## The Annals of Statistics

### Optimal estimation for the functional Cox model

#### Abstract

Functional covariates are common in many medical, biodemographic and neuroimaging studies. The aim of this paper is to study functional Cox models with right-censored data in the presence of both functional and scalar covariates. We study the asymptotic properties of the maximum partial likelihood estimator and establish the asymptotic normality and efficiency of the estimator of the finite-dimensional estimator. Under the framework of reproducing kernel Hilbert space, the estimator of the coefficient function for a functional covariate achieves the minimax optimal rate of convergence under a weighted $L_{2}$-risk. This optimal rate is determined jointly by the censoring scheme, the reproducing kernel and the covariance kernel of the functional covariates. Implementation of the estimation approach and the selection of the smoothing parameter are discussed in detail. The finite sample performance is illustrated by simulated examples and a real application.

#### Article information

Source
Ann. Statist., Volume 44, Number 4 (2016), 1708-1738.

Dates
Revised: January 2016
First available in Project Euclid: 7 July 2016

https://projecteuclid.org/euclid.aos/1467894713

Digital Object Identifier
doi:10.1214/16-AOS1441

Mathematical Reviews number (MathSciNet)
MR3519938

Zentralblatt MATH identifier
1345.62028

#### Citation

Qu, Simeng; Wang, Jane-Ling; Wang, Xiao. Optimal estimation for the functional Cox model. Ann. Statist. 44 (2016), no. 4, 1708--1738. doi:10.1214/16-AOS1441. https://projecteuclid.org/euclid.aos/1467894713

#### 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.
• Breiman, L. and Friedman, J. H. (1985). Estimating optimal transformations for multiple regression and correlation. J. Amer. Statist. Assoc. 80 580–619 (with discussion and with a reply by the authors).
• Cai, T. T. and Yuan, M. (2012). Minimax and adaptive prediction for functional linear regression. J. Amer. Statist. Assoc. 107 1201–1216.
• Carey, J. R., Liedo, P., Müller, H.-G., Wang, J.-L., Senturk, D. and Harshman, L. (2005). Biodemography of a long-lived tephritid: Reproduction and longevity in a large cohort of female Mexican fruit flies, Anastrepha ludens. Exp. Gerontol. 40 793–800.
• Chen, K., Chen, K., Müller, H.-G. and Wang, J.-L. (2011). Stringing high-dimensional data for functional analysis. J. Amer. Statist. Assoc. 106 275–284.
• Cox, D. R. (1972). Regression models and life-tables. J. R. Stat. Soc. Ser. B. Stat. Methodol. 34 187–220.
• Cox, D. R. (1975). Partial likelihood. Biometrika 62 269–276.
• Crambes, C., Kneip, A. and Sarda, P. (2009). Smoothing splines estimators for functional linear regression. Ann. Statist. 37 35–72.
• Ferraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis. Theory and Practice. Springer Series in Statistics. Springer, New York.
• Fleming, T. R. and Harrington, D. P. (1991). Counting Processes and Survival Analysis. Wiley, New York.
• Gu, C. (2013). Smoothing Spline ANOVA Models, 2nd ed. Springer Series in Statistics 297. Springer, New York.
• Hall, P. and Horowitz, J. L. (2007). Methodology and convergence rates for functional linear regression. Ann. Statist. 35 70–91.
• Hastie, T. and Tibshirani, R. (1986). Generalized additive models. Statist. Sci. 1 297–318 (with discussion).
• Hastie, T. J. and Tibshirani, R. (1990). Exploring the nature of covariate effects in proportional hazards model. Biometrics 46 1005–1016.
• Huang, J. (1999). Efficient estimation of the partly linear additive Cox model. Ann. Statist. 27 1536–1563.
• Jacobsen, M. (1984). Maximum likelihood estimation in the multiplicative intensity model: A survey. Int. Stat. Rev. 52 193–207.
• James, G. M. and Hastie, T. J. (2002). Generalized linear models with functional predictors. J. R. Stat. Soc. Ser. B. Stat. Methodol. 63 533–550.
• Johansen, S. (1983). An extension of Cox’s regression model. Int. Stat. Rev. 51 165–174.
• Kong, D., Ibrahim, J., Lee, E. and Zhu, H. (2014). FLCRM: Functional linear Cox regression models. Submitted.
• Müller, H.-G. and Stadtmüller, U. (2005). Generalized functional linear models. Ann. Statist. 33 774–805.
• Qu, S., Wang, J. and Wang, X. (2016). Supplement to “Optimal estimation for the functional Cox model.” DOI:10.1214/16-AOS1441SUPP.
• Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer, New York.
• Riesz, F. and Sz-Nagy, B. (1990). Functional Analysis. Ungar, New York.
• Sasieni, P. (1992a). Information bounds for the conditional hazard ratio in a nested family of regression models. J. R. Stat. Soc. Ser. B. Stat. Methodol. 54 617–635.
• Sasieni, P. (1992b). Nonorthogonal projections and their application to calculating the information in a partly linear Cox model. Scand. J. Stat. 19 215–233.
• Tsiatis, A. A. (1981). A large sample study of Cox’s regression model. Ann. Statist. 9 93–108.
• Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation. Springer, New York. Revised and extended from the 2004 French original, translated by Vladimir Zaiats.
• van der Vaart, A. W. (2000). Asymptotic Statistics. Cambridge Univ. Press, Cambridge.
• van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. With Applications to Statistics. Springer, New York.
• Wahba, G. (1990). Spline Models for Observational Data. CBMS-NSF Regional Conference Series in Applied Mathematics 59. SIAM, Philadelphia, PA.
• Yuan, M. and Cai, T. T. (2010). A reproducing kernel Hilbert space approach to functional linear regression. Ann. Statist. 38 3412–3444.

#### Supplemental materials

• Supplement to “Optimal estimation for the functional Cox model”. Due to space constraint, the derivation of $\mathit{GCV}(\lambda)$ and proofs of lemmas are relegated to the supplementary file [Qu, Wang and Wang (2016)].