## Abstract

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.

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## Notes

Godambe (1960) had earlier studied variances based on the “information sandwich”, but was concerned with inefficient estimators on the model rather than with misspecification. Cox (1961) derived the sandwich in an informal treatment of tests of separate families of hypotheses, later crediting Huber for a rigorous discussion of the distributional result.

Although \(h_t\) is not itself in \(H\), it is of bounded variation and hence may be renormalized to be in \(H\), which is all that is needed in the sequel since the estimating equations are linear in \(h\).

Indeed, the term \({\mathbb {G}}_N [(R-\pi _0)/\pi _0] \psi _{\theta _0,h}\) in (1), which has the same limiting distribution whether the \(\psi _{\theta _0,h}\) are regarded as random or fixed by conditioning (van der Vaart and Wellner 1996, Sect. 2.9), is the normalized error arising from IPW estimation of the Phase I total of the scores. The solution to the sample survey problem, to estimate this unknown total using two phase stratified sampling, is best achieved when the calibration variables used to adjust the sampling weights are highly correlated with the scores.

This result would be of no surprise to a survey sampler. For estimation of a population total using stratified Bernoulli sampling, it is well known that conditioning on the Phase II stratum totals \(\{n_1,\ldots ,n_J\}\) (see Table 1) is equivalent to finite population stratified sampling (Särndal et al. 1992, Sect. 9.8, Example 9.14).

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## Acknowledgments

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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Breslow, N.E., Hu, J. & Wellner, J.A. Z-estimation and stratified samples: application to survival models.
*Lifetime Data Anal* **21**, 493–516 (2015). https://doi.org/10.1007/s10985-014-9317-5

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DOI: https://doi.org/10.1007/s10985-014-9317-5