The Annals of Statistics

Spline-backfitted kernel smoothing of nonlinear additive autoregression model

Li Wang and Lijian Yang

Full-text: Open access

Abstract

Application of nonparametric and semiparametric regression techniques to high-dimensional time series data has been hampered due to the lack of effective tools to address the “curse of dimensionality.” Under rather weak conditions, we propose spline-backfitted kernel estimators of the component functions for the nonlinear additive time series data that are both computationally expedient so they are usable for analyzing very high-dimensional time series, and theoretically reliable so inference can be made on the component functions with confidence. Simulation experiments have provided strong evidence that corroborates the asymptotic theory.

Article information

Source
Ann. Statist. Volume 35, Number 6 (2007), 2474-2503.

Dates
First available in Project Euclid: 22 January 2008

Permanent link to this document
http://projecteuclid.org/euclid.aos/1201012969

Digital Object Identifier
doi:10.1214/009053607000000488

Mathematical Reviews number (MathSciNet)
MR2382655

Zentralblatt MATH identifier
1129.62038

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62G08: Nonparametric regression

Keywords
Bandwidths B spline knots local linear estimator mixing Nadaraya–Watson estimator nonparametric regression

Citation

Wang, Li; Yang, Lijian. Spline-backfitted kernel smoothing of nonlinear additive autoregression model. The Annals of Statistics 35 (2007), no. 6, 2474--2503. doi:10.1214/009053607000000488. http://projecteuclid.org/euclid.aos/1201012969.


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References

  • Bosq, D. (1998). Nonparametric Statistics for Stochastic Processes, 2nd ed. Lecture Notes in Statist. 110. Springer, New York.
  • Chen, R. and Tsay, R. S. (1993). Nonlinear additive ARX models. J. Amer. Statist. Assoc. 88 955–967.
  • de Boor, C. (2001). A Practical Guide to Splines, rev. ed. Springer, New York.
  • Doukhan, P. (1994). Mixing: Properties and Examples. Lecture Notes in Statist. 85. Springer, New York.
  • Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Chapman and Hall, London.
  • Fan, J., Härdle, W. and Mammen, E. (1998). Direct estimation of low-dimensional components in additive models. Ann. Statist. 26 943–971.
  • Fan, J. and Jiang, J. (2005). Nonparametric inferences for additive models. J. Amer. Statist. Assoc. 100 890–907.
  • Härdle, W., Hlávka, Z. and Klinke, S. (2000). XploRe Application Guide. Springer, Berlin.
  • Hastie, T. J. and Tibshirani, R. J. (1990). Generalized Additive Models. Chapman and Hall, London.
  • Hengartner, N. W. and Sperlich, S. (2005). Rate optimal estimation with the integration method in the presence of many covariates. J. Multivariate Anal. 95 246–272.
  • Horowitz, J., Klemelä, J. and Mammen, E. (2006). Optimal estimation in additive regression. Bernoulli 12 271–298.
  • Horowitz, J. and Mammen, E. (2004). Nonparametric estimation of an additive model with a link function. Ann. Statist. 32 2412–2443.
  • Huang, J. Z. (1998). Projection estimation in multiple regression with application to functional ANOVA models. Ann. Statist. 26 242–272.
  • Huang, J. Z. and Yang, L. (2004). Identification of nonlinear additive autoregressive models. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 463–477.
  • Linton, O. B. (1997). Efficient estimation of additive nonparametric regression models. Biometrika 84 469–473.
  • Linton, O. B. and Härdle, W. (1996). Estimation of additive regression models with known links. Biometrika 83 529–540.
  • Linton, O. B. and Nielsen, J. P. (1995). A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika 82 93–100.
  • Mammen, E., Linton, O. and Nielsen, J. (1999). The existence and asymptotic properties of a backfitting projection algorithm under weak conditions. Ann. Statist. 27 1443–1490.
  • Opsomer, J. D. and Ruppert, D. (1997). Fitting a bivariate additive model by local polynomial regression. Ann. Statist. 25 186–211.
  • Pham, D. T. (1986). The mixing property of bilinear and generalized random coefficient autoregressive models. Stochastic Process. Appl. 23 291–300.
  • Robinson, P. M. (1983). Nonparametric estimators for time series. J. Time Ser. Anal. 4 185–207.
  • Sperlich, S., Tjøstheim, D. and Yang, L. (2002). Nonparametric estimation and testing of interaction in additive models. Econometric Theory 18 197–251.
  • Stone, C. J. (1985). Additive regression and other nonparametric models. Ann. Statist. 13 689–705.
  • Stone, C. J. (1994). The use of polynomial splines and their tensor products in multivariate function estimation (with discussion). Ann. Statist. 22 118–184.
  • Tjøstheim, D. and Auestad, B. (1994). Nonparametric identification of nonlinear time series: Projections. J. Amer. Statist. Assoc. 89 1398–1409.
  • Wang, L. and Yang, L. (2006). Spline-backfitted kernel smoothing of nonlinear additive autoregression model. Manuscript. Available at www.arxiv.org/abs/math/0612677.
  • Xue, L. and Yang, L. (2006). Estimation of semiparametric additive coefficient model. J. Statist. Plann. Inference 136 2506–2534.
  • Xue, L. and Yang, L. (2006). Additive coefficient modeling via polynomial spline. Statist. Sinica 16 1423–1446.
  • Yang, L., Härdle, W. and Nielsen, J. P. (1999). Nonparametric autoregression with multiplicative volatility and additive mean. J. Time Ser. Anal. 20 579–604.
  • Yang, L., Sperlich, S. and Härdle, W. (2003). Derivative estimation and testing in generalized additive models. J. Statist. Plann. Inference 115 521–542.