## The Annals of Statistics

### Asymptotic approximation of nonparametric regression experiments with unknown variances

Andrew V. Carter

#### 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.

#### Article information

Source
Ann. Statist., Volume 35, Number 4 (2007), 1644-1673.

Dates
First available in Project Euclid: 29 August 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1188405625

Digital Object Identifier
doi:10.1214/009053606000001613

Mathematical Reviews number (MathSciNet)
MR2351100

Zentralblatt MATH identifier
1147.62034

Subjects
Primary: 62B15: Theory of statistical experiments
Secondary: 62G20: Asymptotic properties 62G08: Nonparametric regression

#### Citation

Carter, Andrew V. Asymptotic approximation of nonparametric regression experiments with unknown variances. Ann. Statist. 35 (2007), no. 4, 1644--1673. doi:10.1214/009053606000001613. https://projecteuclid.org/euclid.aos/1188405625

#### 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.
• Brown, L. D. and Low, M. G. (1996). Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 2384–2398.
• Carter, A. V. (2006). A continuous Gaussian approximation to a nonparametric regression in two dimensions. Bernoulli 12 143–156.
• Dette, H. and Munk, A. (1998). Testing heteroscedasticity in nonparametric regression. J. R. Stat. Soc. Ser. B Stat. Methodol. 60 693–708.
• Donoho, D. L. and Johnstone, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 425–455.
• Donoho, D. L. and Johnstone, I. M. (1998). Minimax estimation via wavelet shrinkage. Ann. Statist. 26 879–921.
• 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.
• Eubank, R. L. and Thomas, W. (1993). Detecting heteroscedasticity in nonparametric regression. J. Roy. Statist. Soc. Ser. B 55 145–155.
• Fan, J. and Yao, Q. (1998). Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85 645–660.
• Grama, I. and Nussbaum, M. (1998). Asymptotic equivalence for nonparametric generalized linear models. Probab. Theory Related Fields 111 167–214.
• Hall, P., Kay, J. W. and Titterington, D. M. (1990). Asymptotically optimal difference-based estimation of variance in nonparametric regression. Biometrika 77 521–528.
• Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998). Wavelets, Approximation and Statistical Applications. Lecture Notes in Statist. 129. Springer, New York.
• Härdle, W. and Tsybakov, A. B. (1988). Robust nonparametric regression with simultaneous scale curve estimation. Ann. Statist. 16 120–135.
• Johnson, O. (2004). Information Theory and the Central Limit Theorem. Imperial College Press, London.
• Le Cam, L. (1964). Sufficiency and approximate sufficiency. Ann. Math. Statist. 35 1419–1455.
• Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer, New York.
• Le Cam, L. and Yang, G. L. (2000). Asymptotics in Statistics, Some Basic Concepts, 2nd ed. Springer, New York.
• Mallat, S. G. (1989). Multiresolution approximations and wavelet orthonormal bases of ${L}^2(\mathbb{R})$. Trans. Amer. Math. Soc. 315 69–87.
• Müller, H.-G. and Stadtmüller, U. (1987). Estimation of heteroscedasticity in regression analysis. Ann. Statist. 15 610–625.
• Pinsker, M. S. (1980). Optimal filtration of square-integrable signals in Gaussian noise. Problems Inform. Transmission 16 52–68.
• Rice, J. (1984). Bandwidth choice for nonparametric regression. Ann. Statist. 12 1215–1230.
• Rohde, A. (2004). On the asymptotic equivalence and rate of convergence of nonparametric regression and Gaussian white noise. Statist. Decisions 22 235–243.
• Ruppert, D., Wand, M. P., Holst, U. and Hössjer, O. (1997). Local polynomial variance-function estimation. Technometrics 39 262–273.
• Zhou, H. H. (2004). Minimax estimation with thresholding and asymptotic equivalence for Gaussian variance regression. Ph.D. dissertation, Cornell Univ.