The Annals of Statistics

Statistical estimation in varying coefficient models

Jianqing Fan and Wenyang Zhang

Full-text: Open access

Abstract

Varying coefficient models are a useful extension of classical linear models. They arise naturally when one wishes to examine how regression coefficients change over different groups characterized by certain covariates such as age. The appeal of these models is that the coef .cient functions can easily be estimated via a simple local regression.This yields a simple one-step estimation procedure. We show that such a one-step method cannot be optimal when different coefficient functions admit different degrees of smoothness. This drawback can be repaired by using our proposed two-step estimation procedure.The asymptotic mean-squared error for the two-step procedure is obtained and is shown to achieve the optimal rate of convergence. A few simulation studies show that the gain by the two-step procedure can be quite substantial.The methodology is illustrated by an application to an environmental data set.

Article information

Source
Ann. Statist., Volume 27, Number 5 (1999), 1491-1518.

Dates
First available in Project Euclid: 23 September 2004

Permanent link to this document
https://projecteuclid.org/euclid.aos/1017939139

Digital Object Identifier
doi:10.1214/aos/1017939139

Mathematical Reviews number (MathSciNet)
MR2001a:62046

Zentralblatt MATH identifier
0977.62039

Subjects
Primary: 62G07: Density estimation
Secondary: 62J12: Generalized linear models

Keywords
Varying coefficient models local linear fit optimal rate of convergence mean-squared errors

Citation

Fan, Jianqing; Zhang, Wenyang. Statistical estimation in varying coefficient models. Ann. Statist. 27 (1999), no. 5, 1491--1518. doi:10.1214/aos/1017939139. https://projecteuclid.org/euclid.aos/1017939139


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References

  • Breiman, L. and Friedman, J. H. (1985). Estimating optimal transformations for multiple regression and correlation (withdiscussion). J. Amer. Statist. Assoc. 80 580-619.
  • Carroll, R. J., Fan, J., Gijbels, I. and Wand M. P. (1997). Generalized partially linear singleindex models. J. Amer. Statist. Assoc. 92 477-489.
  • Chen, R. and Tsay, R. S. (1993). Functional-coefficient autoregressive models. J. Amer. Statist. Assoc. 88 298-308.
  • Cleveland, W. S., Grosse, E. and Shyu, W. M. (1991). Local regression models. In Statistical Models in S (J. M. Chambers, and T. J. Hastie, eds.) 309-376. Wadsworth / Brooks-Cole, Pacific Grove, CA.
  • Fan, J. and Gijbels, I. (1995). Data-driven bandwidthselection in local polynomial fitting: variable bandwidthand spatial adaptation. J. Roy. Statist. Soc. Ser. B 57 371-394.
  • Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Chapman and Hall, London.
  • Fan, J., H¨ardle, W. and Mammen, E. (1998). Direct estimation of additive and linear components for high dimensional data. Ann. Statist. 26 943-971.
  • Fan, J. and Zhang, J. (2000). Two-step estimation of functional linear models withapplications to longitudinal data. J. Roy. Statist. Soc. Ser. B 62. To appear.
  • Friedman, J. H. (1991). Multivariate adaptive regression splines (withdiscussion). Ann. Statist. 19 1-141.
  • Green, P. J. and Silverman, B. W. (1994). Nonparametric Regression and Generalized Linear Models: A Roughness Penalty Approach. Chapman and Hall, London.
  • Gu, C. and Wahba, G. (1993). Smoothing spline ANOVA with component-wise Bayesian "confidence intervals." J. Comput. Graph. Statist. 2 97-117.
  • H¨ardle, W. and Stoker, T. M. (1989). Investigating smooth multiple regression by the method of average derivatives. J. Amer. Statist. Assoc. 84 986-995.
  • Hastie, T. J. and Tibshirani, R. (1990). Generalized Additive Models. Chapman and Hall, London.
  • Hastie, T. J. and Tibishirani, R. J. (1993). Varying-coefficient models. J. Roy. Statist. Soc. Ser. B 55, 757-796.
  • Heckman, J., Ichimura, H., Smith, J. and Todd, P. (1998). Characterizing selection bias using experimental data. Econometrica, 66 1017-1098.
  • Hoover, D. R., Rice, J. A., Wu, C. O. and Yang, L. P. (1997). Nonparametric smoothing estimates of time-varying coefficient models withlongitudinal data. Biometrika 85 809-822.
  • Li, K. C. (1991). Sliced inverse regression for dimension reduction (withdiscussion). J. Amer. Statist. Assoc. 86 316-342.
  • Mack, Y. P., Silverman, B. W. (1982). Weak and Strong uniform consistency of kernel regression estimates. Z. Wahrsch. Verw. Gebiete 61 405-415.
  • Ruppert, D. (1997). Empirical-bias bandwidths for local polynomial nonparametric regression and density estimation. J. Amer. Statist. Assoc. 92 1049-1062.
  • Ruppert, D., Sheather, S. J. and Wand, M. P. (1995). An effective bandwidthselector for local least squares regression. J. Amer. Statist. Assoc. 90 1257-1270.
  • Shumway, R. H. (1988). Apllied Staistical Time Series Analysis. Prentice-Hall, Englewood Cliffs, NJ.
  • Stone, C. J., Hansen, M., Kooperberg, C. and Truong, Y. K. (1997). Polynomial splines and their tensor products in extended linear modeling. Ann. Statist. 25 1371-1470.
  • Stone, M. (1974). Cross-validatory choice and assessment of statistical predictions (with discussion). J. Roy. Statist. Soc. Ser. B 36 111-147.
  • Wahba, G. (1984). Partial spline models for semiparametric estimation of functions of several variables. In Statistical Analysis of Time Series. Proceedings of the Japan-U.S. Joint Seminar, Tokyo 319-329. Institute of Statistical Mathematics, Tokyo.
  • Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing. Chapman and Hall, London.