Asymptotics of sums of regression residuals under multiple ordering of regressors

We prove theorems about the Gaussian asymptotics of anempirical bridge built from residuals of a linear model under multipleregressor orderings. We study the testing of the hypothesis of a linearmodel for the components of a random vector: one of the componentsis a linear combination of the others up to an error that does notdepend on the other components of the random vector. The independentcopies of the random vector are sequentially ordered in ascending orderof several of its components. The result is a sequence of vectors ofhigher dimension, consisting of induced order statistics (concomitants)corresponding to different orderings. For this sequence of vectors, withoutthe assumption of a linear model for the components, we prove alemma of weak convergence of the distributions of an appropriatelycentered and normalized process to a centered Gaussian process withalmost surely continuous trajectories. Assuming a linear relationship ofthe components, standard least squares estimates are used to computeregression residuals, that is, the differences between response values andthe predicted ones by the linear model. We prove a theorem of weakconvergence of the process of sums of of regression residuals under thenecessary normalization to a centered Gaussian process.