Built on

Wilks, S. S. (1938). The large-sample distribution of the likelihood ratio for testing composite hypotheses
Original
1938
Earlier work this paper cites.
Moreau, J. J. (1962). Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement polaires
1962
Earlier work this paper cites.
Andersen, P. K., and Gill, R. D. (1982). Cox’s regression model for counting processes: a large sample study
1982
Earlier work this paper cites.
Gordon, Y. (1988). On Milman’s inequality and random subspaces which escape through a mesh in
1988
Earlier work this paper cites.
Jacobsen M. (1989). Existence and Unicity of MLEs in Discrete Exponential Family Distributions
1989
Earlier work this paper cites.
Tibshirani, R. (1997). The LASSO method for variable selection in the Cox model
1997
Earlier work this paper cites.
Murphy, S. A. and van der Vaart, A. W. (2000). On profile likelihood
2000
Earlier work this paper cites.
Liang, T., and Sur, P. (2020). A Precise High-Dimensional Asymptotic Theory for Boosting and Minimum-L1-Norm Interpolated Classifiers
Original
2002
Earlier work this paper cites.
Fan, J. and Li, R. (2002). Variable selection for Cox’s proportional hazards model and frailty model
2002
Earlier work this paper cites.
Gui, J., and Li, H. (2005). Penalized Cox regression analysis in the high-dimensional and low-sample size settings, with applications to microarray gene expression data
2005
Earlier work this paper cites.
Zhang, H. H., and Lu, W. (2007). Adaptive Lasso for Cox’s proportional hazards model
2007
Earlier work this paper cites.

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2022

A Modern Theory for High-dimensional Cox Regression Models

Ye, Hanxuan, Zhang, Xianyang, Zhou, Huijuan

Understand

The proportional hazards model has been extensively used in many fields such as biomedicine to estimate and perform statistical significance testing on the effects of covariates influencing the survival time of patients.

The classical theory of maximum partial-likelihood estimation (MPLE) is used by most software packages to produce inference, e.g., the coxph function in R and the PHREG procedure in SAS.
In this paper, we investigate the asymptotic behavior of the MPLE in the regime in which the number of parameters p is of the same order as the number of samples n.
The main results are (i) existence of the MPLE undergoes a sharp 'phase transition'; (ii) the classical MPLE theory leads to invalid inference in the high-dimensional regime.

Built on

Wilks, S. S. (1938). The large-sample distribution of the likelihood ratio for testing composite hypotheses
Original
1938
Earlier work this paper cites.
Moreau, J. J. (1962). Décomposition orthogonale d’un espace hilbertien selon deux cônes mutuellement polaires
1962
Earlier work this paper cites.
Andersen, P. K., and Gill, R. D. (1982). Cox’s regression model for counting processes: a large sample study
1982
Earlier work this paper cites.
Gordon, Y. (1988). On Milman’s inequality and random subspaces which escape through a mesh in
1988
Earlier work this paper cites.
Jacobsen M. (1989). Existence and Unicity of MLEs in Discrete Exponential Family Distributions
1989
Earlier work this paper cites.
Tibshirani, R. (1997). The LASSO method for variable selection in the Cox model
1997
Earlier work this paper cites.
Murphy, S. A. and van der Vaart, A. W. (2000). On profile likelihood
2000
Earlier work this paper cites.
Liang, T., and Sur, P. (2020). A Precise High-Dimensional Asymptotic Theory for Boosting and Minimum-L1-Norm Interpolated Classifiers
Original
2002
Earlier work this paper cites.
Fan, J. and Li, R. (2002). Variable selection for Cox’s proportional hazards model and frailty model
2002
Earlier work this paper cites.
Gui, J., and Li, H. (2005). Penalized Cox regression analysis in the high-dimensional and low-sample size settings, with applications to microarray gene expression data
2005
Earlier work this paper cites.
Zhang, H. H., and Lu, W. (2007). Adaptive Lasso for Cox’s proportional hazards model
2007
Earlier work this paper cites.

Similar

Donoho, D. L., Maleki, A., and Montanari, A. (2009). Message-passing algorithms for compressed sensing
2009
Cited alongside, same era.
Huber , P. J. and Ronchetti, E. (2009)
2009
Cited alongside, same era.
Javanmard, A., and Soltanolkotabi, M. (2020). Precise statistical analysis of classification accuracies for adversarial training
Original
2010
Cited alongside, same era.
Thrampoulidis, C., Oymak, S., and Soltanolkotabi, M. (2020). Theoretical insights into multiclass classification: A high-dimensional asymptotic view
Original
2011
Cited alongside, same era.
Bradic, J., Fan, J., and Jiang, J. (2011). Regularization for Cox’s proportional hazards model with NP-dimensionality
2011
Cited alongside, same era.

Then

Donoho, D. and Montanari A. (2016). High dimensional robust M-estimation: Asymptotic variance via approximate message passing
2016
Later among the works it cites.
Fang, E. X., Ning, Y., and Liu, H. (2017). Testing and confidence intervals for high dimensional proportional hazards models
2017
Later among the works it cites.
Dhifallah, O., Thrampoulidis, C., and Lu, Y. M. (2018). Phase retrieval via polytope optimization: Geometry, phase transitions, and new algorithms
Original
2018
Later among the works it cites.
Yu, Y., Bradic, J., and Samworth, R. J. (2018). Confidence intervals for high-dimensional Cox models
Original
2018
Later among the works it cites.
Candès, E. J. and Sur, P. (2018) . The phase transition for the existence of the maximum likelihood estimate in high-dimensional logistic regression
Original

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

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Gaïffas, S., and Guilloux, A. (2012). High-dimensional additive hazards models and the Lasso

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Huang, J., Sun, T., Ying, Z., Yu, Y., and Zhang, C. H. (2013). Oracle inequalities for the lasso in the Cox model

2013

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El Karoui, N. (2013). Asymptotic behavior of unregularized and ridge-regularized high-dimensional robust regression estimators: rigorous results

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2013

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El Karoui, N., Bean, D., Bickel, P. J., Lim, C., and Yu, B. (2013). On robust regression with high-dimensional predictors

2013

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Kong, S., and Nan, B. (2014). Non-asymptotic oracle inequalities for the high-dimensional Cox regression via Lasso

2014

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Thrampoulidis, C., Oymak, S., and Hassibi, B. (2015). Regularized linear regression: A precise analysis of the estimation error

2015

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2018