Confidence bands in generalized linear models

Abstract

Generalized linear models (GLM) include many useful models. This paper studies simultaneous confidence regions for the mean response function in these models. The coverage probabilities of these regions are related to tail probabilities of maxima of Gaussian random fields, asymptotically, and hence, the so-called tube formula is applicable without any modification. However, in the generalized linear models, the errors are often nonadditive and non-Gaussian and may be discrete. This poses a challenge to the accuracy of the approximation by the tube formula in the moderate sample situation. Here two alternative approaches are considered. These approaches are based on an Edgeworth expansion for the distribution of a maximum likelihood estimator and a version of Skorohod’s representation theorem, which are used to convert an error term (which is of order $n^{-1 /2}$ in one-sided confidence regions and of $n^{-1} in two-sided confidence regions) from the Edgeworth expansion to a “bias” term. The bias is then estimated and corrected in two ways to adjust the approximation formula. Examples and simulations show that our methods are viable and complementary to existing methods. An application to insect data is provided. Code for implementing our procedures is available via the software parfit