Inference for spherical location under high concentration

Motivated by the fact that circular or spherical data are often much concentrated around a location .., we consider inference about .. under high concentration asymptotic scenarios for which the probability of any fixed spherical cap centered at .. 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 .., a scalar concentration parameter . and a functional nuisance f. We determine the class of distributions for which high concentration is obtained as . diverges to infinity. For such distributions, we then consider inference (point estimation, confidence zone estimation, hypothesis testing) on .. in asymptotic scenarios where .n diverges to infinity at an arbitrary rate with the sample size n. Our asymptotic investigation reveals that, interestingly, optimal inference procedures on .. show consistency rates that depend on f. Using asymptotics .à la Le Cam,. we show that the spherical mean is, at any f, a parametrically superefficient estimator of .. and that the Watson and Wald tests for H0:..=..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.