The Annals of Statistics

Effect of mean on variance function estimation in nonparametric regression

Lie Wang, Lawrence D. Brown, T. Tony Cai, and Michael Levine

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Abstract

Variance function estimation in nonparametric regression is considered and the minimax rate of convergence is derived. We are particularly interested in the effect of the unknown mean on the estimation of the variance function. Our results indicate that, contrary to the common practice, it is not desirable to base the estimator of the variance function on the residuals from an optimal estimator of the mean when the mean function is not smooth. Instead it is more desirable to use estimators of the mean with minimal bias. On the other hand, when the mean function is very smooth, our numerical results show that the residual-based method performs better, but not substantial better than the first-order-difference-based estimator. In addition our asymptotic results also correct the optimal rate claimed in Hall and Carroll [J. Roy. Statist. Soc. Ser. B 51 (1989) 3–14].

Article information

Source
Ann. Statist., Volume 36, Number 2 (2008), 646-664.

Dates
First available in Project Euclid: 13 March 2008

Permanent link to this document
https://projecteuclid.org/euclid.aos/1205420514

Digital Object Identifier
doi:10.1214/009053607000000901

Mathematical Reviews number (MathSciNet)
MR2396810

Zentralblatt MATH identifier
1133.62033

Subjects
Primary: 62G08: Nonparametric regression 62G20: Asymptotic properties

Keywords
Minimax estimation nonparametric regression variance estimation

Citation

Wang, Lie; Brown, Lawrence D.; Cai, T. Tony; Levine, Michael. Effect of mean on variance function estimation in nonparametric regression. Ann. Statist. 36 (2008), no. 2, 646--664. doi:10.1214/009053607000000901. https://projecteuclid.org/euclid.aos/1205420514


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References

  • Brown, L. D. and Levine, M. (2006). Variance estimation in nonparametric regression via the difference sequence method. Ann. Statist. To appear.
  • Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Chapman and Hall, London.
  • Fan, J. and Yao, Q. (1998). Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85 645–660.
  • Gasser, T. and Müller, H. G. (1979). Kernel estimation of regression functions. Smoothing Techniques for Curve Estimation. Lecture Notes in Math. 757 23–68. Springer, New York.
  • Gasser, T. and Müller, H. G. (1984). Estimating regression functions and their derivatives by the kernel method. Scand. J. Statist. 11 197–211.
  • Gasser, T., Müller, H. G. and Mammitzsch, V. (1985). Kernels for nonparametric curve estimation. J. Roy. Statist. Soc. B 47 238–252.
  • Hall, P. and Carroll, R. J. (1989). Variance function estimation in regression: The effect of estimating the mean. J. Roy. Statist. Soc. Ser. B 51 3–14.
  • Hall, P., Kay, J. and Titterington, D. M. (1990). Asymptotically optimal difference-based estimation of variance in nonparametric regression. Biometrika 77 521–528.
  • Karlin, S. and Studden, W. J. (1966). Tchebycheff Systems: With Applications in Analysis and Statistics. Interscience, New York.
  • Le Cam, L. (1986). Asymptotic Methods in Statistical Decision Theory. Springer, New York.
  • Levine, M. (2006). Bandwidth selection for a class of difference-based variance estimators in the nonparametric regression: A possible approach. Comput. Statist. Data Anal. 50 3405–3431.
  • Müller, H. G. and Stadtmüller, U. (1987). Estimation of heteroscedasticity in regression analysis. Ann. Statist. 15 610–625.
  • Müller, H. G. and Stadtmüller, U. (1993). On variance function estimation with quadratic forms. J. Statist. Plann. Inference 35 213–231.
  • Munk, A., Bissantz, N., Wagner, T. and Freitag, G. (2005). On difference based variance estimation in nonparametric regression when the covariate is high dimensional. J. Roy. Statist. Soc. Ser. B 67 19–41.
  • Neumann, M.(1994). Fully data-driven nonparametric variance estimators. Statistics 25 189–212.
  • Rice, J. (1984). Bandwidth choice for nonparametric kernel regression. Ann. Statist. 12 1215–1230.
  • Ruppert, D., Wand, M. P., Holst, U. and Hössjer, O. (1997). Local polynomial variance function estimation. Technometrics 39 262–273.
  • von Neumann, J.(1941). Distribution of the ratio of the mean squared successive difference to the variance. Ann. Math. Statist. 12 367–395.
  • von Neumann, J. (1942). A further remark concerning the distribution of the ratio of the mean squared successive difference to the variance. Ann. Math. Statist. 13 86–88.