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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.

Non-concave penalized maximum likelihood methods, such as the Bridge, the SCAD, and the MCP, are widely used because they not only perform the parameter estimation and variable selection simultaneously but also are more efficient than the Lasso. They include a tuning parameter which controls a penalty level, and several information criteria have been developed for selecting it. While these criteria assure the model selection consistency, they have a problem in that there are no appropriate rules for choosing one from the class of information criteria satisfying such a preferred asymptotic property. In this paper, we derive an information criterion based on the original definition of the AIC by considering minimization of the prediction error rather than model selection consistency. Concretely speaking, we derive a function of the score statistic that is asymptotically equivalent to the non-concave penalized maximum likelihood estimator and then provide an estimator of the Kullback-Leibler divergence between the true distribution and the estimated distribution based on the function, whose bias converges in mean to zero.Furthermore, through simulation studies, we find that the performance of the proposed information criterion is about the same as or even better than that of the cross-validation.

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In this paper we study the uniform tail-probability estimates of a regularized leastsquares estimator for the linear regression model, by making use of the polynomial type large deviation inequality for the associated statistical random fields, which may not be locally asymptotically quadratic. Our results provide a measure of rate of consistency in variable selection in sparse estimation, which in particular enable us to verify various arguments requiring convergence of moments of estimator-dependent statistics, such as the expected maximum-likelihood for AIC-type and many other moment based model assessment procedure including the C p -type.