Permutation p-value approximation via generalized Stolarsky invariance

Abstract: When it is necessary to approximate a small permutation pvalue p, then simulation is very costly. For linear statistics, a Gaussian approximation p̂1 reduces to the volume of a spherical cap. Using Stolarsky’s (1973) invariance principle from discrepancy theory, we get a formula for the mean of (p̂1− p)2 over all spherical caps. From a theorem of Brauchart and Dick (2013) we get such a formula averaging only over spherical caps of volume exactly p̂1. We also derive an improved estimator p̂2 equal to the average true p-value over spherical caps of size p̂1 containing the original data point x0 on their boundary. This prevents p̂2 from going below 1/N when there are N unique permutations. We get a formula for the mean of (p̂2 − p)2 and find numerically that the root mean squared error of p̂2 is roughly proportional to p̂2 and much smaller than that of p̂1.