Moment convergence in regularized estimation under multiple and mixed-rates asymptotics
In M-estimation under standard asymptotics, the weak convergence combined with the polynomial type large deviation estimate of the associated statistical random field Yoshida (2011) provides us with not only the asymptotic distribution of the associated M-estimator but also the convergence of its moments, the latter playing an important role in theoretical statistics. In this paper, we study the above program for statistical random fields of multiple and also possibly mixedrates type in the sense of Radchenko (2008) where the associated statistical random fields may be nondifferentiable and may fail to be locally asymptotically quadratic. Consequently, a very strong mode of convergence of a wide range of regularized M-estimators is ensured.Our results are applied to regularized estimation of an ergodic diffusion observed at high frequency.
Keywordsregularized estimation moment convergence large deviation inequality sparse estimation stochastic differential equation mixed-rates asymptotics
2000 Mathematics Subject Classification62E20
Unable to display preview. Download preview PDF.
- 1.G. Afendras and M. Markatou, “Optimality of Training/Test Size and Resampling Effectiveness of Cross- Validation Estimators of the Generalization Error”, arXiv:1511.02980 (2015).Google Scholar
- 2.G. Afendras and M. Markatou, “Uniform Integrability of the OLS Estimators, and the Convergence of Their Moments”, arXiv:1511.02962 (2015).Google Scholar
- 15.J. Jacod and P. Protter, Discretization of Processes, in Stoch. Modelling and Appl. Probab. (Springer, Heidelberg, 2012), Vol. 67.Google Scholar
- 19.H. Masuda, “Parametric Estimation of Lévy Processes”, in Lecture Notes in Math., Vol. 2128: Lévy matters. IV (Springer, Cham, 2015), pp. 179–286.Google Scholar
- 20.H. Masuda and Y. Shimizu, “Moment Convergence in Regularized Estimations”, arXiv:1406.6751v2 (2014).Google Scholar
- 25.Y. Shimizu, “Moment Convergence of Regularized Least-Squares Estimator for Linear Regression Model”, Ann. Inst. Statist. Math. (2016).Google Scholar
- 29.Y. Umezu, Y. Shimizu, H. Masuda, and Y. Ninomiya, “AIC for Nonconcave Penalized Likelihood Method”, arXiv:1509.01688v2 2015.Google Scholar
- 30.A. W. van der Vaart, Asymptotic Statistics, in Cambridge Ser. in Statist. and Probab. Math. (Cambridge Univ. Press, Cambridge, 1998), Vol. 3.Google Scholar