M-estimators with non-standard rates of convergence and weakly dependent data

This paper analyzes M-estimators over general objective functions. We do not assume convexity and differentiability of the functions. A new result regarding M-estimators is derived. Unlike most of the former econometric literature, the rate of convergence is not square root n. The rate of convergence is non-standard and depends on the moment bounds of the objective function analyzed. We can actually connect the rate of convergence to the smoothness of the objective function in certain class of functions as described in van der Vaart and Wellner (Weak Convergence and Empirical Processes, Springer, Berlin, 1996). We also simplify this rate of convergence idea and extend to weakly dependent data from iid case. This rate is simple and usable in econometrics literature. We illustrate the techniques by deriving the rate of convergence for LAD estimator for censored regression and maximum score estimator with weakly dependent data.