Chapter 9.2, Problem 8E

Let Y₁, Y₂, … , Y_n denote a random sample from a probability density function f (y), which has unknown parameter θ. If $\hat{\theta }$ is an unbiased estimator of θ, then under very general conditions

$V(\hat{\theta })\ge I(\theta ),\text{where }I(\theta )={\left[nE\left(-\frac{{\partial }^2\ln f(Y)}{\partial {\theta }^2}\right)\right]}^{-1}.$

(This is known as the Cramer–Rao inequality.) If $V(\hat{\theta })=I(\theta )$, the estimator $\hat{\theta }$ is said to be efficient.

1. a Suppose that f (y) is the normal density with mean μ and variance σ². Show that $\bar{Y}$ is an efficient estimator of μ.
2. b This inequality also holds for discrete probability functions p(y). Suppose that p(y) is the Poisson probability function with mean λ. Show that $\bar{Y}$ is an efficient estimator of λ.