Theory and Methods of Statistics

Theorem 5.2.2

Information Lower Bound in Multiparameter Case

Suppose T = T(X) is an estimator of g(θ) with bias b(θ) and finite variance, that is, ${\text{E}}_{\mathit{\theta }}\left[T\right]=g(\mathit{\theta })+b(\mathit{\theta })$ and ${\text{Var}}_{\mathit{\theta }}\left[T\right]<\infty$where the family $\left\{f(x,\mathit{\theta }),\mathit{\theta }\in \Theta \subset {\mathbb{R}}^k\right\}$ satisfies regularity Conditions 1, 2, and 3. Then,

$\begin{array}{l}\hfill {\text{Var}}_{\mathit{\theta }}\left[T\right]\ge \left\{{▽}^T\left[g(\mathit{\theta })+b(\mathit{\theta })\right]\right\}I{(\mathit{\theta })}^{-1}\left\{▽\left[g(\mathit{\theta })+b(\mathit{\theta })\right]\right\}.\end{array}$

Example 5.2.1

In a random sample $\left(X_1,\dots ,X_n\right)$ from N(μ,σ²), ...