Trimmed and Winsorized Means¶
Overview¶
When data contains outliers or comes from heavy-tailed distributions, robust alternatives to the sample mean may perform better.
Trimmed Mean¶
The \(\alpha\)-trimmed mean removes the smallest and largest \(\alpha\) fraction of observations:
\[
\bar{X}_{\text{trim}(\alpha)} = \frac{1}{n - 2\lfloor n\alpha \rfloor} \sum_{i=\lfloor n\alpha \rfloor + 1}^{n - \lfloor n\alpha \rfloor} X_{(i)}
\]
Winsorized Mean¶
The Winsorized mean replaces extreme values with the nearest non-trimmed value instead of removing them.
Comparison¶
| Estimator | Robustness | Efficiency (Normal) |
|---|---|---|
| Sample mean | Low (breakdown = 0) | 100% |
| 10% trimmed mean | Moderate | ~97% |
| Median (50% trim) | High (breakdown = 50%) | ~64% |