© 2001 by Biometrika Trust
The smoothed median and the bootstrap
1 School of Mathematics, University of South Australia, Adelaide, South Australia 5005, Australiabruce.brown{at}unisa.edu.au 2 Centre for Mathematics & its Applications, Australian National University, Canberra, A.C.T.0200, Australia. peter.hall{at}anu.edu.au 3 Statistical Laboratory, University of Cambridge, Cambridge CB3 0WB, U.K. g.a.young{at}statslab.cam.ac.uk
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.
Key Words: Bootstrap; Calibrated percentile method; Median; Percentile-t; Rank methods; Smoothed median
Received November 1999. Revised May 2000