In Chapter 3, we consider semi-parametric inference for semi-survival AC models and propose a likelihood-based approach for estimating the association parameter. A nice equivalent condition for different types of estimating functions is established. Similar idea is used again to construct a score test. Despite that we have seen efficiency gain or power improvement by choosing an appropriate weight function, optimality results are still not available. As mentioned earlier, each term in the product of the likelihood function is neither the conditional likelihood nor the partial likelihood since the probabilities are calculated conditional on an un-nested sequence of conditioning events. Further investigation is needed to elucidate the proposed likelihood, and it is hoped that we develop more understanding for the theoretical properties of the proposed methods.
For establishing the asymptotic normality, the functional delta method is applied for two problems. For the Log-rank statistics in Chapter 4, its expression has been shown to be a statistically differentiable functional that allows us to verify the consistency of the jackknife estimator. This theoretical justification allows us to safely use a computationally simple way for determining the decision rule of the testing procedure. Theoretical property of the jackknife estimator is only proven for the simple case of the Log-rank statistics with no censoring. For other complicated cases, the jackknife method is still a useful tool even though it may lack theoretical justification. Nevertheless finding a tractable and theoretically valid way of constructing confidence intervals or bands still deserves further investigation.
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