Past exam of the options II course of the University of Oxford / 2004 / O / O5 / d / Solution

Put $S={\sum }_i{X}_i$. Independence and the exponential sampling model give the gamma distribution with densityFor $n>k$, substitution $u=\lambda s$ yieldsAt $k=1$, $n\ge 2$, this gives $\mathbb{E}({\bar{X}}^{-1})=n\lambda /(n-1)$. Thereforeis an unbiased estimator of the rate. For $n>2$, the same formula with $k=2$ givesFor the source-allowed value $n=2$, the first moment is still finite, but the second moment contains an integral proportional to ${\int }_0^1{s}^{-1}{e}^{-\lambda s}ds$ and diverges. Hence the estimator has infinite variance when $n=2$; the finite formula applies only to $n>2$.

The sample score is $n/\lambda -S$, and its Fisher information is $I_n(\lambda )=n/{\lambda }^2$. The Cramér-Rao lower bound for unbiased rate estimation is thus ${\lambda }^2/n$. For every $n>2$, the actual variance is strictly larger by the factor $n/(n-2)$. The ratio approaches one as $n$ grows, but equality is not attained at finite $n$. At $n=2$ the finite-variance form of the bound is inapplicable to this estimator; its infinite variance provides no efficiency equality.