On M-Processes and M-Estimation

We relate the asymptotic behavior of M-estimators of the regression parameter in a linear model in which the dimension of the regression parameter may increase with the sample size to the stochastic equicontinuity of an associated M-process. The approach synthesises a number of results for the dimensionally fixed regression model and then extends these results in a direct unified way. The resulting theorems require only mild conditions on the -function and the underlying distribution function. In particular, the results do not require to be smooth and hence can be applied to such estimators as the least absolute deviations estimator. We also treat one-step M-estimation.