Permutation p-value approximation via generalized Stolarsky invariance

When it is necessary to approximate a small permutation $p$-value $p$, then simulation is very costly. For linear statistics, a Gaussian approximation $\hat p_1$ 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 $(\hat p_1-p)^2$ 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 $\hat p_1$. We also derive an improved estimator $\hat p_2$ equal to the average true $p$-value over spherical caps of size $\hat p_1$ containing the original data point $\boldsymbol{x}_0$ on their boundary. This prevents $\hat p_2$ from going below $1/N$ when there are $N$ unique permutations. We get a formula for the mean of $(\hat p_2-p)^2$ and find numerically that the root mean squared error of $\hat p_2$ is roughly proportional to $\hat p_2$ and much smaller than that of $\hat p_1$.