On inference in parametric survival data models

The usual parametric models for survival data are of the following form. Some parametrically specified hazard rate $\alpha(s,\theta)$ is assumed for possibly censored random life times $X_1^0,\ldots,X_n^0$; one observes only $X_i=\min\{X_i^0,c_i\}$ and $\delta_i=I\{X_i^0\le c_i\}$ for certain censoring times $c_i$ that either are given or come from some censoring distribution. We study the following problems: What do the maximum likelihood estimator and other estimators really estimate when the true hazard rate $\alpha(s)$ is different from the parametric hazard rates? What is the limit distribution of an estimator under such outside-the-model circumstances? How can traditional model-based analyses be made model-robust? Does the model-agnostic viewpoint invite alternative estimation approaches? What are the consequences of carrying out model-based and model-robust bootstrapping? How do theoretical and empirical influence functions generalise to situations with censored data? How do methods and results carry over to more complex models for life history data like regression models and Markov chains?