MSE of Variance Estimators¶
Overview¶
We compare the MSE of different variance estimators for Normal data.
Three Estimators¶
| Estimator | Formula | Bias | MSE |
|---|---|---|---|
| MLE (\(n\)) | \(\frac{1}{n}\sum(X_i-\bar{X})^2\) | \(-\sigma^2/n\) | \(\frac{2n-1}{n^2}\sigma^4\) |
| Bessel (\(n-1\)) | \(\frac{1}{n-1}\sum(X_i-\bar{X})^2\) | \(0\) | \(\frac{2\sigma^4}{n-1}\) |
| MSE-optimal (\(n+1\)) | \(\frac{1}{n+1}\sum(X_i-\bar{X})^2\) | \(-\frac{2\sigma^2}{n+1}\) | \(\frac{2\sigma^4}{n+1}\) |
Key Insight¶
The MSE-optimal estimator (dividing by \(n+1\)) has the smallest MSE among estimators of the form \(c \sum(X_i - \bar{X})^2\), demonstrating the bias-variance tradeoff.