| Comprehensive model adequacy checking procedures are discussed for general parametric and semiparametric model specifications, with illustration in a variety of examples containing assumptions on dependence structures, density shapes, functional forms and other model features. We use the efficient score processes developed by Bickel, Ritov and Stoker (2006) as building blocks, from which many omnibus tests can be constructed. This set of omnibus tests include Class I tests with decreasing power along high frequencies, and Class II tests with approximately equal power on limited frequencies. We also give a unified view of a group of asymptotically distribution free tests from the score perspective. This set of tests is essentially derived from a family of inefficient scores, enabling the limit Gaussian processes to have nice variance-covariance structure. Additionally, we propose data-driven tests in the score and spectral domains. Either model selection rules or thresholding methods are invoked to choose the scores or spectra on which to focus. Finally, we consider aggregating different types of tests, primarily combining one Class I test and one Class II test, in the hope of achieving a balance between the two classes. Numerical experiments confirm that both Class I and Class II tests have their own strong and weak aspects, and the aggregated procedures demonstrate a balanced and stable performance; although signal strength (of departures) is a fundamental limiting factor of all such procedures. In summary, a statistical model is warranted only when it passes various diagnostic checks with different but complementary strengths. |
