Anderson-Darling (AD)

The Anderson-Darling test is a modification of the Cramér-von Mises test that gives more weight to the tails of the distribution:

\[ A^2 = -n - \sum_{i=1}^{n} \frac{2i-1}{n} \left[ \ln F(z_i) + \ln(1 - F(z_{n+1-i})) \right] \]

Key characteristics:

  • Tail sensitivity: Emphasizes differences in the tails through weighted integration

  • Distribution testing: Particularly effective for testing specific distributional forms

  • Power: More powerful than KS for tail-heavy distributions

  • Weighting: Uses \(\frac{1}{F(1-F)}\) weighting that emphasizes extremes

When to use:

  • Rare object class distributions (imbalanced detection scenarios)

  • Extreme lighting/weather conditions (nighttime, fog, snow)

  • When drift is expected in the tails of distributions

  • For heavy-tailed distributions (rare events in video streams)

  • Detecting outlier shifts or changes in extreme values

  • When tail behavior is critical for model performance

Limitations:

Can be “hyper-sensitive.” It may trigger alerts for tail-end noise that doesn’t actually impact the model’s predictive performance, leading to high alert volume.