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, ${\text{E}}_{\mathbit{\theta }}\left[T\right]=g\left(\mathbit{\theta }\right)+b\left(\mathbit{\theta }\right)$ and ${\text{Var}}_{\mathbit{\theta }}\left[T\right]<\infty$where the family $\left\{f\left(x,\mathbit{\theta }\right),\mathbit{\theta }\in \Theta \subset {\mathbb{R}}^{k}\right\}$ satisfies regularity Conditions 1, 2, and 3. Then,

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

Example 5.2.1

In a random sample $\left({X}_{1},\dots ,{X}_{n}\right)$ from N(μ,σ2), ...

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