Abstract
We examine and compare the finite sample performance of the competing back-fitting and integration methods for estimating additive nonparametric regression using simulated data. Although, the asymptotic properties of the integration estimator, and to some extent the backfitting, method too, are well understood, its small sample properties are not well investigated. Apart from some small experiments in the above cited papers, there is little hard evidence concerning the exact distribution of the estimates. It is our purpose to provide an extensive finite sample comparison between the backfitting procedure and the integration procedure using simulated data.
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References
Breiman, L. and J.H. Friedman (1985). Estimating optimal transformations for multiple regression and correlation (with discussion).Journal of the American Statistical Association,80, 580–619.
Buja, A., T. Hastic and R. Tibshirani (1989). Lincar smoothers and additive models (with discussion).The Annals of Statistics,17, 453–555.
Deaton, A. and J. Muellbauer (1980).Economics and Consumer Behavior. Cambridge University, Press, Cambridge.
Härdle, W., and P. Hall (1993). On the backfitting algorithm for additive regression models.Statistica Neederlandica,47, 43–57.
Härdle, W., and O.B. Linton (1994). Applied nonparametric methods,The Handbook of Econometrics vol. IV, ch. 38 (R.F. Engle and D.F. McFadden, eds.) Elsevier, Amsterdam.
Hastic, T. and R. Tibshirani (1990).Generalized Additive Models, Chapman and Hall, London.
Ibragimov, I.A. and R.Z. Hasminskii (1980). On nonparametric estimation of regression,Soviet Math. Dokl.,21, 810–814.
Linton, O.B. (1997). Efficient estimation of additive nonparamteric regression models.Biometrika,84, 469–473.
Linton, O.B., R. Chen, N. Wang, and W. Härdle (1995). An analysis of transformation for additive nonparametric regression.Journal of the American Statistical Association,92, 1512–1521.
Linton, O.B. and W. Härdle, (1996). Estimation of additive regression models with known links.Biometrika,83, 529–540.
Linton, O.B., E. Mammen and J. Nielson (1998). The Existence and Asymptotic Properties of a Backfitting Projection Algorithm under weak conditions. Manuscript, Yale University.
Linton, O.B. and J.P. Nielsen (1995). A kernel method of cstimating structured nonparametric regression based on marginal integration.Biometrika,82, 93–100.
Masry, E. and D. Tjøstheim (1995). Nonparametric estimation and identification of nonlinear ARCH time series: strong convergence and asymptotic normality.Econometric Theory,11, 258–289.
Masry, E. and D. Tjøstheim, (1997). Additive nonlinear ARX time series and projection estimates.Econometric Theory,13, 214–252.
Newey, W.K. (1990). Semiparametric efficiency bounds.Journal of Applied Econometrics,5, 99–135.
Newey, W.K. (1994). Kernel estimation of partial means.Econometric Theory,10, 233–253.
Nielsen, J.P. (1996). Multiplicative and additive marker dependent hazard estimation based on marginal integration. Manuscript, PFA Pension.
Nielsen, J.P. and O.B. Linton (1997). An optimization interpretation of integration and backfitting estimators for separable nonparametric models.Journal of the Royal Statistical Society, Series B,60, 217–222.
Opsomer, J.D. and D. Ruppert (1997). Fitting a bivariate additive model by local polynomial regression.The Annals of Statistics,25, 212–243.
Porter, J. (1996). Essays in Semiparametric Econometrics.PhD Thesis, MIT.
Powell, J.L. (1994). Estimation in semiparametric models.The Handbook of Econometrics, vol. IV, ch. 41 (R.F. Engle and D.F. McFadden, eds.) Elsevier, Amsterdam.
Ruppert, D. and M.P. Wand (1995). Multivariate Locally Weighted Least Squares.The Annals of Statistics,22, 1346–1370.
Severance-Lossin, E. and S. Sperlich (1997). Estimation of Derivatives for Additive Separable Models.Discussion Paper, SFB 373, Humboldt-University Berlin, Germany.
Stone, C.J. (1980). Optimal rates of convergence for nonparametric estimators.The Annals of Statistics,8, 1348–1360.
Stone, C.J. (1982). Optimal global rates of convergence for nonparametric regression.The Annals of Statistics,8, 1040–1053.
Stone, C.J. (1985). Additive regression and other nonparametric models.The Annals of Statistics,13, 685–705.
Stone, C.J. (1986). The dimensionality reduction principle for generalized additive models.The Annals of Statistics,14, 592–606.
Tjøstheim, D. and B. Auestad (1994). Nonparametric identification of nonlinear time series: projections.Journal of the American Statistical Association,89, 1398–1409.
Venables, W.N. and B. Ripley (1994).Modern applied statistics with S-Plus. Springer Verlag, New York.
Wand, M.P. and M.C. Jones (1995).Kernel Smoothing. Monographs on Statistics and Applied Probability, vol. 60. Chapman and Hall, London.
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The research was supported by the National Science Foundation, NATO, and Deutsche Forschungsmemeinschaft, SFB 373.
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Sperlich, S., Linton, O.B. & Härdle, W. Integration and backfitting methods in additive models-finite sample properties and comparison. Test 8, 419–458 (1999). https://doi.org/10.1007/BF02595879
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DOI: https://doi.org/10.1007/BF02595879