On Tests Applied to Residuals

The null distribution of a test statistic computed on residuals can be affected by the dependence among the residuals. It might be expected, however, that the dependence decreases as the sample size increases, so that the limiting null distribution is the same as if the test were based on independent observations. The author investigates whether this large sample equivalence holds for tests of correlation, scale difference, regression, and location difference. Particular emphasis is placed on showing when rank tests based on residuals have a strongly asymptotically distribution-free property. U statistic theorems are used to investigate these issues.