Asymptotics of an empirical bridge of regression on induced order statistics

We propose a class of tests for linear regression on concomitants (induced order statistics). These tests are based on sequential sums of regression residuals. We self-center and self-normalize these sums. The resulting process is called an empirical bridge. We prove weak convergence of the empirical bridge in uniform metrics to a centered Gaussian process. The proposed tests are of chi-square type.