TY - JOUR
T1 - Asymptotic behavior of Cox's partial likelihood and its application to variable selection
AU - Li, Runze
AU - Ren, Jian Jian
AU - Yang, Guangren
AU - Yu, Ye
N1 - Funding Information:
The authors would like to thank the editors Professors Raymond J. Carroll and Qiwei Yao for organizing this special issue. Runze Li is grateful to the editors for their invitation and their constructive comments on an earlier version of this paper. Li’s research is supported by National Institute on Drug Abuse grants P50 DA039838, P50 DA036107, and R01 DA039854, National Science Foundation (NSF) grant DMS 1512422, and National Library of Medicine grant T32 LM012415. His research was also partially supported by National Nature Science Foundation of China grants 11690014 and 11690015. Ren’s research is supported by NSF grants DMS 0905772, DMS 1232424, and DMS 1407461. Yang’s research was supported by the NNSFC grant 11471086, the National Social Science Foundation of China grant 16BTJ032, the Fundamental Research Funds for the Central Universities 15JNQM019 and 21615452, the National Statistical Scientific Research Center Projects 2015LD02, the China Scholarship Council 201506785010 and Science and Technology Program of Guangzhou 2016201604030074. All authors equally contributed to this paper and are listed in alphabetic order. Guangren Yang is the corresponding author. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NSF, the NIDA, or the NIH.
Publisher Copyright:
© Institute of Statistical Science. All rights reserved.
PY - 2018/10
Y1 - 2018/10
N2 - For theoretical properties of variable selection procedures for Cox's model, we study the asymptotic behavior of partial likelihood for the Cox model. We find that the partial likelihood does not behave like an ordinary likelihood, whose sample average typically tends to its expected value, a finite number, in probability. Under some mild conditions, we prove that the sample average of partial likelihood tends to infinity at the rate of the logarithm of the sample size, in probability. We apply the asymptotic results on the partial likelihood to study tuning parameter selection for penalized partial likelihood. We find that the penalized partial likelihood with the generalized cross-validation (GCV) tuning parameter proposed in Fan and Li (2002) enjoys the model selection consistency property, despite the fact that GCV, AIC and Cp, equivalent in the context of linear regression models, are not model selection consistent. Our empirical studies via Monte Carlo simulation and a data example confirm our theoretical findings.
AB - For theoretical properties of variable selection procedures for Cox's model, we study the asymptotic behavior of partial likelihood for the Cox model. We find that the partial likelihood does not behave like an ordinary likelihood, whose sample average typically tends to its expected value, a finite number, in probability. Under some mild conditions, we prove that the sample average of partial likelihood tends to infinity at the rate of the logarithm of the sample size, in probability. We apply the asymptotic results on the partial likelihood to study tuning parameter selection for penalized partial likelihood. We find that the penalized partial likelihood with the generalized cross-validation (GCV) tuning parameter proposed in Fan and Li (2002) enjoys the model selection consistency property, despite the fact that GCV, AIC and Cp, equivalent in the context of linear regression models, are not model selection consistent. Our empirical studies via Monte Carlo simulation and a data example confirm our theoretical findings.
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U2 - 10.5705/ss.202016.0401
DO - 10.5705/ss.202016.0401
M3 - Article
C2 - 30294192
AN - SCOPUS:85054530548
VL - 28
SP - 2713
EP - 2731
JO - Statistica Sinica
JF - Statistica Sinica
SN - 1017-0405
IS - 4
ER -