×

zbMATH — the first resource for mathematics

Asymptotic equivalence theory for nonparametric regression with random design. (English) Zbl 1029.62044
Summary: This paper establishes the global asymptotic equivalence between nonparametric regression with random design and white noise under sharp smoothness conditions on an unknown regression or drift function. The asymptotic equivalence is established by constructing explicit equivalence mappings between the nonparametric regression and the white-noise experiments, which provide synthetic observations and synthetic asymptotic solutions from any one of the two experiments with asymptotic properties identical to the true observations and given asymptotic solutions from the other. The impact of such asymptotic equivalence results is that an investigation in one nonparametric problem automatically yields asymptotically analogous results in all other asymptotically equivalent nonparametric problems.

MSC:
62G20 Asymptotic properties of nonparametric inference
62G08 Nonparametric regression and quantile regression
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] BROWN, L. D. and LOW, M. G. (1996). Asy mptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384-2398. · Zbl 0867.62022
[2] BROWN, L. D. and ZHANG, C.-H. (1996). Coupling inequalities for some random design matrices and asy mptotic equivalence of nonparametric regression and white noise. Technical Report 96-020, Dept. Statistics, Rutgers Univ.
[3] BROWN, L. D. and ZHANG, C.-H. (1998). Asy mptotic nonequivalence of nonparametric experiments when the smoothness index is 1/2. Ann. Statist. 26 279-287. · Zbl 0932.62061
[4] DONOHO, D. L. and JOHNSTONE, I. M. (1998). Minimax estimation via wavelet shrinkage. Ann. Statist. 26 879-921. · Zbl 0935.62041
[5] DONOHO, D. L., JOHNSTONE, I. M., KERKy ACHARIAN, G. and PICARD, D. (1996). Density estimation by wavelet thresholding. Ann. Statist. 24 508-539. · Zbl 0860.62032
[6] EFROMOVICH, S. (1998). Simultaneous sharp estimation of functions and their derivatives. Ann. Statist. 26 273-278. · Zbl 0930.62035
[7] EFROMOVICH, S. and SAMAROV, A. (1996). Asy mptotic equivalence of nonparametric regression and white noise model has its limits. Statist. Probab. Lett. 28 143-145. · Zbl 0849.62023
[8] FAN, J. (1991). On the estimation of quadratic functionals. Ann. Statist. 19 1273-1294. · Zbl 0729.62076
[9] LE CAM, L. (1986). Asy mptotic Methods in Statistical Decision Theory. Springer, New York. · Zbl 0605.62002
[10] LE CAM, L. and YANG, G. L. (1990). Asy mptotics in Statistics: Some Basic Concepts. Springer, New York.
[11] NUSSBAUM, M. (1996). Asy mptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 2399-2430. · Zbl 0867.62035
[12] PINSKER, M. S. (1980). Optimal filtering of square integrable signals in Gaussian white noise. Problems Inform. Transmission 16 120-133. · Zbl 0452.94003
[13] TSy BAKOV, A. B. (1998). Pointwise and sup-norm sharp adaptive estimation of functions on the Sobolev classes. Ann. Statist. 26 2420-2469. · Zbl 0933.62028
[14] PHILADELPHIA, PENNSy LVANIA 19104 E-MAIL: lbrown@wharton.upenn.edu T. T. CAI DEPARTMENT OF STATISTICS THE WHARTON SCHOOL UNIVERSITY OF PENNSy LVANIA PHILADELPHIA, PENNSy LVANIA 19104 E-MAIL: tcai@wharton.upenn.edu M. G. LOW DEPARTMENT OF STATISTICS THE WHARTON SCHOOL UNIVERSITY OF PENNSy LVANIA PHILADELPHIA, PENNSy LVANIA 19104 C.-H. ZHANG DEPARTMENT OF STATISTICS RUTGERS UNIVERSITY PISCATAWAY, NEW JERSEY 08854
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.