The smoothed median and the bootstrap

SUMMARY Even in one dimension the sample median exhibits very poor performance when used in conjunction with the bootstrap. For example, both the percentile-t bootstrap and the calibrated percentile method fail to give second-order accuracy when applied to the median. The situation is generally similar for other rank-based methods, particularly in more than one dimension. Some of these problems can be overcome by smoothing, but that usually requires explicit choice of the smoothing parameter. In the present paper we suggest a new, implicitly smoothed version of the k-variate sample median, based on a particularly smooth objective function. Our procedure preserves many features of the conventional median, such as robustness and high efficiency, in fact higher than for the conventional median, in the case of normal data. It is however substantially more amenable to application of the bootstrap. Focusing on the univariate case, we demonstrate these properties both theoretically and numerically.