Sankhya: The Indian Journal of Statistics

1998, Volume 60, Series A, Pt. 1 ,pp. 119-133

SPHERING AND ITS PROPERTIES

By

GUOYING LI and JIAN ZHANG, Academia Sinica, Beijing

SUMMARY. Centering and sphering is an intuitive approach to remove, in a sense, location, scale and correlation structure in data sets and to force us to examine other aspects of them. It is frequently applied in data analyses. This paper is intended to discuss properties of sphering procedures, such as affine (including orthogonal and lower triangular as special cases) equivariance/invariance, application to projection pursuit (PP) and asymptotic behavior. In particular, the three commonly used sphering procedures, named LTS, SRS and JFS, are studied. It is shown that all sphering methods in a PP-after-sphering procedure results in the same optimal projections. It is also shown that the sphering matrix of JFS is inconsistent, whereas, those of SRS and LTS not only are consistent but also have asymptotic distributions.

AMS (1991) subject classification. Primary 62H99.

Key words and phrases. Sphering, equivariance, invariance, asymptotics, projection pursuit.

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