Two-Sample Multivariate Test of Homogeneity - Ali S. Barakat (مجلة جامعة النجاح للأبحاث العلوم الطبيعية)
Given independent multivariate random samples X1, X2,.... , and Y1, Y2,..... , from distributions F and G, a test is desired for Ho: F = G against general alternatives. Consider the k • (n1+n2) possible ways of choosing one observation from the combined samples and then one of its k nearest neighbors, and let Sk be the proportion of these choices in which the point and neighbor are in the same sample. SCHILLING proposed Sk as a test statistic, but did not indicate how to determine k. BARAKAT, QUADE, and SALAMA proposed a test statistic, which is equivalent to a sum of N Wilkoxon rank sums. The limiting distribution of the test has not been found yet. We suggest as a test statistic Tm = S Sh(m,j)و Where h (m,j) = I{jth nearest neighbor of the median m is a y}. The limiting distribution of Tm is normal. A simulation with multivariate normal data suggests that our test is generally more powerful than Schilling’s test using k = 1, 2 or 3.