James–Stein Positive-Part Shrinkage
Inspect radial mean shrinkage while separating realized error from the James–Stein expected-risk theorem.
Free theory articleInspect radial mean shrinkage while separating realized error from the James–Stein expected-risk theorem.
Concept & interpretation
The James–Stein estimator offers a way to shrink a multivariate normal observation towards the origin. It is well known to dominate the standard maximum likelihood estimator (MLE) in terms of expected squared error risk when the dimension is 3 or more. The positive-part variation provides an additional continuous improvement by preventing the shrinkage factor from becoming negative, which would erroneously push the estimate in the opposite direction.
Experiment
You can interactively observe how the observed radius affects the shrinkage factor and the estimates. Open the interactive module at https://riano.app/module/james-stein-positive-part-shrinkage and try the following steps:
- Set Dimension to 5 and Noise sigma to 1.
- Set Observed radius in sigma to 1. Notice that the ordinary factor is 1 - 3/1 = -2.0. This negative factor pushes the ordinary estimate past the origin to -2.0, increasing the realized squared error to 4.0 compared to the MLE error of 1.0.
- Observe that for the same inputs, the positive-part factor clamps to 0.0, yielding a positive-part estimate of 0.0. The realized error for the positive-part estimator is perfectly 0.0 here since it correctly stopped at the origin.
- Change Observed radius in sigma to 0. Notice the ordinary formulas become "Unavailable" due to division by zero, but the positive-part factor extends continuously to 0.0.