Efficiency of the Sample Mean

Overview

The sample mean \(\bar{X}\) achieves the Cramér–Rao lower bound for estimating the mean of a Normal distribution, making it the most efficient unbiased estimator.

CRLB for the Normal Mean

For \(X \sim N(\mu, \sigma^2)\), the Fisher information for \(\mu\) is \(I(\mu) = 1/\sigma^2\), giving:

\[ \text{Var}(\hat{\mu}) \geq \frac{1}{nI(\mu)} = \frac{\sigma^2}{n} = \text{Var}(\bar{X}) \]

Efficiency Under Non-Normality

For non-normal distributions, the sample mean may not be efficient. The asymptotic relative efficiency (ARE) compares estimators:

\[ \text{ARE}(\bar{X}, \text{Median}) = \frac{\text{Var}(\text{Median})}{\text{Var}(\bar{X})} = \frac{\pi}{2} \approx 1.57 \quad \text{(for Normal)} \]