ABSTRACT
Using the martingale methods and the tools for asymptotic statistics which we have studied so far, in this chapter we will prove the consistency and the asymptotic normality of Z-estimators in various parametric models in statistics.
We first give an intuitive explanation of our approach with an example of the i.i.d. model, and then develop the approach up to a general theory for Z-estimators. Logically, readers may start their study from the second section, where the rigorous description actually begins. However, it is highly recommended that readers first read the intuitive explanation quickly to get a clear overview of the approach. The rigorous arguments for the i.i.d. model will be completed at the beginning of the third section.
Next we deal with Markov chain models, method of moment estimators, diffusion process models, and Cox's regression models. Just after the explanation for our treatment of Markov chain models, we give an interim summary of our approach.
Finally, we give a remark to treat the cases where the components of Z-estimators may have different rates of convergence in the last section; the corresponding example of diffusion process models will also be presented.
