Abstract
In this paper, we study the properties of the simultaneous and componentwise splines for the varying coefficient model with repeatedly measured (longitudinal) dependent variable and time invariant covariates. The proposed simultaneous smoothing spline estimators are mainly obtained from the penalized least squares with adjustment for the variations of covariates in the penalized terms. We do this mainly to avoid the penalized terms being influenced by the scales of the covariates and the random smoothing parameters appearing in the estimators, which complicates the derivation of the asymptotic properties of the estimators. It is shown in this study that our estimators have smaller variances than the componentwise ones. Through a Monte Carlo simulation and two empirical examples, the simultaneous smoothing splines are all found to be more accurate in the variances.
Similar content being viewed by others
References
Abramovich, F. and Grinshtein, V. (1999). Derivation of equivalent kernel for general spline smoothing. A systematic approach,Bernoulli,5, 359–379.
Chiang, C. T., Rice, J. A. and Wu, C. O. (2001). Smoothing spline estimation for varying coefficient models with repeatedly measured dependent variable,Journal of the American Statistical Association,96, 605–619.
Fan, J. Q. and Zhang, J. T. (2000). Functional linear models for longitudinal data,Journal of the Royal Statistical Society, Series B,62, 303–322.
Hastie, T. J. and Tibshirani, R. J. (1993). Varying coefficient model,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.
Huang, J. Z., Wu, C. O. and Zhou, L. (2002). Varying-coefficient models and basis function approximations for the analysis repeated measurements,Biometrika,89, 111–128.
Kaslow, R. A., Ostrow, D. G., Detels, R., Phair, J. P., Polk, B. F. and Rinaldo, C. R. (1987). The multicenter AIDS cohort study: Rationale, organization and selected characteristics of the participants,American Journal of Epidemiology,126, 310–318.
Lin, X. and Carroll, R. J. (2000). Semiparametric regression for the mean and rate functions of recurrent events,Journal of the Royal Statistical Society, Series B,62, 711–730.
Nychka, D. (1995). Splines as local smoothers,The Annals of Statistics,23, 1175–1197.
Rice, J. A. and Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves,Journal of the Royal Statistical Society, Series B,53, 233–243.
Umbricht-Schneiter, A., Montoya, I. D., Hoover D. R., Demuch, K. L., Chiang, C.-T., and Preston, K. L. (1999). Naltrexone shortened opioid detoxification with buprenorphine,Drug and Alcohol Dependence,56, 181–190.
Wahba, G. (1990).Spline Models for Observational Data, SIAM, Philadelphia, Pennsylvania.
Wu, C. O. and Chiang, C. T. (2000). Kernel estimation and its applications of a varying coefficient model with longitudinal data,Statistica Sinica,10, 433–456.
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.
Wu, C. O., Yu, K. F. and Chiang, C.-T. (2000). A two-step smoothing method for varying coefficient models with repeated measurements,Annals of the Institute of Statistical Mathematics,52, 519–543.
Zeger, S. L. and Diggle, P. J. (1994). Semiparametric models for longitudinal data with application to CD4 cell numbers in HIV seroconverters,Biometrics,50, 689–699.
Author information
Authors and Affiliations
About this article
Cite this article
Chiang, CT. Comparisons between simultaneous and componentwise splines for varying coefficient models. Ann Inst Stat Math 57, 637–653 (2005). https://doi.org/10.1007/BF02915430
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02915430