An Approximation for the Exact Distribution of the Wilcoxon Test for Symmetry

Abstract
An approximation of the exact distribution of the Wilcoxon test for symmetry by means of an Edgeworth series is derived. Probabilities are computed for sample sizes N = 10(1)26(2)42(3)48,50,100, and the accuracy investigated and compared with that of the normal approximation. The Edgeworth expansion gives a very satisfactory approximation for N ≥ 15 (maximum error < 25×10−5 when taken to terms of order 1/N and < 6×10−5 to terms of order 1/N 2, for a standardized variable (x) in the region 1.75 ≤ x ≤ 3.1). For N = 20 the Edgeworth approximation almost coincides with the exact distribution. For N > 20 where no exact probabilities are readily available, the accuracy of the normal distribution is investigated relative to the Edgeworth expansion to terms of order 1/N 2. The validity of this procedure is established by calculating certain exact probabilities in the interval studied, for values of N = 21, 22 and 25. The Edgeworth expansion to terms of order 1/N is readily computed and gives the same degree of accuracy when N = 15 as that given by the normal distribution when N = 100.