Vol 26, No 2 (2017)

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.

We consider a class of mixture models for positive continuous data and the estimation of an underlying parameter θ of the mixing distribution. With a unified approach, we obtain classes of dominating estimators under squared error loss of an unbiased estimator, which include smooth estimators. Applications include estimating noncentrality parameters of chi-square and F-distributions, as well as ρ²/(1 − ρ²), where ρ is amultivariate correlation coefficient in a multivariate normal set-up. Finally, the findings are extended to situations, where there exists a lower bound constraint on θ.

This paper contains a theorem connected with Y. V. Linnik’s conjecture on conditions of independence of a pair of normal sample statistics, which are continuous and homogeneous. These properties are significant for the problem, as it is seen from the proof.