Exact distribution of a maximally selected Wilcoxon and a new hybrid test of symmetry.

ABSTRACT

Recently, a maximally selected normalized Wilcoxon, whose asymptotic distribution is a Brownian Bridge, was proposed for testing symmetry of a distribution about zero. The test sequentially discards observations whose absolute value is below increasing thresholds. The Wilcoxon is obtained at each threshold, and the maximum is the test statistic. We develop a recursive function for the exact distribution of a modification of the Max Wilcoxon test (MW) and provide critical values and a program for computing the p-value for a sample. A new hybrid test that combines the sign and MW tests is introduced. The power of MW and the new hybrid test are compared with Modarres and Gastwirth's hybrid test (MGH) and the Max McNemar (MM), under the generalized lambda distributions (GLD) family and two normal mixture models. The MW and the new hybrid test outperform the MGH, which is superior to the MM test in the GLD family. In one mixture model, MM is the least powerful test and the remaining three are essentially equivalent. In the second mixture model, when the zero median assumption is nearly valid, the MW test does well; its performance degrades when this assumption is violated. In the latter case, the MM performs better than MW for the same degree of skewness because the MM simultaneously tests both symmetry and zero median. Data from a genetic study of monozygotic twins discordant for major depressive disorder is used to illustrate the new tests.