SELF-CONSISTENT PATTERNS FOR SYMMETRICAL MULTIVARIATE DISTRIBUTIONS

The set of k points that optimally represent a distribution in terms o f mean squared error have been called principal points (Flury 1990). P rincipal points are a special case of self-consistent points. Any give n set of k distinct points in RP induce a partition of RP into Voronoi regions or domains of attraction according to minimal distance. A set of k points are called self-consistent for a distribution if each poi nt equals the conditional mean of the distribution over its respective Voronoi region. For symmetric multivariate distributions, sets of sel f-consistent points typically form symmetric patterns. This paper inve stigates the optimality of different symmetric patterns of self-consis tent points for symmetric multivariate distributions and in particular for the bivariate normal distribution. These results are applied to t he problem of estimating principal points.