Exact distribution of Bartels Statistic
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1 Exact distribution of Bartels Statistic Frederico Caeiro CMA and DM, Universidade Nova de Lisboa, Portugal, Ayana Mateus CMA and DM, Universidade Nova de Lisboa, Portugal, November 13, Exact null distribution of Bartels Statistic Bartels Rank Test (of randomness) is the nonparametric version of von Neumann ratio test. Let R i = rank(x i ), i = 1,..., n, denote the rank of the i-th observation from the sample. Then the test statistic is given by n 1 i=1 RV N = (R i R i+1 ) 2 n i=1 (R i (n + 1)/2) 2. (1) Note that if there are no ties, the denominator is equal to n(n 2 1)/12 and the statistic RV N in (1) is equivalent to the numerator n 1 NM = (R i R i+1 ) 2. i=1 The exact distribution of NM, 4 n 10, under the null hypothesis (null distribution) is given in [1]. Here we present in Table 1 the exact null distribution (left-tail probabilities) of Bartels Randomness statistic for samples of size n = 5,..., 17. Further details can be found in [2, 3, 4] Table 1: Exact null distribution of MN, for n = 5,...,
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