Resampling methods for the nonparametric and generalized Behrens-Fisher problems

Defining a location parameter as a generalization of the median, a robust test is proposed for (a) the nonparametric Behrens-Fisher problem, where the underlying distributions may have different scales and could be skewed, and (b) the generalized Behrens-Fisher problem, where the distributions may even have different shapes. We propose to bootstrap a signed rank statistic based on differences of sample values and derive rigorous bootstrap central limit theorems for its probabilistic justification, allowing for the so-called m-out-of-n bootstrap. The location parameter of interest is the pseudo-median of the distribution of the difference between a control measurement and an observation from the treatment group. It reduces to (a) the shift in the two sample location model and (b) the difference between the centers of symmetry in the nonparametric Behrens-Fisher model, under the additional assumption that the distributions are symmetric. Due to its importance for applications, we also extend our results to an ANOVA design where each treatment is compared with the control group. Finally, we compare our test with competitors on the basis of theory as well as simulation studies. It turns out that our approach yields a substantial improvement for distributions close to the generalized extreme value type, which makes it attractive for applications in engineering as well as finance. Several heteroscedastic data sets from electrical engineering, astro physics, energy research, analytical chemistry and psychology are used to illustrate our solution.