Asymptotically Distribution-Free Goodness-of-Fit Testing: A Unifying View

We outline a general paradigm for constructing asymptotically distribution-free (ADF) goodness-of-fit tests, which can be regarded as a generalization of Khmaladze (1993). This is achieved by a nonorthogonal projection of a class of functions onto the ortho-complement of the extended tangent space (ETS) associated with the null hypothesis. In parallel with the work of Bickel et al. (2006), we obtain transformed empirical processes (TEP) which are the building blocks for constructing omnibus tests such as the usual Kolmogorov-Smirnov type tests and Cramer-von Mise type tests, as well as Portmanteau tests and directional tests. The critical values can be tabulated due to the ADF property. All the tests are capable of detecting local (Pitman) alternative at the root-n scale. We shall illustrate the framework in several examples, mostly in regression model specification testing.