The Annals of Statistics

Efficient estimation of the partly linear additive Cox model

Jian Huang

Full-text: Open access

Abstract

The partly linear additive Cox model is an extension of the (linear) Cox model and allows flexible modeling of covariate effects semiparametrically. We study asymptotic properties of the maximum partial likelihood estimator of this model with right-censored data using polynomial splines. We show that, with a range of choices of the smoothing parameter (the number of spline basis functions) required for estimation of the nonparametric components, the estimator of the finite-dimensional regression parameter is root-$n$ consistent, asymptotically normal and achieves the semiparametric information bound. Rates of convergence for the estimators of the nonparametric components are obtained. They are comparable to the rates in nonparametric regression. Implementation of the estimation approach can be done easily and is illustrated by using a simulated example.

Article information

Source
Ann. Statist., Volume 27, Number 5 (1999), 1536-1563.

Dates
First available in Project Euclid: 23 September 2004

Permanent link to this document
https://projecteuclid.org/euclid.aos/1017939141

Digital Object Identifier
doi:10.1214/aos/1017939141

Mathematical Reviews number (MathSciNet)
MR2000m:62015

Zentralblatt MATH identifier
0977.62035

Subjects
Primary: 62G05: Estimation 62G20: Asymptotic properties
Secondary: 62G07: Density estimation 62P99: None of the above, but in this section

Keywords
Additive regression asymptotic normality right-censored data partial likelihood polynomial splines projection rate of convergence semiparametric information bound

Citation

Huang, Jian. Efficient estimation of the partly linear additive Cox model. Ann. Statist. 27 (1999), no. 5, 1536--1563. doi:10.1214/aos/1017939141. https://projecteuclid.org/euclid.aos/1017939141


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References

  • ANDERSEN, P. K. and GILL, R. D. 1982. Cox's regression model for counting processes: a large sample study. Ann. Statist. 10 1100 1120. Z.
  • ANDERSEN, P. K., BORGAN, O., GILL, R. D. and KEIDING, N. 1993. Statistical Models Based on Counting Processes. Springer, New York. Z.
  • BICKEL, P. J., KLAASSEN, C. A. J., RITOV, Y. and WELLNER, J. A. 1993. Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Univ. Press.
  • BREIMAN, L. and FRIEDMAN, J. H. 1985. Estimating optimal transformations for multiple regression and correlation. J. Amer. Statist. Assoc. 80 580 598. Z.
  • COX, D. R. 1972. Regression models and life-tables. J. Roy. Statist. Soc. Ser. B 34 187 220. Z.
  • DABROWSKA, D. M. 1997. Smoothed Cox regression. Ann. Statist. 25 1510 1540. Z.
  • FLEMING, T. R. and HARRINGTON, D. P. 1991. Counting Processes and Survival Analysis. Wiley, New York. Z.
  • GRAMBSCH, P. M., THERNEAU, T. M. and FLEMING, T. R. 1990. Martingale-based residuals for survival models. Biometrika 77 147 160. Z.
  • GRAMBSCH, P. M., THERNEAU, T. M. and FLEMING T. R. 1995. Diagnostic plots to reveal functional form for covariates in multiplicative intensity models. Biometrics 51 1469 1482. Z.
  • HASTIE, T. and TIBSHIRANI, R. 1986. Generalized additive models. Statist. Sci. 1 297 318. Z.
  • HASTIE, T. and TIBSHIRANI, R. 1990. Exploring the nature of covariate effects in the proportional hazards model. Biometrics 46 1005 1016. Z.
  • HUANG, J. 1996. Efficient estimation for the Cox model with interval censoring. Ann. Statist. 24 540 568. Z.
  • KOOPERBERG, C., STONE, C. and TRUONG, Y. K. 1995. The L2 rate of convergence for hazard regression. Scand. J. Statist. 22 143 158. Z. O'SULLIVAN, F. 1993. Nonparametric estimation in the Cox model. Ann. Statist. 21 124 145. Z.
  • SASIENI, P. 1992a. Information bounds for the conditional hazard ratio in a nested family of regression models. J. Roy. Statist. Soc. Ser. B 54 617 635. Z.
  • SASIENI, P. 1992b. Non-orthogonal projections and their application to calculating the information in a partly linear Cox model. Scand. J. Statist. 19 215 233. Z.
  • SCHUMAKER, L. 1981. Spline Functions: Basic Theory. Wiley, New York. Z.
  • SHEN, X. and WONG, W. H. 1994. Convergence rate of sieve estimates. Ann. Statist. 22 580 615. Z.
  • STONE, C. J. 1985. Additive regression and other nonparametric models. Ann. Statist. 13 689 705. Z.
  • STONE, C. J. 1986a. The dimensionality reduction principle for generalized additive models. Ann. Statist. 14 590 606. Z.
  • STONE, C. J. 1986b. Comment on ``Generalized Additive Models'' by T. Hastie and R. Tibshirani. Statist. Sci. 1 312 314. Z.
  • STONE, C. J. 1994. The use of polynomial splines and their tensor products in multivariate Z. function estimation with discussion. Ann. Statist. 22 118 184. Z.
  • TSIATIS, A. A. 1981. A large sample study of Cox's regression model. Ann. Statist. 9 93 108. Z.
  • VAN DER VAART, A. W. 1991. On differentiable functionals. Ann. Statist. 19 178 204. Z.
  • VAN DER VAART, A. W. and WELLNER, J. A. 1996. Weak Convergence and Empirical Processes. Springer, New York.
  • IOWA CITY, IOWA 52242 E-MAIL: jian@stat.uiowa.edu