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

JOURNAL OF MULTIVARIATE ANALYSIS

卷:176

Article

Jupp, P. E.¹  Kume, A.² 

[1] Univ St Andrews, Sch Math & Stat, St Andrews KY16 9SS, Fife, Scotland

[2] Univ Kent, Sch Math Stat & Actuarial Sci, Canterbury CT2 7FS, Kent, England

关键词: Cartan-Hadamard manifold;    Compositional data;    Directional statistics;    Exponential map;    Probability integral transform;    Shape space;    Simplex;   

DOI  :  10.1016/j.jmva.2019.104579

来源: Elsevier

【 摘 要 】

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, chi, in that for each distribution with continuous positive density on chi, there is a continuous mapping of chi 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 Sigma(5)(2). Their behaviour is investigated by simulation. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free