- Have minimum variance among all other estimators. NB: Proofs of these properties will be done in class

### Minimum Variance Among Estimators In the realm of statistical estimation, one desirable property of an estimator is its **variance**. An estimator with a lower variance is generally preferred because it implies more consistent and reliable estimates across different samples. Among all unbiased estimators, the **Uniform Minimum Variance Unbiased Estimator (UMVUE)** stands out for having the smallest variance. - **UMVUE (Uniform Minimum Variance Unbiased Estimator):** - **Definition:** An unbiased estimator that has the lowest variance among all unbiased estimators for a parameter. - **Significance:** Ensures the most efficient estimation by minimizing uncertainty. ### Key Properties - **Unbiasedness:** The estimator's expected value equals the true parameter value. - **Minimum Variance:** No other unbiased estimator has a lower variance for every possible value of the parameter. ### Example Consider estimating the population mean, \( \mu \). The sample mean \( \bar{X} \) is an unbiased estimator of \( \mu \) and, under certain conditions (e.g., normality), it serves as the UMVUE. ### NB: Proofs of these properties, including the derivation and conditions under which an estimator achieves minimum variance, will be covered in class.