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On the backfitting algorithm for additive regression models

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  • W. Härdle
  • P. Hall

Abstract

We analyse additive regression model fitting via the backfitting algorithm. We show that in the case of a large class of curve estimators, which includes regressograms, simple step‐by‐step formulae can be given for the back‐fitting algorithm. The result of each cycle of the algorithm may be represented succinctly in terms of a sequence of d projections in n‐dimensional space, where d is the number of design coordinates and n is sample size. It follows from our formulae that the limit of the algorithm is simply the projection of the data onto that vector space which is orthogonal to the space of all n‐vectors fixed by each of the projections. The formulae also provide the convergence rate of the algorithm, the variance of the backfitting estimator, consistency of the estimator, and the relationship of the estimator to that obtained by directly minimizing mean squared distance.

Suggested Citation

  • W. Härdle & P. Hall, 1993. "On the backfitting algorithm for additive regression models," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 47(1), pages 43-57, March.
    • Handle: RePEc:bla:stanee:v:47:y:1993:i:1:p:43-57
    DOI: 10.1111/j.1467-9574.1993.tb01405.x
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    Cited by:

    1. Jianbao Chen & Suli Cheng, 2021. "GMM Estimation of a Partially Linear Additive Spatial Error Model," Mathematics, MDPI, vol. 9(6), pages 1-28, March.
    2. Chèze-Payaud, Nathalie & Poggi, Jean-Michel & Portier, Bruno, 1998. "Estimation and test of linearity for a class of additive nonlinear models," Statistics & Probability Letters, Elsevier, vol. 40(2), pages 189-201, September.
    3. Opsomer, Jean D., 2000. "Asymptotic Properties of Backfitting Estimators," Journal of Multivariate Analysis, Elsevier, vol. 73(2), pages 166-179, May.
    4. Cheng, Suli & Chen, Jianbao, 2023. "GMM estimation of partially linear additive spatial autoregressive model," Computational Statistics & Data Analysis, Elsevier, vol. 182(C).
    5. David Conde & Miguel A. Fernández & Cristina Rueda & Bonifacio Salvador, 2021. "Isotonic boosting classification rules," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 15(2), pages 289-313, June.
    6. Hegland, Markus & McIntosh, Ian & Turlach, Berwin A., 1999. "A parallel solver for generalised additive models," Computational Statistics & Data Analysis, Elsevier, vol. 31(4), pages 377-396, October.
    7. Stefan Sperlich & Oliver Linton & Wolfgang Härdle, 1999. "Integration and backfitting methods in additive models-finite sample properties and comparison," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 8(2), pages 419-458, December.

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