The Convergence Rate Of The M Estimat

Statistical models have corresponding assumptions,and the estimation of statis-tors under the assumed framework often has good properties.But as various types of data appear,the assumptions of many statistical models are not well satisfied completely.M-estimation is a point estimation method proposed by Huber in 1964.M-estimation has become one of the most popular methods to solve the problem of robustness of statistical estimation,not only because of its robustness,but also because of its good large sample properties,such as asymptotic normality.In addition,the distance between two distributions is al-ways an important direction in probability and statistics.The Stein method is a particularly effective way to calculate the upper bound of the distance between distributions,especially for the computation of Berry-Esseen bound and Wasser-stein bound under independent and identically distributed conditions.The asymptotic normality of M-estimation under various regularization con-ditions has been extensively studied.In this paper,we obtain the upper bound of M-estimation approaching normal distribution in different cases,which corre-sponds to the upper bound of convergence for Wassersein-1 distance upper bound and bounded Lipschitz distance.In particular,under the classical condition,the convergence rate of the two upper bounds is O(n-1/2),and it is a function of es-timating the mean square error.Different from the assumption that the general scholars need the independent and identical distribution of the random variables for M-estimation,the convergence rate given in this paper is not necessary for the same distribution.Finally,we give the upper bound of a mean square error when M-estimation has no analytical solution.