, Volume 19, Issue 1, pp 166–186 | Cite as

On internally corrected and symmetrized kernel estimators for nonparametric regression

  • Oliver B. LintonEmail author
  • David T. Jacho-Chávez
Original Paper


We investigate the properties of a kernel-type multivariate regression estimator first proposed by Mack and Müller (Sankhya 51:59–72, 1989) in the context of univariate derivative estimation. Our proposed procedure, unlike theirs, assumes that bandwidths of the same order are used throughout; this gives more realistic asymptotics for the estimation of the function itself but makes the asymptotic distribution more complicated. We also propose a modification of this estimator that has a symmetric smoother matrix, which makes it admissible, unlike some other common regression estimators. We compare the performance of the estimators in a Monte Carlo experiment.


Multivariate regression Smoothing matrix Symmetry 

Mathematics Subject Classification (2000)

62G08 62G20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bickel PJ, Ritov Y (1991) Efficient estimation of linear functionals of a probability measure with known marginal distributions. Ann Stat 19:1316–1346 zbMATHCrossRefMathSciNetGoogle Scholar
  2. Cohen A (1966) All admissible linear estimates of the mean vector. Ann Math Stat 37:458–463 zbMATHCrossRefGoogle Scholar
  3. Cox DR, Hinkley DV (1974) Theoretical statistics, 1st edn. Chapman and Hall, London zbMATHGoogle Scholar
  4. Fan J (1992) Design-adaptive nonparametric regression. J Am Stat Assoc 87:998–1004 zbMATHCrossRefGoogle Scholar
  5. Fan J, Gijbels I (1996) Local polynomial modelling and its applications, 1st edn. Chapman and Hall, London zbMATHGoogle Scholar
  6. Gasser T, Müller HG (1979) Kernel estimation of regression functions. In: Smoothing techniques for curve estimation, 1st edn. Lecture notes in mathematics, vol 757. Springer, New York, pp 23–68 Google Scholar
  7. Giné W, Guillou A (2002) Rates of strong uniform consistency for multivariate kernel density estimators. Ann Inst Henri Poincaré 38:907–921 zbMATHCrossRefGoogle Scholar
  8. Hall P, Li Q, Racine JS (2007) Nonparametric estimation of regression functions in the presence of irrelevant regressors. Rev Econ Stat 89(4):784–789 CrossRefGoogle Scholar
  9. Hastie TJ, Tibshirani RJ (1990) Generalized additive models, 1st edn. Monographs on statistics and applied probability, vol 43. Chapman and Hall, London zbMATHGoogle Scholar
  10. Hastie TJ, Tibshirani RJ (2000) Bayesian backfitting. Stat Sci 15(3):196–223 zbMATHCrossRefMathSciNetGoogle Scholar
  11. Hengartner NW, Sperlich S (2005) Rate optimal estimation with the integration method in the presence of many covariates. J Multivar Anal 95:246–272 zbMATHCrossRefMathSciNetGoogle Scholar
  12. Johnston GJ (1979) Smooth nonparametric regression analysis. PhD dissertation, University of North Carolina at Chapel Hill Google Scholar
  13. Jones MC, Davies SJ, Park BU (1994) Versions of kernel-type regression estimators. J Am Stat Assoc 89:825–832 zbMATHCrossRefMathSciNetGoogle Scholar
  14. Ker AP (2004) Nonparametric estimation of possibly similar densities. Cardon Research paper 2004-06 Google Scholar
  15. Kim W, Linton O, Hengartner N (1999) A computationally efficient oracle estimator for additive nonparametric regression with bootstrap confidence intervals. J Comput Graph Stat 8:278–297 CrossRefMathSciNetGoogle Scholar
  16. Li Q, Racine JS (2007) Nonparametric econometrics: theory and practice. Princeton University Press, Princeton zbMATHGoogle Scholar
  17. Linton OB, Nielsen JP (1995) A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika 82:93–100 zbMATHCrossRefMathSciNetGoogle Scholar
  18. Mack YP, Müller HG (1989) Derivative estimation in nonparametric regression with random predictor variable. Sankhya 51:59–72 zbMATHGoogle Scholar
  19. Pagan A, Ullah A (1999) Nonparametric econometrics, 1st edn. Themes in modern econometrics. Cambridge University Press, Cambridge Google Scholar
  20. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in C: the art of scientific computing, 2nd edn. Cambridge University Press, Cambridge Google Scholar
  21. Robinson PM (1983) Nonparametric estimation for time series models. J Time Ser Anal 4:185–208 zbMATHCrossRefMathSciNetGoogle Scholar
  22. Sam AG, Ker AP (2006) Nonparametric regression under alternative data environments. Stat Probab Lett 76(10):1037–1046 zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2009

Authors and Affiliations

  1. 1.Department of EconomicsLondon School of EconomicsLondonUK
  2. 2.Department of EconomicsIndiana UniversityBloomingtonUSA

Personalised recommendations