INFERENCE FOR SPHERICAL LOCATION UNDER HIGH CONCENTRATION

• 成果类型: Article
• 署名作者: Paindaveine, Davy; Verdebout, Thomas
• 署名单位: Universite Libre de Bruxelles; Universite de Toulouse; Universite Toulouse 1 Capitole
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/19-AOS1918
• 发表日期: 2020
• 页码: 2982-2998
• 关键词: mean direction; regression; statistics; tests

摘要

Motivated by the fact that circular or spherical data are often much concentrated around a location theta, we consider inference about theta under high concentration asymptotic scenarios for which the probability of any fixed spherical cap centered at theta converges to one as the sample size n diverges to infinity. Rather than restricting to Fisher-von Mises-Langevin distributions, we consider a much broader, semiparametric, class of rotationally symmetric distributions indexed by the location parameter theta, a scalar concentration parameter kappa and a functional nuisance f. We determine the class of distributions for which high concentration is obtained as kappa diverges to infinity. For such distributions, we then consider inference (point estimation, confidence zone estimation, hypothesis testing) on theta in asymptotic scenarios where kappa(n) diverges to infinity at an arbitrary rate with the sample size n. Our asymptotic investigation reveals that, interestingly, optimal inference procedures on theta show consistency rates that depend on f. Using asymptotics a la Le Cam, we show that the spherical mean is, at any f, a parametrically superefficient estimator of theta and that the Watson and Wald tests for H-0 : theta =theta(0) enjoy similar, nonstandard, optimality properties. We illustrate our results through simulations and treat a real data example. On a technical point of view, our asymptotic derivations require challenging expansions of rotationally symmetric functionals for large arguments of f.