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Smooth backfitting in practice

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  • Jens Perch Nielsen
  • Stefan Sperlich

Abstract

Summary. Compared with the classical backfitting of Buja, Hastie and Tibshirani, the smooth backfitting estimator (SBE) of Mammen, Linton and Nielsen not only provides complete asymptotic theory under weaker conditions but is also more efficient, robust and easier to calculate. However, the original paper describing the SBE method is complex and the practical as well as the theoretical advantages of the method have still neither been recognized nor accepted by the statistical community. We focus on a clear presentation of the idea, the main theoretical results and practical aspects like implementation and simplification of the algorithm. We introduce a feasible cross‐validation procedure and apply it to the problem of data‐driven bandwidth choice for the SBE. By simulations it is shown that the SBE and our cross‐validation work very well indeed. In particular, the SBE is less affected by sparseness of data in high dimensional regression problems or strongly correlated designs. The SBE has reasonable performance even in 100‐dimensional additive regression problems.

Suggested Citation

  • Jens Perch Nielsen & Stefan Sperlich, 2005. "Smooth backfitting in practice," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(1), pages 43-61, February.
    • Handle: RePEc:bla:jorssb:v:67:y:2005:i:1:p:43-61
    DOI: 10.1111/j.1467-9868.2005.00487.x
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    Cited by:

    1. Marcella Cartledge & Luke Taylor, 2022. "Incentive pay and decision quality: evidence from NCAA football coaches," Applied Economics, Taylor & Francis Journals, vol. 54(30), pages 3505-3520, June.
    2. de Uña Álvarez, Jacobo & Roca Pardiñas, Javier, 2009. "Additive models in censored regression," Computational Statistics & Data Analysis, Elsevier, vol. 53(9), pages 3490-3501, July.
    3. Joel L. Horowitz, 2012. "Nonparametric additive models," CeMMAP working papers CWP20/12, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    4. Song, Qiongxia & Yang, Lijian, 2010. "Oracally efficient spline smoothing of nonlinear additive autoregression models with simultaneous confidence band," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2008-2025, October.
    5. Fengler, M.R. & Mammen, E. & Vogt, M., 2015. "Specification and structural break tests for additive models with applications to realized variance data," Journal of Econometrics, Elsevier, vol. 188(1), pages 196-218.
    6. Théophile T. Azomahou & Raouf Boucekkine & Bity Diene, 2009. "A closer look at the relationship between life expectancy and economic growth," International Journal of Economic Theory, The International Society for Economic Theory, vol. 5(2), pages 201-244, June.
    7. Su, Liangjun & Lu, Xun, 2013. "Nonparametric dynamic panel data models: Kernel estimation and specification testing," Journal of Econometrics, Elsevier, vol. 176(2), pages 112-133.
    8. Jing Wang & Lijian Yang, 2009. "Efficient and fast spline-backfitted kernel smoothing of additive models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(3), pages 663-690, September.
    9. Martins-Filho, Carlos & yang, ke, 2007. "Finite sample performance of kernel-based regression methods for non-parametric additive models under common bandwidth selection criterion," MPRA Paper 39295, University Library of Munich, Germany.
    10. Peter Malec, 2016. "A Semiparametric Intraday GARCH Model," Cambridge Working Papers in Economics 1633, Faculty of Economics, University of Cambridge.
    11. Joel L. Horowitz, 2012. "Nonparametric additive models," CeMMAP working papers 20/12, Institute for Fiscal Studies.
    12. Stefan Sperlich, 2014. "On the choice of regularization parameters in specification testing: a critical discussion," Empirical Economics, Springer, vol. 47(2), pages 427-450, September.
    13. Berthold R. Haag, 2008. "Non‐parametric Regression Tests Using Dimension Reduction Techniques," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(4), pages 719-738, December.
    14. Qi Li & Jeffrey Scott Racine, 2006. "Nonparametric Econometrics: Theory and Practice," Economics Books, Princeton University Press, edition 1, volume 1, number 8355.
    15. Suneel Babu Chatla, 2023. "Nonparametric inference for additive models estimated via simplified smooth backfitting," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 75(1), pages 71-97, February.
    16. Xia Cui & Heng Peng & Songqiao Wen & Lixing Zhu, 2013. "Component Selection in the Additive Regression Model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(3), pages 491-510, September.
    17. Huang, Zhensheng & Zhang, Riquan, 2009. "Efficient estimation of adaptive varying-coefficient partially linear regression model," Statistics & Probability Letters, Elsevier, vol. 79(7), pages 943-952, April.
    18. Lin, Huazhen & Pan, Lixian & Lv, Shaogao & Zhang, Wenyang, 2018. "Efficient estimation and computation for the generalised additive models with unknown link function," Journal of Econometrics, Elsevier, vol. 202(2), pages 230-244.
    19. Rui Li & Yuanyuan Zhang, 2021. "Two-stage estimation and simultaneous confidence band in partially nonlinear additive model," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(8), pages 1109-1140, November.
    20. Su, Liangjun & Ullah, Aman, 2008. "Local polynomial estimation of nonparametric simultaneous equations models," Journal of Econometrics, Elsevier, vol. 144(1), pages 193-218, May.
    21. Rodríguez-Álvarez, María Xosé & Roca-Pardiñas, Javier & Cadarso-Suárez, Carmen, 2011. "A new flexible direct ROC regression model: Application to the detection of cardiovascular risk factors by anthropometric measures," Computational Statistics & Data Analysis, Elsevier, vol. 55(12), pages 3257-3270, December.
    22. Holger Dette & Matthias Guhlich & Natalie Neumeyer, 2015. "Testing for additivity in nonparametric quantile regression," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 67(3), pages 437-477, June.
    23. Holger Dette & Juan Carlos Pardo‐Fernández & Ingrid Van Keilegom, 2009. "Goodness‐of‐Fit Tests for Multiplicative Models with Dependent Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(4), pages 782-799, December.
    24. Juhyun Park & Burkhardt Seifert, 2010. "Local additive estimation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(2), pages 171-191, March.
    25. Li, Degui & Linton, Oliver & Lu, Zudi, 2015. "A flexible semiparametric forecasting model for time series," Journal of Econometrics, Elsevier, vol. 187(1), pages 345-357.

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