Within the nonparametric regression model with unknown regression function l and independent, symmetric errors, a new multiscale signed rank statistic is introduced and a conditional multiple test of the simple hypothesis l=0 against a nonparametric alternative is proposed. This test is distribution-free and exact for finite samples even in the heteroscedastic case. It adapts in a certain sense to the unknown smoothness of the regression function under the alternative, and it is uniformly consistent against alternatives whose sup-norm tends to zero at the fastest possible rate. The test is shown to be asymptotically optimal in two senses: It is rate-optimal adaptive against Hölder classes. Furthermore, its relative asymptotic efficiency with respect to an asymptotically minimax optimal test under sup-norm loss is close to 1 in case of homoscedastic Gaussian errors within a broad range of Hölder classes simultaneously.
@article{1211819567,
author = {Rohde, Angelika},
title = {Adaptive goodness-of-fit tests based on signed ranks},
journal = {Ann. Statist.},
volume = {36},
number = {1},
year = {2008},
pages = { 1346-1374},
language = {en},
url = {http://dml.mathdoc.fr/item/1211819567}
}
Rohde, Angelika. Adaptive goodness-of-fit tests based on signed ranks. Ann. Statist., Tome 36 (2008) no. 1, pp. 1346-1374. http://gdmltest.u-ga.fr/item/1211819567/