Inference for spherical location under high concentration

Abstract

Motivated by the fact that circular or spherical data are often much concentrated around a location $\mathit{\theta }$, we consider inference about $\mathit{\theta }$ under high concentration asymptotic scenarios for which the probability of any fixed spherical cap centered at $\mathit{\theta }$ 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 $\mathit{\theta }$, 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 $\mathit{\theta }$ 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 $\mathit{\theta }$ 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 $\mathit{\theta }$ and that the Watson and Wald tests for ${\mathcal{H}}_0;\mathit{\theta }={\mathit{\theta }}_0$ 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 $f$.