The Annals of Statistics

Functional single index models for longitudinal data

Ci-Ren Jiang and Jane-Ling Wang

Full-text: Open access

Abstract

A new single-index model that reflects the time-dynamic effects of the single index is proposed for longitudinal and functional response data, possibly measured with errors, for both longitudinal and time-invariant covariates. With appropriate initial estimates of the parametric index, the proposed estimator is shown to be $\sqrt{n}$-consistent and asymptotically normally distributed. We also address the nonparametric estimation of regression functions and provide estimates with optimal convergence rates. One advantage of the new approach is that the same bandwidth is used to estimate both the nonparametric mean function and the parameter in the index. The finite-sample performance for the proposed procedure is studied numerically.

Article information

Source
Ann. Statist., Volume 39, Number 1 (2011), 362-388.

Dates
First available in Project Euclid: 3 December 2010

Permanent link to this document
https://projecteuclid.org/euclid.aos/1291388379

Digital Object Identifier
doi:10.1214/10-AOS845

Mathematical Reviews number (MathSciNet)
MR2797850

Zentralblatt MATH identifier
1209.62073

Subjects
Primary: 62G08: Nonparametric regression 62G05: Estimation
Secondary: 62G20: Asymptotic properties

Keywords
Asymptotic theory cross-validation dimension reduction functional data MAVE smoothing

Citation

Jiang, Ci-Ren; Wang, Jane-Ling. Functional single index models for longitudinal data. Ann. Statist. 39 (2011), no. 1, 362--388. doi:10.1214/10-AOS845. https://projecteuclid.org/euclid.aos/1291388379


Export citation

References

  • Bai, Y., Fung, W. K. and Zhu, Z. Y. (2009). Penalized quadratic inference functions for single-index models with longitudinal data. J. Multivariate Anal. 100 152–161.
  • Friedman, J. H. and Stuetzle, W. (1981). Projection pursuit regression. J. Amer. Statist. Assoc. 76 817–823.
  • Hall, P. (1989). On projection pursuit regression. Ann. Statist. 17 573–588.
  • Härdle, W., Hall, P. and Ichimura, H. (1993). Optimal smoothing in single-index models. Ann. Statist. 21 157–178.
  • Härdle, W. and Stoker, T. (1989). Investigating smooth multiple regression by the method of average derivatives. J. Amer. Statist. Assoc. 84 986–995.
  • Ichimura, H. (1993). Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. J. Econometrics 58 71–120.
  • Jiang, C.-R. and Wang, J.-L. (2010). Covariate adjusted functional principal components analysis for longitudinal data. Ann. Statist. 38 1194–1226.
  • 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.
  • Kong, E. and Xia, Y. (2007). Variable selection for the single-index model. Biometrika 94 217–229.
  • Li, K.-C. (1991). Sliced inverse regression for dimension reduction. J. Amer. Statist. Assoc. 86 316–327.
  • Naik, P. and Tsai, C.-L. (2000). Partial least squares estimator for single-index models. J. R. Stat. Soc. Ser. B Stat. Methodol. 62 763–771.
  • Rice, J. A. and Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. J. Roy. Statist. Soc. Ser. B 53 233–243.
  • Wu, C. O. and Chiang, C. T. (2000). Kernel smoothing on varying coecient models with longitudinal dependent variable. Statist. Sinica 10 433–456.
  • Xia, Y. (2006). Asymptotic distributions for two estimators of the single-index model. Econometric Theory 22 1112–1137.
  • Xia, Y. (2007). A constructive approach to the estimation of dimension reduction directions. Ann. Statist. 35 2654–2690.
  • Xia, Y., Tong, H., Li, W. K. and Zhu, L.-X. (2002). An adaptive estimation of dimension reduction space. J. R. Stat. Soc. Ser. B Stat. Methodol. 64 363–410.
  • Yao, F. (2007). Asymptotic distributions of nonparametric regression estimators for longitudinal of functional data. J. Multivariate Anal. 98 40–56.
  • Yao, F., Müller, H.-G. and Wang, J.-L. (2005). Functional data analysis for sparse longitudinal data. J. Amer. Statist. Assoc. 100 577–590.
  • 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.