Inference for Spherical Location under High Concentration

Motivated by the fact that circular or spherical data are often much concentrated around a location θ, we consider inference about θunder high concentration asymptotic scenarios for which the probability of any fixed spherical cap centered at θ 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 θ, a scalar concentration parameter κ and a functional nuisance f. We determine the class of distributions for which high concentration is obtained as κ diverges to infinity. For such distributions, we then consider inference (point estimation, confidence zone estimation, hypothesis testing) on θ in asymptotic scenarios where κn diverges to infinity at an arbitrary rate with the sample size n. Our asymptotic investigation reveals that, interestingly, optimal inference procedures on θ 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 super-efficient estimator of θ and that the Watson and Wald tests for H0 :θ = θ0 enjoy similar, non-standard, optimality properties. Our results are illustrated by Monte Carlo simulations. On a technical point of view, our asymptotic derivations require challenging expansions of rotationally symmetric functionals for large arguments of the nuisance function f.