Statistica Sinica 6(1996), 375-388

ASYMPTOTICS OF AN ESTIMATOR OF A ROBUST

SPREAD FUNCTIONAL

Ola Hössjer, Peter J. Rousseeuw and Christophe Croux

Lund Institute of Technology, Universitaire Instelling Antwerpen

and Université Libre de Bruxelles

Abstract: The functional S(F)=med_X~F(med_Y~F|X—Y|) proposed by Rousseeuw and Croux (1993) exists for any distribution , because no moments are needed. It measures spread in the general sense, without reference to a central point. A natural estimator of S(F) is S_n= med_imed _j,j≠i|z_i—z_j&# 124 where z₁,...,z_n are i.i.d. observations from F. In this paper we prove that S_n is asymptotically normal, and verify that the normality already holds for small samples. Moreover, one can use a constant multiple of S_n to estimate a scale parameter in a (possibly nongaussian) parametric model.

Key words and phrases: Breakdown point, influence function, order statistics, scale estimation.