WIAS Preprint No. 1355, (2008)

Adaptive goodness-of-fit tests based on signed ranks


  • Rohde, Angelika

2010 Mathematics Subject Classification

  • 62G10 62G20 62G35


  • Exact multiple testing, exponential inequality, multiscale statistic, relative asymptotic efficiency, signed ranks, sharp asymptotic adaptivity




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 one in case of homoscedastic Gaussian errors within a broad range of Hölder classes simultaneously.

Appeared in

  • Ann. Statist. 36 (2008), pp. 1346--1374.

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