Asymptotics of an empirical bridge of regression on induced order statistics

We develop a class of statistical tests for analysis of multivariate data. These statistical tests verify the hypothesis of a linear regression model. To solve the question of the applicability of the regression model, one needs a statistical test to determine whether the actual multivariate data corresponds to this model. If the data does not correspond to the model, then the latter should be corrected. The developed statistical tests are based on an ordering of data array by some null variable. With this ordering, all observed variables become concomitants (induced order statistics). Statistical tests are based on functionals of the process of sequential (under the introduced ordering) sums of regression residuals. We prove a theorem on weak convergence of this process to a centered Gaussian process with continuous trajectories. This theorem is the basis of an algorithm for analysis of multivariate data for matching a linear regression model. The proposed statistical tests have several advantages compared to the commonly used statistical tests based on recursive regression residuals. So, unlike the latter, the statistics of the new tests are invariant to a change in ordering from direct to reverse. The proof of the theorem is based on the Central Limit Theorem for induced order statistics by Davydov and Egorov (2000).