The bias-corrected sample variance $$ \hat{\sigma}^2 = \dfrac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2 $$ is commonly used in natural sciences to estimate the variance $ \sigma^2 $ of a Gaussian random variable $ X $ from an i.i.d. sample $ X_1, \dots, X_n $.
One of the earliest things I learned in statistical inference classes is that bias is only one of many relevant properties when comparing estimators. My understanding is that mean square error (MSE) is generally a better measure of estimator performance than bias alone.
Why then is the minimum MSE estimator $$ \hat{\sigma}_{MMSE}^2 = \dfrac{1}{n+1} \sum_{i=1}^n (X_i - \bar{X})^2 $$ not the standard choice for this problem?
