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

In a regular one-parameter model with joint density $f_{\theta }(x)$, write the score function as $U_{\theta }={\partial }_{\theta }\log f_{\theta }(X)$ and the Fisher information as $I_n(\theta )={\mathbb{E}}_{\theta }[U_{\theta }^2]$. Assume a parameter-independent support, differentiability and sufficient domination to interchange parameter differentiation and integration, and $0<I_n(\theta )<\infty$. If $T$ has finite variance and is unbiased for a differentiable target $g(\theta )$, the Cramér-Rao lower bound isIndeed, differentiating total mass gives ${\mathbb{E}}_{\theta }{U}_{\theta }=0$, and differentiating ${\mathbb{E}}_{\theta }T=g(\theta )$ gives ${\mathrm{Cov}}_{\theta }(T,U_{\theta })=g^{\prime }(\theta )$. Cauchy-Schwarz inequality proves the bound. For $n$ independent identically distributed observations, information adds: $I_n=n{I}_1$. In particular for the parameter itself, $g(\theta )=\theta$, the bound is $1/I_n$. The assumptions are material: the same formula cannot simply be imposed on models with parameter-dependent support.