• Bernoulli
  • Volume 12, Number 1 (2006), 143-156.

A continuous Gaussian approximation to a nonparametric regression in two dimensions

Andrew V. Carter

Full-text: Open access


Estimating the mean in a nonparametric regression on a two-dimensional regular grid of design points is asymptotically equivalent to estimating the drift of a continuous Gaussian process on the unit square. In particular, we provide a construction of a Brownian sheet process with a drift that is almost the mean function in the nonparametric regression. This can be used to apply estimation or testing procedures from the continuous process to the regression experiment as in Le~Cam's theory of equivalent experiments. Our result is motivated by first looking at the amount of information lost in binning the data in a density estimation problem.

Article information

Bernoulli, Volume 12, Number 1 (2006), 143-156.

First available in Project Euclid: 28 February 2006

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

asymptotic equivalence of experiments density estimation nonparametric regression


Carter, Andrew V. A continuous Gaussian approximation to a nonparametric regression in two dimensions. Bernoulli 12 (2006), no. 1, 143--156.

Export citation


  • [1] Brown, L.D. and Low, M.G. (1996) Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist., 24, 2384-2398.
  • [2] 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.
  • [3] 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.
  • [4] 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.
  • [5] Carter, A.V. (2002) Deficiency distance between multinomial and multivariate normal experiments. Ann. Statist., 30, 708-730.
  • [6] Donoho, D.L. and Johnstone, I.M. (1999) Asymptotic minimaxity of wavelet estimators with sampled data. Statist. Sinica, 9, 1-32.
  • [7] Donoho, D.L., Johnstone, I.M., Kerkyacharian, G. and Picard, D. (1995) Wavelet shrinkage: Asymoptopia? J. Roy. Statist. Soc. Ser. B, 57, 301-369.
  • [8] Dudley, R.M. (2002) Real Analysis and Probability. Cambridge: Cambridge University Press.
  • [9] Efromovich, S. (1999) Nonparametric Curve Estimation: Methods, Theory, and Applications. New York: Springer-Verlag.
  • [10] Fan, J. and Marron, J.S. (1994) Fast implementation of nonparametric curve estimators. J. Comput. Graph. Statist., 3, 35-56.
  • [11] Grama, I. and Nussbaum, M. (1998) Asymptotic equivalence for nonparametric generalized linear models. Probab. Theory Related Fields, 111, 167-214.
  • [12] Hall, P. and Wand, M.P. (1996) On the accuracy of binned kernel density estimators. J. Multivariate Anal., 56, 165-184.
  • [13] Le Cam, L. (1986) Asymptotic Methods in Statistical Decision Theory. New York: Springer-Verlag.
  • [14] Nussbaum, M. (1996) Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist., 24, 2399-2430.
  • [15] Rohde, A. (2004) On the asymptotic equivalence and rate of convergence of nonparametric regression and Gaussian white noise. Statist. Decisions, 22, 235-243.
  • [16] Silverman, B.W. (1982) Algorithm AS 176: Kernel density estimation using the fast Fourier transform. Appl. Statist., 31, 93-99.
  • [17] Strasser, H. (1985) Mathematical Theory of Statistics: Statistical Experiments and Asymptotic Decision Theory. Berlin: Walter de Gruyter.