The best you can do, Cramér–Rao for an exponential

Question

For n independent draws from an exponential distribution, you want to estimate the mean lifetime 	heta = 1/\lambda. Compute the Cramér–Rao lower bound on the variance of any unbiased estimator of 	heta, and check whether the sample mean attains it.

Accepted Answer

Set up the log-density in 	heta. Writing the density as f(x;	heta) = 	frac{1}{	heta}e^{-x/	heta}, one observation has \ln f = -\ln	heta - \frac{x}{	heta}, \qquad \frac{\partial \ln f}{\partial	heta} = -\frac{1}{	heta} + \frac{x}{	heta^2}. Fisher information. Differentiate again: \dfrac{\partial^2 \ln f}{\partial	heta^2} = \dfrac{1}{	heta^2} - \dfrac{2x}{	heta^3}. Take -\mathbb{E}[\cdot] using \mathbb{E}[X] = 	heta: I1(	heta) = -\left(\frac{1}{	heta^2} - \frac{2	heta}{	heta^3}ight) = -\left(\frac{1}{	heta^2} - \frac{2}{	heta^2}ight) = \frac{1}{	heta^2}. The bound. For …

The best you can do, Cramér–Rao for an exponential

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