TY - JOUR
T1 - Measures of goodness of fit obtained by almost-canonical transformations on Riemannian manifolds
AU - Jupp, P. E.
AU - Kume, Alfred
PY - 2019/12/10
Y1 - 2019/12/10
N2 - 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, X, in that for each distribution with continuous positive density on X, there is a continuous mapping of 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 S2, (ii) isotropic Mardia–Dryden distributions on the shape space Σ52 Their behaviour is investigated by simulation.
AB - 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, X, in that for each distribution with continuous positive density on X, there is a continuous mapping of 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 S2, (ii) isotropic Mardia–Dryden distributions on the shape space Σ52 Their behaviour is investigated by simulation.
KW - Cartan-Hadamard manifold
KW - Compositional data
KW - Directional statistics
KW - Exponential map
KW - Probability integral transform
KW - Shape space
KW - Simplex
U2 - 10.1016/j.jmva.2019.104579
DO - 10.1016/j.jmva.2019.104579
M3 - Article
VL - In press
JO - Journal of Multivariate Analysis
JF - Journal of Multivariate Analysis
SN - 0047-259X
ER -