Asymptotics of sums of regression residuals under multiple ordering of regressors
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We prove theorems about the Gaussian asymptotics of an empirical bridge built from linear model regressors with multiple regressor ordering. We study the testing of the hypothesis of a linear model for the components of a random vector: one of the components is a linear combination of the others up to an error that does not depend on the other components of the random vector. The results of observations of independent copies of a random vector are sequentially ordered in ascending order of several of its components. The result is a sequence of vectors of higher dimension, consisting of induced order statistics (concomitants) corresponding to different orderings. For this sequence of vectors, without the assumption of a linear model for the components, we prove a lemma of weak convergence of the distributions of an appropriately centered and normalized process to a centered Gaussian process with almost surely continuous trajectories. Assuming a linear relationship of the components, standard least squares estimates are used to compute regression residuals, the difference between response values and the predicted ones by the linear model. We prove a theorem of weak convergence of the process of regression residuals under the necessary normalization to a centered Gaussian process. Then we prove a theorem of the same convergence for the empirical bridge, a self-centered and self-normalized process of regression residuals.
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