Lifetime Data Analysis

, Volume 21, Issue 4, pp 493–516 | Cite as

Z-estimation and stratified samples: application to survival models



The infinite dimensional Z-estimation theorem offers a systematic approach to joint estimation of both Euclidean and non-Euclidean parameters in probability models for data. It is easily adapted for stratified sampling designs. This is important in applications to censored survival data because the inverse probability weights that modify the standard estimating equations often depend on the entire follow-up history. Since the weights are not predictable, they complicate the usual theory based on martingales. This paper considers joint estimation of regression coefficients and baseline hazard functions in the Cox proportional and Lin–Ying additive hazards models. Weighted likelihood equations are used for the former and weighted estimating equations for the latter. Regression coefficients and baseline hazards may be combined to estimate individual survival probabilities. Efficiency is improved by calibrating or estimating the weights using information available for all subjects. Although inefficient in comparison with likelihood inference for incomplete data, which is often difficult to implement, the approach provides consistent estimates of desired population parameters even under model misspecification.


Semiparametric models Proportional hazards Additive hazards Calibration of sampling weights Model misspecification Survey sampling 



Wellner’s research was supported in part by National Science Foundation Grant DMS-1104832 and National Institute of Allegery and Infectious Diseases Grant 2R01 AI291968-04. Dedicated to Niels Keiding on the occasion of his 70th birthday.


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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of BiostatisticsUniversity of WashingtonSeattleUSA
  2. 2.Department of StatisticsUniversity of WashingtonSeattleUSA

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