Almost fully efficient and robust simultaneous estimation of location and scale parameters: A minimum distance approach

Abstract

Simultaneous robust estimates of location and scale parameters are derived from minimizing a minimum distance criterion function. The criterion function measures the squared distance between the pth power ( p > 0) of the empirical distribution function and the pth power of the imperfectly determined model distribution function over the real line. We show that the estimator is uniquely defined, asymptotically bivariate normal and has positive breakdown. When p = 0.56 the estimator is almost fully efficient at the normal model. Efficiencies are 0.9999 and 0.9998 for the location and scale parameters, respectively. Some other choices of p values produce highly efficient and robust estimates as well. It is shown that the location estimator has maximum breakdown point 0.5 independent of p when the scale is known. If the true and target models are both symmetric, then the location estimator is consistent for the center of the symmetry even if the true model is imperfectly determined.