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On the asymptotic efficiency of directional models checks for regression. (English) Zbl 1383.62128
Ferger, Dietmar (ed.) et al., From statistics to mathematical finance. Festschrift in honour of Winfried Stute. Cham: Springer (ISBN 978-3-319-50985-3/hbk; 978-3-319-50986-0/ebook). 71-87 (2017).
Summary: In a landmark paper, W. Stute [Ann. Stat. 25, No. 2, 613–641 (1997; Zbl 0926.62035)] introduced smooth and directional tests for homoskedastic regression models based on weighted averages of estimated principal components of the CUSUM of residual concomitants process. This article shows that Stute’s [loc. cit.] directional test is asymptotically uniformly most powerful in a semiparametric context, considering the joint distribution of the regression error term and the covariate as a nuisance parameter. Moreover, the directional test is shown to be asymptotically equivalent to the classical $$t$$-ratio under homoskedasticity. This article also studies the heteroskedastic case, where the directional test based on the CUSUM of standarized residual concomitants [W. Stute et al., ibid. 26, No. 5, 1916–1934 (1998; Zbl 0930.62044)] is asymptotically equivalent to the semiparametric efficient test, which is the $$t$$-ratio using the generalized least squares estimator.
For the entire collection see [Zbl 1383.62010].

##### MSC:
 62G10 Nonparametric hypothesis testing 62G20 Asymptotic properties of nonparametric inference 62E20 Asymptotic distribution theory in statistics
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