Exact and Approximate Probabilities for the Null Distribution of Bartels Randomness Test

In this work we revisit the statistical properties of the Bartels randomness test. The exact distribution of the statistic, under the randomization hypothesis, can only be obtained when the sample size (n) is small, since it requires the full set of permutations of the first n positive integers. Here, we present the exact null distribution without ties, for samples of size 10 ≤ n ≤ 17, extending the results available in the literature. Since the null distribution is asymptotically normally distributed, but at a slow rate, Bartels concluded that the null distribution is well approximated by a Beta distribution, for samples of size 10 ≤ n ≤ 100. We present a new approximation, based on the Edgeworth series, for the null distribution of the Bartels randomness statistic. The precision of this new approximation is also discussed.