Testing Regression Models via Multiple Orderings of Regressors: OSD 2024

A new method for testing linear regression dependencies in multivariate data is proposed. The null hypothesis states that one of the data columns (response) depends linearly on the other columns (regressors) up to additive noise. The regression residuals are ordered in ascending order of the different columns, and empirical bridges are calculated for each sequence of such ordered regression residuals. The test statistic is the sum of integrals of the squares of the empirical bridges. The idea of the test is to detect deviations from linear dependence of the response on any of the regressors. Theorems on the limiting behavior of the empirical bridges and the test statistic are proved under the assumption of true null hypothesis. The calculation of the limiting distribution is based on the Smirnov's formula. Theorems are illustrated with simulated examples.