• Bernoulli
  • Volume 5, Number 5 (1999), 927-949.

Nonparametric estimation of quadratic regression functionals

Li-Shan Huang and Jianqing Fan

Full-text: Open access


Quadratic regression functionals are important for bandwidth selection of nonparametric regression techniques and for nonparametric goodness-of-fit test. Based on local polynomial regression, we propose estimators for weighted integrals of squared derivatives of regression functions. The rates of convergence in mean square error are calculated under various degrees of smoothness and appropriate values of the smoothing parameter. Asymptotic distributions of the proposed quadratic estimators are considered with the Gaussian noise assumption. It is shown that when the estimators are pseudo-quadratic (linear components dominate quadratic components), asymptotic normality with rate n-1/2 can be achieved.

Article information

Bernoulli, Volume 5, Number 5 (1999), 927-949.

First available in Project Euclid: 12 February 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

asymptotic normality equivalent kernel local polynomial regression


Huang, Li-Shan; Fan, Jianqing. Nonparametric estimation of quadratic regression functionals. Bernoulli 5 (1999), no. 5, 927--949.

Export citation


  • [1] Birgé, L. and Massart, P. (1995) Estimation of integral functionals of a density. Ann. Statist., 23, 11-29.
  • [2] Bickel, P.J. and Ritov, Y. (1988) Estimating integrated squared density derivatives: sharp best order of convergence estimates. Sankhya Ser. A, 50, 381-393.
  • [3] Doksum, K. and Samarov, A. (1995) Nonparametric estimation of global functionals and a measure of the explanatory power of covariates in regression. Ann. Statist., 23, 1443-1473.
  • [4] Donoho, D.L. and Nussbaum, M. (1990) Minimax quadratic estimation of a quadratic functional. J. Complexity, 6, 290-323.
  • [5] Efromovich, S. and Low, M. (1996) On Bickel and Ritov's conjecture about adaptive estimation of the integral of the square of density derivative. Ann. Statist., 24, 682-686.
  • [6] Fan, J. (1991) On the estimation of quadratic functionals. Ann. Statist., 19, 1273-1294.
  • [7] Fan, J. and Gijbels, I. (1995) Data-driven bandwidth selection in local polynomial fitting: variable bandwidth and spatial adaptation. J. Roy. Statist. Soc. Ser. B, 57, 371-394.
  • [8] Fan, J. and Gijbels, I. (1996) Local Polynomial Modelling and Its Applications. London: Chapman & Hall.
  • [9] Fan, J. and Huang, L.-S. (1996) Rates of convergence for the pre-asymptotic substitution bandwidth selector. Technical report no. 912, Department of Statistics, Florida State University.
  • [10] Hall, P. and Marron, J.S. (1987) Estimation of integrated squared density derivatives. Statist. Probab. Lett., 6, 109-115.
  • [11] Hall, P. and Marron, J.S. (1991) Lower bounds for bandwidth selection in density estimation. Probab. Theory Related Fields, 90, 149-173.
  • [12] Hall, P., Kay, J.W. and Titterington, D.M. (1990) Asymptotically optimal difference-based estimation of variance in nonparametric regression. Biometrika, 77, 521-528.
  • [13] Jones, M.C. and Sheather, S.J. (1991) Using non-stochastic terms to advantage in kernel-based estimation of integrated squared density derivatives. Statist. Probab. Lett., 11, 511-514.
  • [14] Khatri, C.G. (1980) Quadratic forms in normal variables. In P.R. Krishnaiah (ed.), Handbook of Statistics, Vol. I, pp. 443-469. New York: North-Holland.
  • [15] Laurent, B. (1996) Efficient estimation of integral functionals of a density. Ann. Statist., 24, 659-681.
  • [16] Laurent, B. (1997) Estimation of integral functionals of a density and its derivatives. Bernoulli, 3, 181-211.
  • [17] Rice, J. (1984) Bandwidth choice for nonparametric regression. Ann. Statist., 12, 1215-1230.
  • [18] Ruppert, D. and Wand, M.P. (1994) Multivariate locally weighted least squares regression. Ann. Statist., 22, 1346-1370.
  • [19] Ruppert, D., Sheather, S.J. and Wand, M.P. (1995) An effective bandwidth selector for local least squares regression. J. Amer. Statist. Assoc., 90, 1257-1270.
  • [20] Stone, C.J. (1980) Optimal rates of convergence for nonparametric estimators. Ann. Statist., 8, 1348-1360.
  • [21] Whittle, P. (1964) On the convergence to normality of quadratic forms in independent random variables. Theory Probab. Appl., 9, 103-108.