Asymptotic approximation of nonparametric regression experiments with unknown variances
Andrew V. Carter
Source: Ann. Statist. Volume 35, Number 4 (2007), 1644-1673.
Abstract
Asymptotic equivalence results for nonparametric regression experiments have always assumed that the variances of the observations are known. In practice, however the variance of each observation is generally considered to be an unknown nuisance parameter. We establish an asymptotic approximation to the nonparametric regression experiment when the value of the variance is an additional parameter to be estimated or tested. This asymptotically equivalent experiment has two components: the first contains all the information about the variance and the second has all the information about the mean. The result can be extended to regression problems where the variance varies slowly from observation to observation.
Primary Subjects: 62B15
Secondary Subjects: 62G20, 62G08
Keywords: Asymptotic equivalence of experiments; nonparametric regression; variance estimation
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Permanent link to this document: http://projecteuclid.org/euclid.aos/1188405625
Digital Object Identifier: doi:10.1214/009053606000001613
Mathematical Reviews number (MathSciNet):
MR2351100
Zentralblatt MATH identifier:
1147.62034
References
Brown, L., Cai, T., Low, M. and Zhang, C.-H. (2002). Asymptotic equivalence theory for nonparametric regression with random design. Ann. Statist. 30 688--707.
Mathematical Reviews (MathSciNet):
MR1922538
Digital Object Identifier: doi:10.1214/aos/1028674838
Project Euclid: euclid.aos/1028674838
Brown, L. D. and Low, M. G. (1996). Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384--2398.
Mathematical Reviews (MathSciNet):
MR1425958
Digital Object Identifier: doi:10.1214/aos/1032181159
Project Euclid: euclid.aos/1032181159
Carter, A. V. (2006). A continuous Gaussian approximation to a nonparametric regression in two dimensions. Bernoulli 12 143--156.
Mathematical Reviews (MathSciNet):
MR2202326
Project Euclid: euclid.bj/1141136654
Dette, H. and Munk, A. (1998). Testing heteroscedasticity in nonparametric regression. J. R. Stat. Soc. Ser. B Stat. Methodol. 60 693--708.
Mathematical Reviews (MathSciNet):
MR1649535
Digital Object Identifier: doi:10.1111/1467-9868.00149
JSTOR: links.jstor.org
Donoho, D. L. and Johnstone, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 425--455.
Mathematical Reviews (MathSciNet):
MR1311089
Zentralblatt MATH:
0815.62019
Digital Object Identifier: doi:10.1093/biomet/81.3.425
JSTOR: links.jstor.org
Donoho, D. L. and Johnstone, I. M. (1998). Minimax estimation via wavelet shrinkage. Ann. Statist. 26 879--921.
Mathematical Reviews (MathSciNet):
MR1635414
Digital Object Identifier: doi:10.1214/aos/1024691081
Project Euclid: euclid.aos/1024691081
Donoho, D. L., Johnstone, I. M., Kerkyacharian, G. and Picard, D. (1995). Wavelet shrinkage: Asymptopia? (with discussion). J. Roy. Statist. Soc. Ser. B 57 301--369.
Mathematical Reviews (MathSciNet):
MR1323344
JSTOR: links.jstor.org
Eubank, R. L. and Thomas, W. (1993). Detecting heteroscedasticity in nonparametric regression. J. Roy. Statist. Soc. Ser. B 55 145--155.
Mathematical Reviews (MathSciNet):
MR1210427
JSTOR: links.jstor.org
Fan, J. and Yao, Q. (1998). Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85 645--660.
Mathematical Reviews (MathSciNet):
MR1665822
Zentralblatt MATH:
0918.62065
Digital Object Identifier: doi:10.1093/biomet/85.3.645
JSTOR: links.jstor.org
Grama, I. and Nussbaum, M. (1998). Asymptotic equivalence for nonparametric generalized linear models. Probab. Theory Related Fields 111 167--214.
Mathematical Reviews (MathSciNet):
MR1633574
Digital Object Identifier: doi:10.1007/s004400050166
Hall, P., Kay, J. W. and Titterington, D. M. (1990). Asymptotically optimal difference-based estimation of variance in nonparametric regression. Biometrika 77 521--528.
Mathematical Reviews (MathSciNet):
MR1087842
Digital Object Identifier: doi:10.1093/biomet/77.3.521
JSTOR: links.jstor.org
Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998). Wavelets, Approximation and Statistical Applications. Lecture Notes in Statist. 129. Springer, New York.
Mathematical Reviews (MathSciNet):
MR1618204
Härdle, W. and Tsybakov, A. B. (1988). Robust nonparametric regression with simultaneous scale curve estimation. Ann. Statist. 16 120--135.
Mathematical Reviews (MathSciNet):
MR0924860
Digital Object Identifier: doi:10.1214/aos/1176350694
Project Euclid: euclid.aos/1176350694
Johnson, O. (2004). Information Theory and the Central Limit Theorem. Imperial College Press, London.
Mathematical Reviews (MathSciNet):
MR2109042
Zentralblatt MATH:
1061.60019
Le Cam, L. (1964). Sufficiency and approximate sufficiency. Ann. Math. Statist. 35 1419--1455.
Mathematical Reviews (MathSciNet):
MR0207093
Digital Object Identifier: doi:10.1214/aoms/1177700372
Project Euclid: euclid.aoms/1177700372
Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer, New York.
Mathematical Reviews (MathSciNet):
MR0856411
Zentralblatt MATH:
0605.62002
Le Cam, L. and Yang, G. L. (2000). Asymptotics in Statistics, Some Basic Concepts, 2nd ed. Springer, New York.
Mathematical Reviews (MathSciNet):
MR1784901
Zentralblatt MATH:
0952.62002
Mallat, S. G. (1989). Multiresolution approximations and wavelet orthonormal bases of $L^2(\mathbbR)$. Trans. Amer. Math. Soc. 315 69--87.
Mathematical Reviews (MathSciNet):
MR1008470
Digital Object Identifier: doi:10.2307/2001373
JSTOR: links.jstor.org
Müller, H.-G. and Stadtmüller, U. (1987). Estimation of heteroscedasticity in regression analysis. Ann. Statist. 15 610--625.
Mathematical Reviews (MathSciNet):
MR0888429
Digital Object Identifier: doi:10.1214/aos/1176350364
Project Euclid: euclid.aos/1176350364
Pinsker, M. S. (1980). Optimal filtration of square-integrable signals in Gaussian noise. Problems Inform. Transmission 16 52--68.
Mathematical Reviews (MathSciNet):
MR0624591
Zentralblatt MATH:
0452.94003
Rice, J. (1984). Bandwidth choice for nonparametric regression. Ann. Statist. 12 1215--1230.
Mathematical Reviews (MathSciNet):
MR0760684
Digital Object Identifier: doi:10.1214/aos/1176346788
Project Euclid: euclid.aos/1176346788
Rohde, A. (2004). On the asymptotic equivalence and rate of convergence of nonparametric regression and Gaussian white noise. Statist. Decisions 22 235--243.
Mathematical Reviews (MathSciNet):
MR2125610
Ruppert, D., Wand, M. P., Holst, U. and Hössjer, O. (1997). Local polynomial variance-function estimation. Technometrics 39 262--273.
Mathematical Reviews (MathSciNet):
MR1462587
Digital Object Identifier: doi:10.2307/1271131
JSTOR: links.jstor.org
Zhou, H. H. (2004). Minimax estimation with thresholding and asymptotic equivalence for Gaussian variance regression. Ph.D. dissertation, Cornell Univ.
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