×

Asymptotic equivalence for nonparametric regression with multivariate and random design. (English) Zbl 1142.62023

Summary: We show that nonparametric regression is asymptotically equivalent, in Le Cam’s sense [L. Le Cam and G. Lo Yang, Asymptotics in statistics. Some basic concepts. 2nd ed. (2000; Zbl 0952.62002)], to a sequence of Gaussian white noise experiments as the number of observations tends to infinity. We propose a general constructive framework, based on approximation spaces, which allows asymptotic equivalence to be achieved, even in the cases of multivariate and random designs.

MSC:

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62B15 Theory of statistical experiments

Citations:

Zbl 0952.62002

References:

[1] Bass, R. F. and Gröchenig, K. (2004). Random sampling of multivariate trigonometric polynomials. SIAM J. Math. Anal. 36 773-795. · Zbl 1096.94008 · doi:10.1137/S0036141003432316
[2] Brown, L. D., Cai, T., Low, M. G. and Zhang, C.-H. (2002). Asymptotic equivalence theory for nonparametric regression with random design. Ann. Statist. 30 688-707. · Zbl 1029.62044 · doi:10.1214/aos/1028674838
[3] Brown, L. D., Carter, A. V., Low, M. G. and Zhang, C.-H. (2004). Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift. Ann. Statist. 32 2074-2097. · Zbl 1062.62083 · doi:10.1214/009053604000000012
[4] Brown, L. D. and Low, M. G. (1996). Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384-2398. · Zbl 0867.62022 · doi:10.1214/aos/1032181159
[5] Brown, L. D. and Zhang, C.-H. (1998). Asymptotic nonequivalence of nonparametric experiments when the smoothness index is 1/2. Ann. Statist. 26 279-287. · Zbl 0932.62061 · doi:10.1214/aos/1030563986
[6] Cai, T. and Brown, L. D. (1999). Wavelet estimation for samples with random uniform design. Statist. Probab. Lett. 42 313-321. · Zbl 0940.62037 · doi:10.1016/S0167-7152(98)00223-5
[7] Carter, A. (2006). A continuous Gaussian process approximation to a nonparametric regression in two dimensions. Bernoulli 12 143-156. · Zbl 1098.62042
[8] Cohen, A. (2000). Wavelet methods in numerical analysis. In Handbook of Numerical Analysis (P. G. Ciarlet, ed.) 7 417-711. North-Holland, Amsterdam. · Zbl 0976.65124
[9] Cohen, A., Daubechies, I. and Vial, P. (1993). Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal. 1 54-81. · Zbl 0795.42018 · doi:10.1006/acha.1993.1005
[10] Dalalyan, A. and Reiß, M. (2007). Asymptotic statistical equivalence for ergodic diffusions: the multidimensional case. Probab. Theory Related Fields 137 25-47. · Zbl 1105.62004 · doi:10.1007/s00440-006-0502-7
[11] De Boor, C. (2001). A Practical Guide to Splines , rev. ed. Springer, New York. · Zbl 0987.65015
[12] Donoho, D. L. and Johnstone, I. M. (1999). Asymptotic minimaxity of wavelet estimators with sampled data. Statist. Sinica 9 1-32. · Zbl 1065.62518
[13] Gaiffas, S. (2007). Sharp estimation in sup norm with random design. Statist. Probab. Lett. 77 782-794. · Zbl 1114.62046 · doi:10.1016/j.spl.2006.11.017
[14] Grama, I. and Nussbaum, M. (1998). Asymptotic equivalence for nonparametric generalized linear models. Probab. Theory Related Fields 111 167-214. · Zbl 0953.62039 · doi:10.1007/s004400050166
[15] Hoffmann, M. and Lepski, O. (2002). Random rates in anisotropic regression (with discussion). Ann. Statist. 30 325-396. · Zbl 1012.62042 · doi:10.1214/aos/1021379858
[16] Johnstone, I. M. and Silverman, B. W. (2004). Boundary coiflets for wavelet shrinkage in function estimation. J. Appl. Probab. 41A 81-98. · Zbl 1049.62041 · doi:10.1239/jap/1082552192
[17] Le Cam, L. and Yang, G. L. (2000). Asymptotics in Statistics. Some Basic Concepts , 2nd ed. Springer, New York. · Zbl 0952.62002
[18] Meyer, Y. (1995). Wavelets and Operators . Cambridge Univ. Press. · Zbl 0776.42019
[19] Nussbaum, M. (1996). Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399-2430. · Zbl 0867.62035 · doi:10.1214/aos/1032181160
[20] Rohde, A. (2004). On the asymptotic equivalence and rate of convergence of nonparametric regression and Gaussian white noise. Statist. Decis. 22 235-243. · Zbl 1057.62033 · doi:10.1524/stnd.22.3.235.57063
[21] Sweldens, W. and Piessens, R. (1993). Wavelet sampling techniques. In 1993 Proceedings of the Statistical Computing Section 20-29. Amer. Statist. Assoc. · Zbl 0755.42019
[22] Wojtaszczyk, P. (1997). A Mathematical Introduction to Wavelets . Cambridge Univ. Press. · Zbl 0865.42026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.