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 “à la Le Cam,” we show that the spherical mean is, at any $f$, a parametrically superefficient estimator of ${\pmb{\theta }}$ and that the Watson and Wald tests for $\mathcal{H}_{0}:{\pmb{\theta }}={\pmb{\theta }}_{0}$ 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 $f$.