Using Kurtosis for Selecting One-Sample T-Test or Wilcoxon Signed-Rank Test

Aims / Objectives: To introduce a statistical test, which is a mixture of the one-sample t-test and Wilcoxon signed-rank test and depends on the sample kurtosis, using data from a symmetric univariate distribution with finite variance.

Study Approach: Computer simulation of coverage probabilities and of power calculations from either the one-sample t-test or the Wilcoxon signed-rank test, based on kurtosis, using the statistical software R.

Methodology: Data are generated with differing sample sizes from the Normal, Uniform, student-t with small degrees of freedom, and Laplace distributions. Coverage probabilities and power calculations are compared using the one-sample t-test, the Wilcoxon signed-rank test, and three proposed mixture tests which select the one-sample t-test or the Wilcoxon signed-rank test based on the sample kurtosis being significantly low or significantly high or either. Nonstandard values of t, the probability of a Type I error, are selected to account for the discrete nature of the Wilcoxon signed-rank test, allowing fair comparisons among the Wilcoxon signed-rank test, the t-test, and the three mixture tests.

Results: The false positive rate and power calculations are simulated for these nine distributions for both two-sided and one-sided tests, allowing comparisons among these five testing procedures.

Conclusion: When a small dataset is sampled from a symmetric distribution, then in comparison to the t-test, the Wilcoxon signed-rank test is equal in preference for the Normal distribution and is in fact more preferable for the non-Normal distributions tested herein. For small sample sizes, the mixture test based on high kurtosis is preferred over the t-test, but otherwise the t-test is preferred over all three mixture tests.