Assessing the influence of individual observations on a goodness-of-fit test based on nonparametric regression

When it is reasonable to assume that a model exhibits smoothness, goodness-of-fit statistics can be constructed that have higher power to detect deviations from a specified parametric model than tests based only on the empirical distribution. Recently Azzalini, Bowman and Härdle (1989) proposed a pseudo-likelihood ratio test using nonparametric (kernel) regression based on this principle. In this paper a diagnostic is developed to assess the influence of individual cases on the test, and on a more robust version of the test. An approximation to the diagnostic is also proposed. It is shown that the diagnostic can expose potentially incorrect inferences concerning model checking due to unusual individual cases. Although the focus is on linear and nonlinear regression models, the ideas also apply to other regression families, such as generalized linear models.