One of the central themes of nonparametric testing theory is the two-sample problem. The famous Mann-Whitney test, equivalent to the Wilcoxon rank sum test, is a solution of the two-sample problem, and is considered nowadays as one of the breakthroughs of twentieth century statistics. Xie and Priebe (2002) proposed a new solution of the two-sample problem, and their test has higher efficacy than most other nonparametric two-sample tests existing in the literature, including the Mann-Whitney test. The gain in efficacy for their test compared to the classical Mann-Whitney test was seen to be phenomenal in some cases (e.g., when the underlying density is strongly skewed) and was seen to be substantial in other cases (e.g., when the underlying density is either asymmetric bimodal or heavily kurtotic). But their test was not data-adaptive, since the test statistic that they proposed had parameters which were functions of the unknown underlying distribution function (i.e., the distribution function of both the samples under the null hypothesis). We propose a methodology which makes the two-sample solution proposed by Xie and Priebe (2002) data-adaptive. We show via theoretical results and simulation studies that the test statistic proposed in our data-adaptive methodology has approximately the same efficacy and power as the one proposed in Xie and Priebe (2002), especially for large sample sizes. We also illustrate our methodology by analyzing magnetic resonance brain imaging data related to normally aging subjects and patients with Alzheimer's disease.

This is joint work with Carey E. Priebe.