## Weighted likelihood for semiparametric models and two-phase stratified samples, with application to Cox regression.(English)Zbl 1142.62014

In the two-phase stratified sampling at the first phase a large sample is drawn and measurements are made on an auxiliary variable $$U$$ that is correlated with the target variable. At the second phase measurements on the target variable $$X$$ are made for a subsample drawn from within strata defined by the auxiliary variable. The authors consider estimates based on Horvitz-Thompson (inverse probability) weighted likelihood (WL) equations for the distribution of $$(X,U)$$. The asymptotic normality of parametric and nonparametric WL estimates normed by $$\sqrt{n}$$ is demonstrated. The results are applied to Bernoulli sampling with estimated weights and to the Cox proportional hazards model.

### MSC:

 62G05 Nonparametric estimation 62D05 Sampling theory, sample surveys 62G20 Asymptotic properties of nonparametric inference 62N02 Estimation in survival analysis and censored data
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### References:

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