On Strong Invariant Coordinate System (SICS) Functionals

In modern multivariate statistical analysis, affine invariance or equivariance for statistical procedures are properties of paramount interest and importance. A statistical procedure lacking such a property can sometimes acquire it if carried out not on the original data but rather on suitably transformed data, in some cases accompanied by a retransformation back to the original coordinate system. Three types of relevant transformation, weak covariance (WC), transformation-retransformation (TR), and strong invariant coordinate system (SICS), are treated in Serfling (2010), where the the WC and TR types are seen to be essentially equivalent, and the SICS type is introduced as a very productive special case of the TR type. It also is seen that any TR type can serve to convert the spatial multivariate quantile function into an affine equivariant version. However, the single TR functional that has been used in practice for this purpose (Chakraborty and Chaudhuri, 1996, and Chakraborty, Chaudhuri, and Oja, 1998), has proved difficult to understand theoretically. In fact, this particular TR functional happens to be a particular case of SICS type and is better understood by consideringing this aspect. The present paper studies SICS functionals with some generality, developing their basic properties, constructing particular families of them, illustrating them explicitly for the bivariate case, and examining their roles in several application contexts. AMS 2000 Subject Classification: Primary 62H99 Secondary 62G99.