Journal of the Korean Statistical Society

The Korean Statistical Society (한국통계학회)
권호 표지

ON MARGINAL INTEGRATION METHOD IN NONPARAMETRIC REGRESSION

  • Published : 2004.12.01
Download PDF
⟨ Previous Next ⟩

Abstract

In additive nonparametric regression, Linton and Nielsen (1995) showed that the marginal integration when applied to the local linear smoother produces a rate-optimal estimator of each univariate component function for the case where the dimension of the predictor is two. In this paper we give new formulas for the bias and variance of the marginal integration regression estimators which are valid for boundary areas as well as fixed interior points, and show the local linear marginal integration estimator is in fact rate-optimal when the dimension of the predictor is less than or equal to four. We extend the results to the case of the local polynomial smoother, too.

Keywords

References

  1. BUJA, A., HASTIE, T. AND TIBSHIRANI, R. (1989). 'Linear smoothers and additive models (with discussion)', The Annals of Statistics, 17, 453-555 https://doi.org/10.1214/aos/1176347115
  2. FAN, J. AND GIJBELS, I. (1996). Local Polynomial Modelling and Its Applications, Chapman & Hall, London
  3. FRIEDMAN, J. H. AND STUETZLE, W. (1981). 'Projection pursuit regression', Journal of the American Statistical Association, 76, 817-823 https://doi.org/10.2307/2287576
  4. HASTIE, T. AND TIBSHIRANI, R. (1990). Generalized Additive Models, Chapman & Hall, New York
  5. LINTON, O. B. AND NIELSEN, J. P. (1995). 'A kernel method of estimating structured nonparametric regression based on marginal integration', Biometrika, 82, 93-100 https://doi.org/10.1093/biomet/82.1.93
  6. MAMMEN, E., LINTON, O. AND NIELSEN, J. (1999). 'The existence and asymptotic properties of a backfitting projection algorithm under weak conditions', The Annals of Statistics, 27, 1442-1490
  7. OPSOMER, J. D. (2000). 'Asymptotic properties of backfitting estimators', Journal of Multivariate Analysis, 73, 166-179 https://doi.org/10.1006/jmva.1999.1868
  8. OPSOMER, J. D. AND RUPPERT, D. (1997). 'Fitting a bivariate additive model by local polynomial regression', The Annals of Statistics, 25, 186-211 https://doi.org/10.1214/aos/1034276626
  9. SEVERANCE-LOSSIN, E. AND SPERLICH, S. (1999). 'Estimation of derivatives for additive separable models', Statistics, 33, 241-265 https://doi.org/10.1080/02331889908802693
  10. SPERLICH, S., LINTON, O. AND H\"ARDLE, W. (1999). 'Integration and backfitting methods in additive models-finite sample properties and comparisons', Test, 8, 419-458