Kernel Regression Estimators for Nonparametric Model Calibration in Survey Sampling

  • N. G. CadiganEmail author
  • J. Chen


This paper introduces new kernel regression estimators with strictly non-negative smoothing weights that are iteratively adjusted. One estimator shares the “optimal” asymptotic bias and variance of the local linear regressor. Other estimators have zero sum of residuals, a desirable property in many applications. In a survey sampling context these estimators can easily be adjusted so that they are internally bias calibrated, which is a property with intuitive appeal. We demonstrate in simulations that one of the estimators with zero sum residuals has bias and variance properties that are very close to “optimal”. In addition, we propose a potentially useful refinement to the usual orders of asymptotic approximations for bias and variance of kernel regression smoothers. The smoothers are illustrated using two examples from fisheries applications, one of which involves data from a stratified random bottom-trawl survey.

AMS Subject Classification

62G08 62G20 


Kernel smoothing Residuals Nonparametric regression Nadaraya-Watson estimator Mack-Müller estimator Local linear estimator 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Breidt, F.J., Opsomer, J.D., 2000. Local polynomial regression estimators in survey sampling. Ann. Statist., 28, 1026–1053.MathSciNetCrossRefGoogle Scholar
  2. Breidt, F.J., Claeskens, G., Opsomer, J.D., 2005. Model-assisted estimation in complex surveys using penalised splines. Biometrika, 92, 831–846.MathSciNetCrossRefGoogle Scholar
  3. Chambers, R.L., Dorfman, A.H., Wehrly, T.E., 1993. Bias robust estimation in finite populations using nonparametric calibration. J. Amer. Statist. Assoc., 88, 268–277.MathSciNetzbMATHGoogle Scholar
  4. Chen, J., Thompson, M.E., Wu, C., 2004. Estimation of fish abundance indices based on scientific research trawl surveys. Biometrics, 60, 116–123.MathSciNetCrossRefGoogle Scholar
  5. Chu, C.-K., Marron, J.S., 1991. Choosing a kernel regression estimator. Statist. Sci., 6, 404–436.MathSciNetCrossRefGoogle Scholar
  6. Cochran, W.G., 1997. Sampling Techniques. 3rd edition, John Wiley, New York.zbMATHGoogle Scholar
  7. DFO, 2006. A Harvest Strategy Compliant with the Precautionary Approach, DFO Can. Sci. Advis. Sec. Sci. Advis. Rep. 2006/023. Available at: [accessed April 2009].Google Scholar
  8. Di Marzioa, M., Taylor, C.C., 2007. On boosting kernel regression. J. Statist. Plann. Inference., 138, 2483–2498.MathSciNetCrossRefGoogle Scholar
  9. Doubleday, W.G., 1981. Manual on groundfish surveys in the Northwest Atlantic. NAFO Sci. Coun. Studies, 2, 7–55.CrossRefGoogle Scholar
  10. Dwyer, K.S., Morgan, M.J., Maddock Parsons, D., Brodie, W.B., Healey, B.P., 2007. An assessment of American plaice in NAFO Div. 3LNO., NAFO SCR Doc. 07/56, Serial No. N5408.Google Scholar
  11. Eubank, R.L., 1999. Nonparametric Regression and Spline Smoothing. 2nd edition, Marcel Dekker, Inc., New York.zbMATHGoogle Scholar
  12. Fan, J., 1992. Design-adaptive nonparametric regression. J. Amer. Statist. Assoc., 87, 998–1004.MathSciNetCrossRefGoogle Scholar
  13. Fan, J., 1993. Local linear regression smoothers and their minimax efficiencies. Ann. Statist., 21, 196–216.MathSciNetCrossRefGoogle Scholar
  14. Firth, D., Bennett, K.E., 1998. Robust models in probability sampling. J. Roy. Statist. Soc. Ser. B, 60, 3–21.MathSciNetCrossRefGoogle Scholar
  15. Fuller, W.A., 2002. Regression Estimation for Survey Samples. Surv. Method., 28, 5–23.Google Scholar
  16. Gasser, T., Müller, H.-G., 1979. Kernel estimation of regression functions. In Smoothing Techniques for Curve Estimation, Gasser, T., Rosenblatt, M. (editors), Berlkin, Springer-Verlag, pp. 23–68.CrossRefGoogle Scholar
  17. Gunderson, D.R., 1993. Surveys of Fisheries Resources. John Wiley, New York.Google Scholar
  18. Hall, P., Marron, J.S., Neumann, M.H., Titterington, D.M., 1997. Curve estimation when the design density is low. Ann. Statist., 25, 756–770.MathSciNetCrossRefGoogle Scholar
  19. Hall, P., Turlach, B.A., 1997. Interpolation methods for adapting to sparse design in nonparametric regression. J. Amer. Statist. Assoc., 92, 466–476.MathSciNetCrossRefGoogle Scholar
  20. Härdle, W., 1990. Applied Nonparametric Regression. Cambridge University Press, New York.CrossRefGoogle Scholar
  21. Hart, J.D., Wehrly, T.E., 1992. Kernel regression when the boundary region is large, with an application of testing the adequacy of polynomial models. J. Amer. Statist. Assoc., 87, 1018–1024.MathSciNetCrossRefGoogle Scholar
  22. Hastie, T., Loader, C., 1993. Local regression: automatic kernel carpentry. Statist. Sci., 8, 120–143.CrossRefGoogle Scholar
  23. Jones, M.C., Davies, S.J., Park, B.U., 1994, Versions of kernel-type regression estimators. J. Amer. Statist. Assoc., 89, 825–832.MathSciNetCrossRefGoogle Scholar
  24. Jones, M.C., Linton, O., Nielsen, J.P., 1995. A simple bias reduction method for density estimation. Biometrika, 82, 327–338.MathSciNetCrossRefGoogle Scholar
  25. Leung, D.H.-Y., 2005. Cross-Validation in Nonparametric Regression With Outliers. Ann. Statist., 33, 2291–2310.MathSciNetCrossRefGoogle Scholar
  26. Mack, Y.P., Müller, H.-G., 1989. Derivative estimation in nonparametric regression with random predictor variable. SankhyaSer. A, 51, 59–72.MathSciNetzbMATHGoogle Scholar
  27. Mammen, E., Marron, J.S., 1997. Mass recentered kernel smoothers. Biometrika, 84, 765–777.MathSciNetCrossRefGoogle Scholar
  28. Montanari, G.E., Ranalli, M.G. 2005. Nonparametric model calibration estimation in survey sampling. J. Amer. Statist. Assoc., 100, 1429–1442.MathSciNetCrossRefGoogle Scholar
  29. Müller, H.-G., Song, K.-S., 1993. Identity reproducing multivariate nonparametric regression. J. Multivariate Anal., 46, 237–253.MathSciNetCrossRefGoogle Scholar
  30. Müller, H.-G., 1997. Density adjusted kernel smoothers for random design nonparametric regression. Statist. Probab. Lett., 36, 161–172.MathSciNetCrossRefGoogle Scholar
  31. Nadaraya, E.A., 1964. On estimating regression. Theory Probab. Appl., 9, 141–142.CrossRefGoogle Scholar
  32. Opsomer, J.D., Miller, C.P., 2005. Selecting the amount of smoothing in nonparametric regression estimation for complex surveys. J. Nonparametr. Stat., 17, 593–611.MathSciNetCrossRefGoogle Scholar
  33. Opsomer, J.D., Breidt, F.J., Moisen, G.G., Kauermann, G., 2007. Model-assisted estimation of forest resources with generalized additive models. J. Amer. Statist. Assoc., 102, 400–409.MathSciNetCrossRefGoogle Scholar
  34. Park, B.U., Kim, W.C., Jones, M.C. (1997a), On identity reproducing nonparametric regression estimators. Statist. Probab. Lett., 32, 279–290.MathSciNetCrossRefGoogle Scholar
  35. Park, B.U., Kim, W.C., Ruppert, D., Jones, M.C., Signorini, D.F., Kohn, R., 1997b. Simple transformation techniques for improved non-parametric regression. Scand. J. Statist., 24, 145–163.MathSciNetCrossRefGoogle Scholar
  36. Quinn, T.J., Deriso, R.B., 1999. Quantitative Fish Dynamics. Oxford University Press, New York.Google Scholar
  37. Rodríguez, J.G., Salas, R., 2005. A probabilistic nonparametric estimator. Papeles de Trabajo, No. 2/05. Instituto de Estudios Fiscales.Google Scholar
  38. Särndal, C., Swensson, B., Wretman, J., 1992. Model Assisted Survey Sampling. Springer-Verlag, New York.CrossRefGoogle Scholar
  39. Seifert, B., Gasser, T., 1996. Finite-sample variance of local polynomials: analysis and solutions. J. Amer. Statist. Assoc., 91, 267–275.MathSciNetCrossRefGoogle Scholar
  40. Shelton, P.A., Rice, J.C., 2002. Limits to overfishing: reference points in the context of the Canadian perspective on the precautionary approach, DFO Can. Sci. Adv. Sec. Res. Doc. 2002/084. Available at: [accessed April 2009].Google Scholar
  41. Silverman, B.W., 1986. Density Estimation for Statistics and Data Analysis. Chapman and Hall, New York.CrossRefGoogle Scholar
  42. Smith, S.J., 1990. Use of statistical models for the estimation of abundance from groundfish trawl survey data. Can. J. Fish. Aquat. Sci., 47, 894–903.CrossRefGoogle Scholar
  43. Watson, G.S., 1964. Smooth regression analysis. Sankhya Ser. A, 26, 359–372.MathSciNetzbMATHGoogle Scholar

Copyright information

© Grace Scientific Publishing 2010

Authors and Affiliations

  1. 1.Science Branch, Fisheries and Oceans CanadaNorthwest Atlantic Fisheries CenterSt. John’sCanada
  2. 2.University of British ColumbiaVancouverCanada

Personalised recommendations