Asymptotics of an empirical bridge of a 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. The sums form an empirical bridge of the regression model by self-centering and self-normalising. We prove weak convergence of the empirical bridge in uniform metrics to a centered Gaussian process. The tests are of chi-squared type.