Mean location and sample mean location on manifolds: Asymptotics, tests, confidence regions

In a previous investigation we studied some asymptotic properties of the sa mple mean location on submanifolds of Euclidean space. The sample mean loca tion generalizes least squares statistics to smooth compact submanifolds of Euclidean space. In this paper these properties are put into use. Tests for hypotheses about mean location are constructed and confidence regions for mean location are indicated. We study the asymptotic distribution of the test statistic. The problem of comparing mean locations for two samples is analyzed. Special attention is paid to observations on Stiefel manifolds including th e orthogonal group O(p) and spheres Sk-1, and special orthogonal groups SO( p). The results also are illustrated with our experience with simulations. (C) 1998 Academic Press.