-maximal points, which are uniform on [0, m] with mean m/2. So \mathbb{E}[X1 \mid X{(n)} = m] = \frac{1}{n}\,m + \frac{n-1}{n}\cdot\frac{m}{2} = \frac{2m + (n-1)m}{2n} = \frac{(n+1)m}{2n}. Therefore T^\star = \mathbb{E}[2X1 \mid X{(n)}] …

Rao-Blackwellizing a crude guess for a uniform ceiling

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