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 $f$.

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高浓度下球体位置的推断

受圆形或球形数据通常集中在 $\pmb\theta$ 位置这一事实的启发，我们考虑在“高浓度”渐近场景下对 $\pmb\theta$ 的推断，其中任何固定球冠的概率集中在在 $\pmb\theta$ 收敛到 1，因为样本大小 $n$ 发散到无穷大。我们不限于 Fisher-von Mises-Langevin 分布，而是考虑更广泛的、半参数的、由位置参数 $\pmb\theta$、标量浓度参数 $\kappa$ 和功能干扰 $ 索引的旋转对称分布类别f$。我们确定在 $\kappa$ 发散到无穷大时获得高浓度的分布类别。对于这样的分布，我们然后考虑推理（点估计、置信区间估计、假设检验）在 $\pmb\theta$ 上的渐近场景中，其中 $\kappa_n$ 以任意速率发散到无穷大，样本大小为 $n$。我们的渐近调查显示，有趣的是，$\pmb\theta$ 上的最佳推理过程显示出依赖于 $f$ 的一致性率。使用渐近线“\`a la Le Cam”，我们证明球面均值在任何 $f$ 处都是 $\pmb\theta$ 的参数超高效估计量，并且 Watson 和 Wald 测试对 $\mathcal{ H}_0:{\pmb\theta}={\pmb\theta}_0$ 享有类似的、非标准的、最优性的特性。我们通过模拟来说明我们的结果并处理一个真实的数据示例。从技术角度来看，我们的渐近推导需要对 $f$ 的大参数进行具有挑战性的旋转对称泛函扩展。