Abstract
Motivated by the fact that circular or spherical data are often much concentrated around a location $\boldsymbol{\theta}$, we consider inference about $\boldsymbol{\theta}$ under high concentration asymptotic scenarios for which the probability of any fixed spherical cap centered at $\boldsymbol{\theta}$ 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 $\boldsymbol{\theta}$, 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 $\boldsymbol{\theta}$ 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 $\boldsymbol{\theta}$ 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 $\boldsymbol{\theta}$ and that the Watson and Wald tests for $\mathcal{H}_{0} ; \boldsymbol{\theta} = \boldsymbol{\theta}_{0}$ 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 $f$.
