Inference for spherical location under high concentration

Abstract: Motivated by the fact that circular or spherical data are often much concentrated around a location $\pmb\theta$, we consider inference about $\pmb\theta$ under "high concentration" asymptotic scenarios for which the probability of any fixed spherical cap centered at $\pmb\theta$ 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 $\pmb\theta$, 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 $\pmb\theta$ 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 $\pmb\theta$ 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 super-efficient estimator of $\pmb\theta$ and that the Watson and Wald tests for $\mathcal{H}_0:{\pmb\theta}={\pmb\theta}_0$ enjoy similar, non-standard, 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 $f$.