An efficient nonparametric test for bivariate two-sample location problem

We propose a new test for the two-sample bivariate location problem. The proposed test statistic has a U-statistic representation with a degenerate kernel. The limiting distribution is found for the proposed test statistic. The power of the test is compared using Monte Carlo simulation to the tests of Blumen [I. Blumen, A new bivariate sign-test for location, Journal of the American Statistical Association 53 (1958) 448-456], Mardia [K.V. Mardia, A non-parametric test for the bivariate two-sample location problem, Journal of the Royal Statistical Society, Series B 29 (1967) 320-342], Peters and Randles [D. Peters, R.H. Randles, A bivariate signed-rank test for the two-sample location problem, Journal of the Royal Statistical Society, Series B 53 (1991) 493-504], LaRocque, Tardif and van Eeden [D. LaRocque, S. Tardif, C. van Eeden, An affine-invariant generalization of the Wilcoxon signed-rank test for the bivariate location problem, Australian and New Zealand Journal of Statistics 45 (2003) 153-165], and Baringhaus and Franz [L. Baringhaus, C. Franz, On a new multivariate two-sample test, Journal of Multivariate Analysis 88 (2004) 190-206]. Under the bivariate normal and bivariate t distributions the proposed test was more powerful than the competitors for almost every change in location. Under the other distributions the proposed test reached the desired power of one at a faster rate than the other tests in the simulation study. Application of the test is presented using bivariate data from a synthetic and a real-life data set.