Sankhya: The Indian Journal of Statistics

1999, Volume 61, Series A, Pt. 3, 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 it 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.

AMS (1991) subject classification. Primary 62F35, secondary 62G05, 62H12.

Key words and phrases. Multivariate location estimates, high Breakdown point, robustness, monotonicity.

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