Scalar Cramér–Rao lower bound

Given: AC; an open scalar parameter interval and a fixed $\theta \in \mathrm{\Theta }$; a measurable real estimator $T$ with ${\mathbb{E}}_{\eta }T=\psi (\eta )$ locally, finite ${\mathbb{E}}_{\theta }{T}^2$, and differentiable $\psi$; a score $S_{\theta }$ with ${\mathbb{E}}_{\theta }{S}_{\theta }=0$, $I(\theta )={\mathbb{E}}_{\theta }{S}_{\theta }^2\in (0,\mathrm{\infty })$; and the score-covariance identity ${\mathrm{Cov}}_{\theta }(T,S_{\theta })={\psi }^{\mathrm{\prime }}(\theta )$. Alternatively, assume the sufficient common-support density and integrable-envelope route stated above.

The Axiom of Choice: AC says every family of nonempty sets has a choice function; it supplies the density representatives used by the score definition.

Score and Fisher information: on positive support, $S_{\theta }={\mathrm{\partial }}_{\theta }\log p_{\theta }$ and $I(\theta )={\mathbb{E}}_{\theta }{S}_{\theta }^2$.

Mean-zero score and the information identity: under the displayed common-support and derivative-envelope hypotheses, the score has mean zero and the information identity holds.

Statistics and estimators: an estimator is a measurable statistic with the target measurable space specified.

Moments, variance, and covariance on a probability space: for square-integrable real random variables, variance is $\mathbb{E}[(T-\mathbb{E}T{)}^2]$ and covariance is $\mathbb{E}[(T-\mathbb{E}T)(S-\mathbb{E}S)]$.

The space $L^p(\mu )$ as the quotient by null functions and Measure-null sets and almost-everywhere statements relative to a measure: $L^2(P_{\theta })$ consists of square-integrable real functions modulo equality $P_{\theta }$-almost everywhere.

The Lebesgue integral is linear on $L^1(\mu )$ and A nonnegative measurable function has integral $0$ exactly when it vanishes almost everywhere: expectations are linear on integrable functions, and a nonnegative function with integral zero vanishes almost everywhere.

Real and complex inner product spaces, with the inner product linear in the first argument: a real inner product is a symmetric bilinear positive-definite pairing.

Cauchy–Schwarz: $\mathrm{\mid }⟨u,v⟩\mathrm{\mid }\le \parallel u\parallel \parallel v\parallel$, with equality exactly for linearly dependent vectors: in a real inner product space, Cauchy–Schwarz holds, with equality exactly when the two vectors are linearly dependent.

The mean value theorem, as the case $g(x)=x$ of Cauchy's: for $f$ continuous on $[a,b]$ with $a<b$ and differentiable on $(a,b)$ there is $c\in (a,b)$ with $f(b)-f(a)=f^{\mathrm{\prime }}(c)(b-a)$ and Dominated convergence: a differentiable parameter function has its difference quotient bounded by a derivative bound on the segment, and an integrable majorant permits the limit to pass through the integral.

Likelihood densities: $p_{\eta }$ is a selected RN density representative of the sampling law.

A sigma-finite signed measure that is absolutely continuous with respect to a sigma-finite positive measure has a unique almost-everywhere density: the selected RN density satisfies $P_{\eta }(A)={\int }_A{p}_{\eta }d\mu$.

The measure with density $f$ relative to $\mu$ and Integrating against a density agrees with integrating the product and Integrable real and complex functions, and their integrals: for the density law $P_{\eta }=p_{\eta }d\mu$, the density identity holds for nonnegative $g$. For real integrable $g$, applying the nonnegative identity to $g^{+}$ and $g^{-}$ gives the signed identity; applying it to $\mathrm{\mid }g\mathrm{\mid }$ verifies that $g{p}_{\eta }$ is integrable.

Unbiased estimators and UMVU estimators: unbiasedness means integrability and ${\mathbb{E}}_{\eta }T=\psi (\eta )$ at every parameter; finite-variance unbiasedness also requires $T\in L^2(P_{\eta })$ at every parameter.