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
H\"{o}lder 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
H\"{o}lder classes simultaneously.Comment: Published in at http://dx.doi.org/10.1214/009053607000000992 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org