  In this paper we study goodness-of-fit testing of single-index models. The
large sample behavior of certain score-type test statistics is investigated. As
a by-product, we obtain asymptotically distribution-free maximin tests for a
large class of local alternatives. Furthermore, characteristic function based
goodness-of-fit tests are proposed which are omnibus and able to detect peak
alternatives. Simulation results indicate that the approximation through the
limit distribution is acceptable already for moderate sample sizes.
Applications to two real data sets are illustrated.
