6.4.6—Unbiased estimates

An estimator T is unbiased for θ when E(T)=θ. Unbiasedness concerns long-run average, not whether one estimate is close to θ.

Check the expectation algebra and compare variance separately; an unbiased estimator can be noisy, while a biased estimator may have lower variance.

The sample mean is unbiased for μ under random sampling, while dividing by n rather than n−1 gives a biased estimate of population variance.

Unbiased does not mean error-free or best for every loss function.