Goodness-of-fit test with nuisance regression and scale

In the linear model Yi = xi′ β + σei, i=1,…,n, with unknown (β, σ), β∈{\open R}p, σ>0, and with i.i.d. errors e1,…,en having a continuous distribution F, we test for the goodness-of-fit hypothesis H0:F(e)≡F0(e/σ), for a specified symmetric distribution F0, not necessarily normal. Even the finite sample null distribution of the proposed test criterion is independent of unknown (β,σ), and the asymptotic null distribution is normal, as well as the distribution under local (contiguous) alternatives. The proposed tests are consistent against a general class of (nonparametric) alternatives, including the case of F having heavier (or lighter) tails than F0. A simulation study illustrates a good performance of the tests.