We consider one of the most important problems in directional statistics, namely the problem of testing the null hypothesis that the spike direction $${\pmb \theta }$$ θ of a Fisher–von Mises–Langevin distribution on the p -dimensional unit hypersphere is equal to a given direction $${\pmb \theta }_0$$ θ 0 . After a reduction through invariance arguments, we derive local asymptotic normality (LAN) results in a general high-dimensional framework where the dimension $$p_n$$ p n goes to infinity at an arbitrary rate with the sample size n , and where the concentration $$\kappa _n$$ κ n behaves in a completely free way with n , which offers a spectrum of problems ranging from arbitrarily easy to arbitrarily challenging ones. We identify various asymptotic regimes, depending on the convergence/divergence properties of $$(\kappa _n)$$ ( κ n ) , that yield different contiguity rates and different limiting experiments. In each regime, we derive Le Cam optimal tests under specified $$\kappa _n$$ κ n and we compute, from the Le Cam third lemma, asymptotic powers of the classical Watson test under contiguous alternatives. We further establish LAN results with respect to both spike direction and concentration, which allows us to discuss optimality also under unspecified $$\kappa _n$$ κ n . To investigate the non-null behavior of the Watson test outside the parametric framework above, we derive its local asymptotic powers through martingale CLTs in the broader, semiparametric, model of rotationally symmetric distributions. A Monte Carlo study shows that the finite-sample behaviors of the various tests remarkably agree with our asymptotic results.

中文翻译：

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检测高维球体上的信号方向：非零和 Le Cam 最优性结果

我们考虑方向统计中最重要的问题之一，即检验零假设的问题，即 p 维单位上的 Fisher–von Mises–Langevin 分布的尖峰方向 $${\pmb \theta }$$ θ超球面等于给定方向 $${\pmb \theta }_0$$ θ 0 。在通过不变性参数减少之后，我们推导出局部渐近正态性 (LAN) 结果在一个通用的高维框架中，其中维度 $$p_n$$pn 以任意速率趋于无穷大，样本大小为 n，其中浓度 $ $\kappa _n$$ κ n 与 n 的行为完全自由，这提供了从任意容易到任意具有挑战性的一系列问题。我们根据 $$(\kappa _n)$$ ( κ n ) 的收敛/发散特性确定各种渐近机制，产生不同的邻接率和不同的限制实验。在每个机制中，我们在指定的 $$\kappa _n$$ κ n 下推导出 Le Cam 最优测试，并且我们从 Le Cam 第三引理计算连续替代下经典 Watson 测试的渐近幂。我们进一步建立了关于尖峰方向和浓度的 LAN 结果，这使我们能够在未指定的 $$\kappa _n$$ κ n 下讨论最优性。为了研究上述参数框架之外的 Watson 检验的非零行为，我们通过更广泛的、半参数的旋转对称分布模型中的鞅 CLT 推导出其局部渐近幂。蒙特卡罗研究表明，各种测试的有限样本行为与我们的渐近结果非常一致。