Robust Differentiable Functionals for the Additive Hazards Model

DOI: 10.4236/ojs.2015.56064   PDF   HTML   XML   3,255 Downloads   3,697 Views  


In this article, we present a new family of estimators for the regression parameter β in the Additive Hazards Model which represents a gain in robustness not only against outliers but also against unspecific contamination schemes. They are consistent and asymptotically normal and furthermore, and they have a nonzero breakdown point. In Survival Analysis, the Additive Hazards Model proposes a hazard function of the form , where  is a common nonparametric baseline hazard function and z is a vector of independent variables. For this model, the seminal work of Lin and Ying (1994) develops an estimator for the regression parameter β which is asymptotically normal and highly efficient. However, a potential drawback of that classical estimator is that it is very sensitive to outliers. In an attempt to gain robustness, álvarez and Ferrarrio (2013) introduced a family of estimators for β which were still highly efficient and asymptotically normal, but they also had bounded influence functions. Those estimators, which are developed using classical Counting Processes methodology, still retain the drawback of having a zero breakdown point.

Share and Cite:

E. Álvarez, E. and Ferrario, J. (2015) Robust Differentiable Functionals for the Additive Hazards Model. Open Journal of Statistics, 5, 631-644. doi: 10.4236/ojs.2015.56064.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Kalbfleisch, J.D. and Prentice, R.L. (1980) The Statistical Analysis of Failure Time Data. Wiley, New York.
[2] Fleming, T.R. and Harrington, D.P. (1991) Counting Processes and Survival Analysis. Wiley, New York.
[3] Andersen, P.K., Borgan, O., Gill, R.D. and Keiding, N. (1993) Statistical Models Based on Counting Processes. Springer-Verlag, New York.
[4] Aalen, O.O., Borgan, O. and Gjessing, H.K. (2008) Survival and Event History Analysis. A Process Point of View. Springer, New York.
[5] Cox, D.R. (1972) Regression Models and Life-Tables (with Discussion). Journal of the Royal Statistical Society: Series B (Statistical Methodology), 34, 187-220.
[6] Aalen, O.O. (1980) A Model for Nonparametric Regression Analysis of Counting Processes. In: Klonecki, N., Koesh, A. and Rosinski, J., Eds., Lecture Notes in Statistics, 2th Edition, Springer, New York, 1-25.
[7] Bednarski, T. (1989) On Sensitivity of Cox’s Estimator. Statistics & Decisions, 7, 215-228.
[8] Sasieni, P. (1993) Maximum Weighted Partial Likelihood Estimators for the Cox Model. Journal of the American Statistical Association, 88, 144-152.
[9] Sasieni, P. (1993) Some New Estimators for Cox Regression. Annals of Statistics, 21, 1721-1759.
[10] Bednarski, T. (1993) Robust Estimation in Cox’s Regression Model. Scandinavian Journal of Statistics, 20, 213-225.
[11] álvarez, E.E. and Ferrario, J. (2013) Robust Estimation in the Additive Hazards Model. Comm. Statist. Theory Methods, in press.
[12] Maronna, R.A., Martin, R.D. and Yohai, V.J. (2006) Robust Statistics: Theory and Methods. Wiley Series in Probability and Statistics, John Wiley & Sons, Hoboken.
[13] Huber, P.J. and Ronchetti, E.M. (2009) Robust Statistics. Wiley, New York.
[14] Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J. and Stahel, W.A. (1986) Robust Statistics: The Approach Based on Influence Functions. Wiley, New York.
[15] Ferrario, J. (2015) Estimación Robusta en Modelos de Supervivencia con Función de Hazard Aditiva. PhD Thesis, Univesidad Nacional de La Plata, La Plata, Argentina.
[16] Lin, D.Y. and Ying, Z. (1994) Semiparametric Analisis of the Additive Risk Model. Biometrika, 81, 61-71.
[17] Bednarski, T., Clarke, B.R. and Kolkiewicz, W. (1991) Statistical Expansions and Locally Uniform Fréchét Differentiability. Journal of the Australian Mathematical Society, 50, 88-97.

comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.