No. 1084

Title:

Moment convergence of M-estimators

Author(s):

Nishiyama, Yoichi (The Institute of Statistical Mathematics)

Key words:

M-estimator, rate of convergence, moment convergence.

Abstract:

    This paper extends the rate of convergence theorem of $M$-estimators presented by van der Vaart and Wellner (1996, Springer) who gave a result of the form $ r_{n}(\widehat{\theta}_{n}-\theta_{0}) =O_{P}(1) $ to a result of the form $ \sup_{n}E|r_{n}(\widehat{\theta}_{n}-\theta_{0})|^{p}< \infty $ for any $ p \geq 1 $. This result is useful for deriving the moment convergence of the rescaled residual. Some examples in the i.i.d. case and the Gaussian white noise model are discussed.