PRINCIPAL POINTS AND SELF-CONSISTENT POINTS OF SYMMETRICAL MULTIVARIATE DISTRIBUTIONS

The k principal points xi(1), ..., xi(k) of a random vector X are the points that approximate the distribution of X by minimizing the expect ed squared distance of X to the nearest of the xi(j). A given set of k points y(1), ..., y(k) partition R(p) into domains of attraction D-1, ..., D-k respectively, where D,consists of all points x is an element of R(p) such that \\x - y(j)\\ < \\x - y(j)\\, l not equal j. If E[x parallel to X is an element of D-j] = y(j) for each j, then y(1), ..., y(k) are k self-consistent points of X (\\.\\ is the Euclidian norm). Principal points are a special case of self-consistent points. Princi pal points and sell-consistent points are cluster means of a distribut ion and represent a generalization of the population mean from one to several points. Principal points and self-consistent points are studie d for a class of strongly symmetric multivariate distributions. A dist ribution is strongly symmetric if the distribution of the principal co mponents (Z(1), ..., Z(p))' is invariant up to sign changes, i.e., (Z( 1), ..., Z(p))' has the same distribution as (+/-Z(1), ..., +/-Z(p))'. Elliptical distributions belong to the class of strongly symmetric di stributions. Several results are given for principal points and self-c onsistent points of strongly symmetric multivariate distributions. One result relates self-consistent points to principal component subspace s. Another result provides a sufficient condition for any set of self- consistent points lying on a line to be symmetric to the mean of the d istribution.