Risk comparison of some shrinkage M-estimators in linear models

The problem of robust estimation of a (linear) regression parameter (vector), in the presence of nuisance scale parameter, is considered when it is a priori suspected that the regression could be restricted to a linear subspace. Asymptotic properties of variants of Stein-rule M-estimators (including the positive-rule shrinkage M-estimators) are studied. Under an asymptotic distributional quadratic risk criterion, their relative dominance picture is explored, analytically as well as by simulation. An extensive sampling experiment is used to examine the small sample characteristics of the proposed estimators over a wide-range of data sampling designs and distributions. Our simulation experiments have provided strong evidence that corroborates with the asymptotic theory. Two examples are provided to illustrate the performance of the estimators in real-life situations.