Maximally Selected Rank Statistics

A common statistical problem is the assessment of the predictive power of a quantitative variable for some dependent variable. A maximally selected rank statistic regarding the quantitative variable provides a test and implicitly an estimate of a cutpoint as a simple classification rule. Restricting the selection to an arbitrary given inner part of the support of the quantitative variable, we show that the asymptotic null distribution of the maximally selected rank statistic is the distribution of the supremum of the absolute value of a standardized Gaussian process on an interval. The asymptotic argument holds also in the case of tied or censored observations. We compare Monte Carlo results with an approximation of the asymptotic distribution under the null hypothesis. In addition, we investigate the behaviour of the test procedure and of the familiar Spearman rank test for independence, under some alternatives. Moreover, we discuss some aspects of the problem of estimating an underlying cutpoint.