ABSTRACT
Like the M-estimators in Chapter 5, the R-estimators are implicitly defined statistical functionals. Although for R-estimators a differentiable statistical functional approach can be formulated, it may call for bounded score functions that do not include the important cases of normal scores, log-rank scores, and other unbounded ones. Moreover the ranks are integer valued random variables, so that even for smooth score functions, the rank scores may not possess smoothness to a very refined extent. For this reason the treatment of Chapter 5 may not go through entirely for R-estimators of location or regression parameters. hence, to encompass a larger class of rank-based estimators, we will follow the tracks based on the asymptotic linearity of rank statistics in shift or regression parameters: This approach has been popularized in the past two decades, and some accounts of the related developments are also available in some other contemporary text books on nonparametrics. However, our aim is to go beyond these reported developments onto the second-order representations and to relax some of the currently used regularity conditions. In this way we aim to provide an up-to-date and unifying treatment of the asymptotic representations for R-estimators of location and regression parameters.
