📂Mathematical Statistics

Unbiased Estimators and the Cramér-Rao Bound

Theorem

Regularity Conditions:

• (R0): The probability density function $f$ is injective with respect to $\theta$. It satisfies the following equation. $$\theta \ne \theta ' \implies f \left( x_{k} ; \theta \right) \ne f \left( x_{k} ; \theta ' \right)$$
• (R1): The probability density function $f$ has the same support for all $\theta$.
• (R2): The true value $\theta_{0}$ is an interior point of $\Omega$.
• (R3): The probability density function $f$ is twice differentiable with respect to $\theta$.
• (R4): The integral $\int f (x; \theta) dx$ is twice differentiable with respect to $\theta$, even when interchanging the order of integration.

Let’s define a likelihood function $L (\theta | \mathbf{X} ) := \prod_{k=1}^{n} f \left( x_{k} | \theta \right)$ when a random sample $X_{1} , \cdots , X_{n}$ that satisfies the Regularity Conditions (R0)-(R4) is drawn from $f (x|\theta)$. If $W \left( \mathbf{X} \right) = W \left( X_{1} , \cdots , X_{n} \right)$ is an unbiased estimator for $\tau \left( \theta \right)$, then having a Cramér-Rao Lower Bound $\text{RC}$ for $W \left( \mathbf{X} \right)$ is equivalent to the following condition being true for some function $a(\theta)$. $$a \left( \theta \right) \left[ W \left( \mathbf{X} \right) - \tau (\theta) \right] = {{ \partial \log L (\theta | \mathbf{X}) } \over { \partial \theta }}$$

Explanation

In summary, it means that when $W \left( \mathbf{X} \right) - \tau (\theta)$ is proportional to ${{ \partial \log L (\theta | \mathbf{X}) } \over { \partial \theta }}$, $\operatorname{Var} W \left( \mathbf{X} \right) = \text{RC}$ follows. The proof of the theorem¹ itself is not very difficult but is omitted here as it relies heavily on logic that is conveniently used only in the specific textbook.