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Goodness-of-fit test with nuisance regression and scale. (English) Zbl 1042.62036
Summary: In the linear model \(Y_i={\mathbf x}_i'\beta+\sigma e_i\), \(i= 1,\dots, n\), with unknown \((\beta,\sigma)\), \(\beta\in\mathbb{R}^p\), \(\sigma> 0\), and with i.i.d. errors \(e_1,\dots, e_n\) having a continuous distribution \(F\), we test for the goodness-of-fit hypothesis \({\mathbf H}_0: F(e)\equiv F_0(e/\sigma)\), 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 \((\beta,\sigma)\), 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 good performance of the tests.

62G08 Nonparametric regression and quantile regression
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
65C60 Computational problems in statistics (MSC2010)
AS 181
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