A Bivariate Permutation Test for Analysis of Three Interval Data Samples
Let X, Y, and Z be three random variables with unknown continuous cumulative distribution functions F, G, and H. D.R. Whitney1） proposed the (U,V) statistics as a bivariate extension of Wilcoxon's U statistic in 1951. It is well known that Whitney's (U,V) tests based on these statistics are particularly powerful against the following two types of alternatives, respectively: (1) F(t)>G(t) and F(t)>H(t) (for all t), (2) G(t)>F(t）>H(t) (for all t). A bivariate permutation (U,V) test that is an extension of Whitney's (U,V) test is proposed for samples in which observations are specified only by intervals with known probability density functions. Here, the two statistics, U and V, are based on generalized signs instead of ranks. The proposed bivariate permutation (U,V) test is not rough even in small samples, because the value of the generalized sign can take on real value densely. In the same way, we can construct a multivariate permutation test for many samples in which observations are specified only by intervals. Computer programs were developed for determining the critical region of (U,V) in a given sample of nx x's, nｙ y's and nz z's.