The Cramer-Rao Bound

The Cramer-Rao bound (Cramer-Rao Lower Bound, CRLB) establishes a lower bound on the variance of every unbiased estimator of the parameter $\theta$ (to maintain notation consistent with the literature, $\beta$ in our case).

Let $X$ be a multidimensional random variable and $\theta$ an unknown deterministic parameter. Let $f^{\vartheta}_{x} (X)$ be the probability density of $X$ given $\theta$. Assume that this probability density exists and is twice differentiable with respect to $\theta$.

Theorem 1 (Cramer-Rao inequality)   Let $T(\cdot)$ be an unbiased estimator of the scalar parameter $\vartheta$, and suppose that the observation space $X$ is independent of $\theta$. Then (under some regularity assumptions...)

$$E^\theta \left[ \left( T(X) - \theta \right)^2 \right] \ge \left[ I_n(\theta) \right]^{-1}$$ (4.1)

where $I_n(\theta)=E^{\theta} \left[ \left( \frac{\partial \ln f^{\theta}_{x} (X) }{\partial \theta} \right)^2 \right]$ (Fisher information).

Since the parameter $\theta$ is unknown, the Cramer-Rao theorem only makes it possible to determine whether the estimator is optimal.