Abstract
The problem to establish the asymptotic distribution of statistical estimators as well as the moment convergence of such estimators has been recognized as an important issue in advanced theories of statistics. This problem has been deeply studied for M-estimators for a wide range of models by many authors. The purpose of this paper is to present an alternative and apparently simple theory to derive the moment convergence of Z-estimators. In the proposed approach the cases of parameters with different rate of convergence can be treated easily and smoothly and any large deviation type inequalities necessary for the same result for M-estimators do not appear in this approach. Applications to the model of i.i.d. observation, Cox’s regression model as well as some diffusion process are discussed.
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Acknowledgments
The authors thank an anonymous referee for her or his helpful comments. This work was supported by Italian MIUR, Grant 2009 (I.N.) and by Grant-in-Aid for Scientific Research (C), 24540152, from Japan Society for the Promotion of Science (Y.N.).
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Negri, I., Nishiyama, Y. Moment convergence of Z-estimators. Stat Inference Stoch Process 20, 387–397 (2017). https://doi.org/10.1007/s11203-016-9146-0
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DOI: https://doi.org/10.1007/s11203-016-9146-0