A QUALITY INDEX BASED ON DATA DEPTH AND MULTIVARIATE RANK-TESTS

Let F and G be the distribution functions of two given populations on R(p), p greater-than-or-equal-to 1. We introduce and study a parameter Q = Q(F, G), which measures the overall ''outlyingness'' of population G relative to population F. The parameter Q can be defined using any concept of data depth. Its value ranges from 0 to 1, and is .5 when F and G are identical. We show that within the class of elliptical distributions when G departs from F in location or G has a larger spread, or both, the value of Q dwindles down from .5. Hence Q can be used to detect the loss of accuracy or precision of a manufacturing process, and thus it should serve as an important measure in quality assurance. This in fact is the reason why we refer to Q as a quality index in this article. In addition to studying the properties of Q, we provide an exact rank test for testing Q = .5 vs. Q < .5. This can be viewed as a multivariate analog of Wilcoxon's rank sum test. The tests proposed here have power against location change and scale increase simultaneously. We introduce some estimates of Q and investigate their limiting distributions when F = G. We also consider a version of Q and its estimates, which are defined after correcting the location shift of G. In this case Q is used to measure scale increase only.