Model diagnosis for parametric regression in high-dimensional spaces

We study tools for checking the validity of a parametric regressionmodel. When the dimension of the regressors is large, many ofthe existing tests face the curse of dimensionality or requiresome ordering of the data. Our tests are based on the residualempirical process marked by proper functions of the regressors.They are able to detect local alternatives converging to thenull at parametric rates. Parametric and nonparametric alternativesare considered. In the latter case, through a proper principalcomponent decomposition, we are able to derive smooth directionaltests which are asymptotically distribution-free under the nullmodel. The new tests take into account precisely the ‘geometryof the model’. A simulation study is carried through andan application to a real dataset is illustrated.