Abstract: Although the adequacy of linearity is well researched in the statistical literature, few studies examine this topic from the viewpoint of a measure of association. Inspired by the well-known distance covariance (dCov), we propose two omnibus tests for the goodness-of-fit of linearity. Methodologically, our tests do not include any tuning parameters and are conveniently implemented. The theoretical details are of independent interest, mainly because the kernel induced by the dCov is not smooth. We investigate the convergence of our tests under null, fixed, and local alternative hypotheses, and devise a bootstrap scheme to approximate their null distributions, showing that its consistency is justified. Numerical studies demonstrate the effectiveness of our proposed tests relative to that of several existing tests.

Key words and phrases: Bootstrap, distance covariance, goodness-of-fit test, linearity.