Moment convergence of Z-estimators

The problem to establish not only the asymptotic distribution results for statistical estimators but also the moment convergence of the estimators has been recognized as an important issue in advanced theories of statistics. There is an authorised theory dealing with this problem for some M-estimators by Ibragimov and Has'minskii (1981). A large deviation inequality, which was a crucial point of Ibragimov and Has'minskii's (1981) theory, has been proved with a good generality by Yoshida (2011). The purpose of this paper is to present an alternative, simple theory to derive the moment convergence of Z-estimators; any large deviation type inequalities do not appear in our approach. Moreover, a merit of our approach is that the cases of parameters with different rate of convergence can be treated easily and smoothly. Applications to some di usion process models and Cox's regression model are discussed.