Sobolev tests of goodness of fit of distributions on compact Riemannian manifolds

Abstract

Classes of coordinate-invariant omnibus goodness-of-fit tests on compact Riemannian manifolds are proposed. The tests are based on Giné's Sobolev tests of uniformity. A condition for consistency is given. The tests are illustrated by an example on the rotation group SO(3).

Copulae on products of compact Riemannian manifolds

“…Such mappings t are the basis of Sobolev tests of uniformity [7,15], several-sample tests [29], tests of symmetry [17], tests of independence [18], and tests of goodness of fit [14]. They can also be used as follows to construct copulae.…”

“…. , (x n , y n ) on X ×Y and a corresponding fitted copula f (·, ·;θ) in this family, the quality of the fit can be assessed using the following version of the Sobolev goodnessof-fit tests of [14]. It is based on the fact that, for distributions on X ×Y with uniform marginals, uniformity is equivalent to independence.…”

“…It is based on the fact that, for distributions on X ×Y with uniform marginals, uniformity is equivalent to independence. This implies that, for measuing the fit of a copula, the weighted tests of uniformity used in the goodness-of-fit tests of [14] can be replaced by weighted versions of the tests of independence considered in [18]. Thus, if t : X → L 2 (X ) and u : Y → L 2 (Y) are functions of the form (4), an appropriate statistic for measuring goodness of fit is…”

A kernel-based classifier on a Riemannian manifold

“…Pennec, 2006 The aim of the present paper is to generalize the Euclidean kernel rule for the classification of observations to the situation where the data belong to a Riemannian manifold. Stimulated by multiple applications, there is presently a growing literature on statistical inference on manifolds, including the estimation of location parameters Patrangenaru, 2003, 2005), density and regression estimation (Hendriks, 1990;Hendriks et al, 1993;Lee and Ruymgaart, 1996;Pelletier, 2005Pelletier, , 2006, and goodness-of-fit tests (see Jupp (2005) for recent results and further references). However, few is known about classification on a manifold.…”

A decomposition for invariant tests of uniformity on the sphere

“…As it is underlined in the recent paper [6], the problem of finding systematical methods for building goodness of fit tests on the sphere and other manifolds remains widely opened. Giné established in [5] a general framework for testing uniformity on a wide family of sample spaces including the sphere.…”