DATA SPHERING: SOME PROPERTIES AND APPLICATIONS

JIAN ZHANG

Institute of Systems Science, Academia Sinica, Beijing 100080, China

(Received September 30, 1996; revised May 15, 1997)

Abstract. Centring-then-sphering is a very important pretreatment in data analysis. The purpose of this paper is to study the asymptotic behavior of the sphering matrix based on the square root decomposition (SRD for short) and its applications. A sufficient condition is given under which SRD has nondegenerate asymptotic distribution. As examples, some commonly used and affine equivariant estimates of the dispersion matrix are shown to satisfy this condition. The case when the population dispersion matrix varies is also treated. Applications to projection pursuit (PP) are presented. It is shown that for elliptically symmetric distributions the PP index after centring-then-sphering is independent of the underlying population location and dispersion.

Key words and phrases: Sphering, dispersion matrix, invariance, asymptotics, projection pursuit.