Cramer-Rao bound for an exponential mean

Question

Let X1, \dots, Xn be i.i.d. Exponential with mean 	heta (density f(x;	heta) = 	frac1	heta e^{-x/	heta} for x \ge 0), for example inter-trade waiting times. Compute the Fisher information for 	heta, state the Cramer-Rao bound, and determine whether \hat{	heta} = \bar{X} is efficient.

Accepted Answer

Score and Fisher information. The log-density of one observation is \log f(x; 	heta) = -\log	heta - \frac{x}{	heta}. Differentiating with respect to 	heta gives the score s(x; 	heta) = -\frac{1}{	heta} + \frac{x}{	heta^2} = \frac{x - 	heta}{	heta^2}. (Sanity check: E[s] = 	frac{E[X] - 	heta}{	heta^2} = 0.) The Fisher information is the variance of the score, using \operatorname{Var}(X) = 	heta^2: I(	heta) = \operatorname{Var}\!\left(\frac{X - 	heta}{	heta^2}ight) = \frac{\operatorname{Var}(X)}{	heta^4} = \frac{	heta^2}{	heta^4} = \frac{1}{	heta^2}. The bound vs the sampl…

Cramer-Rao bound for an exponential mean

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