Sankhy¯a: The Indian Journal of Statistics 1999, Volume 61, Series A, Pt. 3, pp. 362-380 ON MULTIVARIATE MONOTONIC MEASURES OF LOCATION WITH HIGH BREAKDOWN POINT By SUJIT K. GHOSH North Carolina State University, Raleigh and DEBAPRIYA SENGUPTA Indian Statistical Institute, Calcutta SUMMARY. The purpose of this article is to propose a new scheme for robust multivariate ranking by introducing a not so familiar notion called monotonicity. Under this scheme, as in the case of classical outward ranking, we get an increasing sequence of regions diverging away from a central region (may be a single point) as nucleus. The nuclear region may be defined as the median region. Monotonicity seems to be a natural property which is not easily obtainable. Several standard statistics such weighted mean, coordinatewise median and the L 1 -median have been studied. We also present the geometry of constructing general monotonic measures of location in arbitrary dimensions and indicate its trade-off with other desirable properties. The article concludes with discussions on finite sample breakdown points and related issues. 1. Introduction Robust handling of multivariate data typically refers to the following: (a) finding a robust measure of location, (b) finding a robust measure of dispersion matrix and (c) detection of possible outliers. The central purpose however, is to create an increasing sequence of regions (depicting increasing degree of outwardness) depending on the geometry of the data cloud. As a consequence we get a center outward ranking of a multivariate data (see, Liu (1990)). The method of construction through ellipsoidal regions (required by (a) and (b)) becomes therefore, one of the many similar techniques. There is a great deal of literature on finding out descriptive multivariate location measures with high finite sample breakdown point. These measures are loosely classified Paper received. October 1997; revised December 1998. AMS (1991) subject classification. Primary 62F35, secondary 62G05, 62H12 Key words and phrases. Multivariate location estimates, high Breakdown point, robustness, monotonicity.