Abstract
Motivated by a study on factors affecting the level of photosynthetic activity in a natural ecosystem, we propose nonlinear varying-coefficient models, in which the relationship between the predictors and the response variable is allowed to be nonlinear. One-step local linear estimators are developed for the nonlinear varying-coefficient models and their asymptotic normality is established leading to point-wise asymptotic confidence bands for the coefficient functions. Two-step local linear estimators are also proposed for cases where the varying-coefficient functions admit different degrees of smoothness; bootstrap confidence intervals are utilized for inference based on the two-step estimators. We further propose a generalized F-test to study whether the coefficient functions vary over a covariate. We illustrate the proposed methodology via an application to an ecology data set and study the finite sample performance by Monte Carlo simulation studies.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Breiman, L., and Friedman, J. H. (1985), “Estimating Optimal Transformations for Multiple Regression and Correlation,” Journal of the American Statistical Association, 80, 580–619.
Cai, Z., Fan, J., and Li, R. (2000), “Efficient Estimation and Inferences for Varying-Coefficient Models,” Journal of the American Statistical Association, 95, 888–902.
Cleveland, W., Grosse, E., and Shyu, W. (1992), “Local Regression Models,” in Statistical Models in S, eds. J. M. Chambers and T. J. Hastie, Pacific Grove: Wadsworth & Brooks/Cole, pp. 309–376.
Fan, J., and Gijbels, I. (1996), Local Polynomial Modelling and Its Applications, London: Chapman and Hall.
Fan, J., and Li, R. (2004), “New Estimation and Model Selection Procedures for Semiparametric Modeling in Longitudinal Data Analysis,” Journal of the American Statistical Association, 99, 710–723.
Fan, J., and Zhang, W. (1999), “Statistical Estimation in Varying Coefficient Models,” Annals of Statistics, 27, 1491–1518.
Fan, J., and Zhang, W. (2008), “Statistical Methods with Varying Coefficient Models,” Statistics and Its Interface, 1, 179–195.
Fan, J., Zhang, C., and Zhang, J. (2001), “Generalized Likelihood Ratio Statistics and Wilks Phenomenon,” Annals of Statistics, 29, 153–193.
Fan, J., Yao, Q., and Cai, Z. (2003), “Adaptive Varying-Coefficient Linear Models,” Journal of the Royal Statistical Society, Series B, 65, 57–80.
Friedman, J. H. (1991), “Multivariate Adaptive Regression Splines,” Annals of Statistics, 19, 1–67.
Garrett, R. H., and Grisham, C. M. (2005), Biochemistry (3rd ed.), Belmont: Thomson Learning.
Geisser, S. (1975), “The Predictive Sample Reuse Method with Applications,” Journal of the American Statistical Association, 70, 320–328.
Green, P. J., and Silverman, B. W. (1994), Nonparametric Regression and Generalized Linear Models: a Roughness Penalty Approach, London: Chapman and Hall.
Gu, C., and Wahba, G. (1993), “Smoothing Spline Anova with Component-Wise Bayesian “Confidence Intervals”,” Journal of Computational and Graphical Statistics, 2, 97–117.
Härdle, W., and Stoker, T. M. (1989), “Investigating Smooth Multiple Regression by the Method of Average Derivatives,” Journal of the American Statistical Association, 84, 986–995.
Hastie, T., and Tibshirani, R. (1990), Generalized Additive Models, New York: Chapman and Hall.
Hastie, T., and Tibshirani, R. (1993), “Varying-Coefficient Models,” Journal of the Royal Statistical Society, Series B, 55, 757–796.
Hoover, D. R., Rice, J. A., Wu, C. O., and Yang, L.-P. (1998), “Nonparametric Smoothing Estimates of Time-Varying Coefficient Models with Longitudinal Data,” Biometrika, 85, 809–822.
Kauermann, G., and Tutz, G. (1999), “On Model Diagnostics Using Varying Coefficient Models,” Biometrika, 86, 119–128.
Li, K.-C. (1991), “Sliced Inverse Regression for Dimension Reduction,” Journal of the American Statistical Association, 86, 316–327.
Monteith, J. L. (1972), “Solar Radiation and Productivity in Tropical Ecosystems,” Journal of Applied Ecology, 9, 747–766.
Osborne, M. R. (1992), “Fisher’s Method of Scoring,” International Statistical Review/Revue Internationale de Statistique, 60, 99–117.
Seber, G. A. F., and Wild, C. J. (2003), Nonlinear Regression, New York: Wiley-Interscience.
Stone, C. J., Hansen, M. H., Kooperberg, C., and Truong, Y. K. (1997), “Polynomial Splines and Their Tensor Products in Extended Linear Modeling: 1994 Wald Memorial Lecture,” Annals of Statistics, 25, 1371–1470.
Wahba, G. (1984), “Partial Spline Models for the Semiparametric Estimation of Functions of Several Variables,” in Statistical Analysis of Time Series, Proceedings of the Japan U.S. Joint Seminar, Tokyo: Institute of Statistical Mathematics, pp. 319–329.
Wu, C. O., Chiang, C.-T., and Hoover, D. R. (1998), “Asymptotic Confidence Regions for Kernel Smoothing of a Varying-Coefficient Model with Longitudinal Data,” Journal of the American Statistical Association, 93, 1388–1402.
Xia, Y., and Li, W. K. (1999), “On Single-Index Coefficient Regression Models,” Journal of the American Statistical Association, 94, 1275–1285.
Author information
Authors and Affiliations
Corresponding author
Additional information
The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIDA, NCI, NIDDK or the NIH.
Electronic Supplementary Material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Kürüm, E., Li, R., Wang, Y. et al. Nonlinear Varying-Coefficient Models with Applications to a Photosynthesis Study. JABES 19, 57–81 (2014). https://doi.org/10.1007/s13253-013-0157-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13253-013-0157-7