Moment convergence in regularized estimation under multiple and mixed-rates asymptotics

TL;DR

This paper extends moment convergence results for M-estimators to complex scenarios involving multiple and mixed-rate asymptotics, including non-differentiable and non-quadratic cases, with applications to high-frequency diffusion data.

Contribution

It develops a framework for moment convergence of regularized M-estimators under complex asymptotic regimes, broadening applicability beyond classical assumptions.

Findings

Established strong convergence modes for regularized M-estimators.

Applied results to high-frequency ergodic diffusion estimation.

Extended theoretical tools to non-differentiable and mixed-rate cases.

Abstract

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 mixed-rates type in the sense of Radchenko (2008) where the associated statistical random fields may be non-differentiable 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. The results are applied to regularized estimation of an ergodic diffusion observed at high frequency.