When Cramer-Rao breaks, the uniform maximum

Question

Let X1, \dots, Xn be i.i.d. Uniform(0, 	heta). A naive Cramer-Rao calculation suggests a variance floor of order 	heta^2/n. Show that an unbiased estimator based on the sample maximum beats it, and explain why the bound does not bind.

Accepted Answer

The naive bound. For Uniform(0,	heta) the density is f(x;	heta) = 1/	heta on [0,	heta], so \log f = -\log	heta and the "score" is -1/	heta, giving a formal Fisher information I(	heta) = 1/	heta^2 and a naive floor 	ext{"CRLB"} = \frac{1}{n\,I(	heta)} = \frac{	heta^2}{n}. The estimator that beats it. Let M = X{(n)} = \maxi Xi. Its CDF is P(M \le m) = (m/	heta)^n on [0,	heta], so E[M] = \frac{n}{n+1}	heta, \qquad \operatorname{Var}(M) = \frac{n}{(n+1)^2(n+2)}	heta^2. Correct the bias with \hat	heta = 	frac{n+1}{n}M, which is unbiased, E[\hat	heta] = 	heta. Its variance is \op…

When Cramer-Rao breaks, the uniform maximum

Your answer

More Statistics questions