📘 7.3 Summary – Unbiased Estimators and Standard Error

🔍 Unbiased Estimator

An estimator \( \hat{\theta} \) is unbiased if its expected value equals the parameter it estimates:

\[ \mathbb{E}(\hat{\theta}) = \theta \]

The bias is defined as \( \mathbb{E}(\hat{\theta}) - \theta \). For an unbiased estimator, this equals 0.

📉 Minimum Variance Unbiased Estimator (MVUE)

Among all unbiased estimators, the one with the smallest variance is called the MVUE.

For normal populations, \( \bar{X} \) is the MVUE for \( \mu \).

📏 Standard Error of an Estimator

The standard error (SE) measures how much a point estimate varies from sample to sample. For the sample mean:

\[ \text{SE}(\bar{X}) = \frac{s}{\sqrt{n}} \]