Over- And Underfitting The M-Estimates

The effects of over- and underfitting the regression model is studied for M-estimators. Applying nowadays already classic tool, namely the asymptotic linearity of M-statistics, the Bahadur representation of M-estimators in over- and underfitted model is found. It allows to establish conditions under which this representation and the asymptotic representation of M-estimators evaluated in the properly fitted model, are equivalent. A part of the conditions is of the same character as the analogous conditions for the least squares. Namely, we need orthogonality of explanatory variables. Of course, since the objective functions of the M estimators are more complicated than the objective function of the least squares (namely the parabola), we need some other condition to guarantee that the estimates in underfitted model estimate properly corresponding part of the vector of regression coefficients.