How precise is the maximum likelihood rate estimate?

Question

Inter-arrival times X1, \dots, Xn are i.i.d. Exponential with rate \lambda. The maximum likelihood estimator is \hat\lambda = 1/\bar X. Find the large-sample distribution of \hat\lambda, and show it is asymptotically efficient. What is its asymptotic variance (the factor multiplying 1/n)?

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The general MLE result. Under regularity conditions the maximum likelihood estimator is consistent and asymptotically normal, reaching the Cramer-Rao floor in the limit: \sqrt{n}\,(\hat\lambda - \lambda) \xrightarrow{d} N\!\left(0, \frac{1}{I(\lambda)}\right), where I(\lambda) is the Fisher information per observation. Attaining 1/I(\lambda) is exactly what "asymptotically efficient" means. Fisher information. For the exponential, \log f = \log\lambda - \lambda x, and the second derivative in \lambda is -1/\lambda^2, so I(\lambda) = \frac{1}{\lambda^2}. Therefore \sqrt{n}\,(\hat\lambda - \lam…

How precise is the maximum likelihood rate estimate?

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