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.
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription.
Read more about accessing full-text
References
Donoho, D. L. (1994). Statistical estimation and optimal recovery. Ann. Statist. 22 238–270.
Dümbgen, L. (2002). Application of local rank tests to nonparametric regression. J. Nonparametr. Statist. 14 511–537.
Dümbgen, L. and Johns, R. B. (2004). Confidence bands for isotonic median curves using sign-tests. J. Comput. Graph. Statist. 13 519–533.
Dümbgen, L. and Spokoiny, V. G. (2001). Multiscale testing of qualitative hypotheses. Ann. Statist. 29 124–152.
Dümbgen, L. and Walther, G. (2008). Multiscale inference about a density. Ann. Statist. 36. To appear.
Ermakov, M. S. (1990). Minimax detection of a signal in a white Gaussian noise. Theory Probab. Appl. 35 667–679.
Eubank, R. L. and Hart, J. D. (1992). Testing goodness-of-fit in regression via order selection criteria. Ann. Statist. 20 1412–1425.
Fan, J. (1996). Test of significance based on wavelet thresholding and Neyman’s truncation. J. Amer. Statist. Assoc. 91 674–688.
Fan, J. and Huang, J.-S. (2001). Goodness-of-fit tests for parametric regression models. J. Amer. Statist. Assoc. 96 640–652.
Fan, J., Zhang, C. and Zhang, J. (2001). Generalized likelihood ratio statistics and Wilks phenomenon. Ann. Statist. 29 153–193.
Hájek, J. and Šidak, Z. (1967). Theory of Rank Tests. Academic Press, New York.
Hart, J. D. (1997). Nonparametric Smoothing and Lack-of-Fit Tests. Springer, New York.
Horowitz, J. and Spokoiny, V. (2001). An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica 69 599–631.
Horowitz, J. and Spokoiny, V. (2002). An adaptive, rate-optimal test of linearity for median regression models. J. Amer. Statist. Assoc. 97 822–835.
Ingster, Y. I. (1982). Minimax nonparametric detection of signals in white Gaussian noise. Problems Inform. Transmission 18 130–140.
Mathematical Reviews (MathSciNet):
MR689340
Ingster, Y. I. (1987). Minimax testing of nonparametric hypotheses on a distribution density in Lp-metrics. Theory Probab. Appl. 31 333–337.
Ingster, Y. I. (1993). Asymptotically minimax hypothesis testing for nonparametric alternatives. I–III. Math. Methods Statist. 2 85–114, 171–189, 249–268.
Ledwina, T. and Kallenberg, W. C. M. (1995). Consistency and Monte Carlo simulation of a data-driven version of smooth goodness-of-fit tests. Ann. Statist. 23 1594–1608.
Ledwina, T. (1994). Data-driven version of Neyman’s smooth test of fit. J. Amer. Statist. Assoc. 89 1000–1005.
Leonov, S. L. (1999). Remarks on extremal problems in nonparametric curve estimation. Statist. Probab. Lett. 43 169–178.
Lepski, O. V. (1993). On asymptotically exact testing of nonparametric hypotheses. CORE Discussion Paper No. 9329, Univ. Catholique de Louvain.
Lepski, O. V. and Tsybakov, A. B. (2000). Asymptotically exact nonparametric hypothesis testing in sup-norm and at a fixed point. Probab. Theory Related Fields 117 17–48.
Spokoiny, V. (1996). Adaptive hypothesis testing using wavelets. Ann. Statist. 24 2477–2498.
Spokoiny, V. (1998). Adaptive and spatially adaptive testing a nonparametric hypothesis. Math. Methods Statist. 7 254–273.
Sz. Nagy, B. (1941). Über Integralungleichungen zwischen einer Funktion und ihrer Ableitung. Acta Sci. Math. 10 64–74.
van der Vaart, A. (1998). Asymptotic Statistics. Cambridge Univ. Press.
