DKW Simultaneous CDF Band

The Dvoretzky-Kiefer-Wolfowitz (DKW) inequality provides a rigorous way to bound the difference between an empirical cumulative distribution function (CDF) and the true underlying CDF. It yields a distribution-free confidence band that holds simultaneously for all values in the domain, which is a much stronger guarantee than constructing individual confidence intervals at separate points.

Investigating the Confidence Band

Open the DKW Simultaneous CDF Band module to visualize how different datasets interact with the confidence band.

1. Set Sample to "Spread" and Size to 8. The blue empirical CDF steps along the true orange curve, with D = 0.1.
2. Set Sample to "Clustered" (points 0.10 through 0.17). The empirical CDF climbs to 1.0 very early, while the true CDF continues linearly. The gap D reaches 0.83 at x = 0.17, which exceeds the band radius of 0.480, so the true CDF escapes the band.
3. Set Size to 1. The band radius widens significantly (ε ≈ 1.358), forcing the clipped band bounds to their logical limits of 0 and 1.
4. Change Alpha to 0.01. Notice the required epsilon increases to guarantee the higher 99% simultaneous coverage probability.
5. Use Probe x to inspect specific points; notice how the gap at any single point is just one local cross-section of the total supremum distance D.