Neyman–Pearson Randomized Test

The Neyman–Pearson lemma provides the optimal way to test a simple null hypothesis against a simple alternative hypothesis. It asserts that the test with the highest power for a given size α is a likelihood-ratio test. For discrete distributions, achieving the exact size α often requires randomizing the decision at the boundary.

Experiment

Follow these ordered instructions using the interactive controls to observe how the randomized decision rule is constructed:

1. Adjust the Alpha twelfths slider control to 4 (which equals an α of 4/12).
2. Observe the mathematical results in the Rejection Probabilities section: φ₂ becomes 1, φ₁ becomes 1/2, and φ₀ remains 0.
3. Select an Observed outcome using its slider control to view the specific rejection probability for that particular data point.

Try a full-page version at https://riano.app/module/neyman-pearson-randomized-test to observe how boundary cases like α = 0 or α = 1 completely constrain or relax the test.