The variance floor for a mean built from a rate

Question

Waiting times X1, \dots, Xn are i.i.d. Exponential with rate \lambda, and the model is written in terms of \lambda. You want to estimate the mean waiting time, 	heta = g(\lambda) = 1/\lambda. Find the Cramer-Rao lower bound for an unbiased estimator of the mean 	heta, and check whether the sample mean attains it.

Accepted Answer

Fisher information for the rate. The exponential density is f(x;\lambda) = \lambda e^{-\lambda x}, so \log f = \log\lambda - \lambda x and the score is 1/\lambda - x. The information for one observation is I(\lambda) = -\mathbb{E}\Big[\frac{\partial^2}{\partial\lambda^2}\log f\Big] = -\mathbb{E}\Big[-\frac{1}{\lambda^2}\Big] = \frac{1}{\lambda^2}. Transform to the mean. With g(\lambda) = 1/\lambda, g'(\lambda) = -\frac{1}{\lambda^2}, \qquad [g'(\lambda)]^2 = \frac{1}{\lambda^4}. So 	ext{CRLB} = \frac{[g'(\lambda)]^2}{n\,I(\lambda)} = \frac{1/\lambda^4}{n/\lambda^2} = \frac{1}{n\lambda^2} = \…

The variance floor for a mean built from a rate

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