Measures of goodness of fit obtained by almost-canonical transformations on Riemannian manifolds

The standard method of transforming a continuous distribution on the line to the uniform distribution on \([0,1]\) is the probability integral transform. Analogous transforms exist on compact Riemannian manifolds, \(\scr X\), in that for each distribution with continuous positive density on \(\scr X\), there is a continuous mapping of \(\scr X\) to itself that transforms the distribution into the uniform distribution. In general, this mapping is far from unique. This paper introduces the construction of an almost-canonical version of such a probability integral transform. The construction is extended to shape spaces, Cartan–Hadamard manifolds, and simplices. The probability integral transform is used to derive tests of goodness of fit from tests of uniformity. Illustrative examples of these tests of goodness of fit are given involving (i) Fisher distributions on \(S^2\), (ii) isotropic Mardia–Dryden distributions on the shape space \(Σ^5_2\). Their behaviour is investigated by simulation.