Testing the link when the index is semiparametric—a comparative study

Generalized structured models are popular in applied statistics. They can circumvent the curse of dimensionality and provide results that are easy to interpret. However, there are two major concerns that need to be addressed before they are applied. Firstly, the credibility of the specified structure, such as additivity, and secondly, the specification of the link function need to be assessed. The focus is on the latter issue. In many cases it is feasible to estimate a nonparametric link, but the effort is often not justified. In contrast parametric links enable the use of likelihood-based estimates, which are asymptotically efficient, and which perform excellently in practice, particularly for small samples. Several statistics for testing the credibility of parametric link specifications are introduced. Estimation and implementation are discussed, and the performance of the statistics is compared in an intensive simulation study. Applications to real data are also described.