Goodness-of-fit test with nuisance regression and scale

Abstract

In the linear model Y _i = x _i′ β + σe _i, i=1,…,n, with unknown (β, σ), β∈{\open R}^p, σ>0, and with i.i.d. errors e ₁,…,e _n having a continuous distribution F, we test for the goodness-of-fit hypothesis H ₀:F(e)≡F ₀(e/σ), for a specified symmetric distribution F ₀, 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 F ₀. A simulation study illustrates a good performance of the tests.