Invariant Tests for Uniformity on Compact Riemannian Manifolds Based on Sobolev Norms
基于 Sobolev 范数的紧黎曼流形均匀性不变检验AI 解读 ↓
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基于 Sobolev 范数的紧黎曼流形均匀性不变检验
- 研究问题
- 如何在圆、球面及一般紧黎曼流形上构造分布均匀性的不变检验，并统一已有方法及分析渐近和局部最优性质？
- 方法与思路
- 作者以拉普拉斯特征函数展开经验分布与均匀分布之差，构造系数平方的加权和，并要求同一特征空间内权重相等。
- 材料陈述的贡献
- 摘要称提出包含多种已有方法的 Sobolev 检验类，给出渐近理论、部分局部最优性质和球面及半球面上的新检验。
- 阅读关注点
- 应核对权重的收敛约束与不变性要求、检验的极限分布和局部备择设定，以及经验分布弱收敛结果如何用于证明。
- 阅读边界
- 仅据版本未注明的 OpenAlex 摘要；权重选择、局部最优性的范围及有限样本表现未完整说明，不能推断对所有备择均最优。
原文摘要
Several invariant tests for uniformity of a distribution on the circle, the sphere and the hemisphere have been proposed by Rayleigh, Watson, Bingham, Ajne, Beran and others. In this paper a class of invariant tests for uniformity on compact Riemannian manifolds containing many of the known ones is presented and studied (the asymptotic theory as well as some local optimality properties for this class of tests are given). The examples include two new tests, one for the sphere and the other for the hemisphere. Let $X$ be a compact Riemannian manifold, $\mu$ the normalized volume element (the uniform distribution of $X$) and $\nu_n$ the empirical distribution corresponding to a sequence of i.i.d. $X$-valued random variables. The statistics in which these tests are based are just convergent weighted sums of the squares of the Fourier coefficients of $\nu_n(\omega) - \mu$ with respect to any orthonormal basis of $L_2(X, \mu)$ consisting of eigenfunctions of the Laplacian. An additional condition is imposed on the weights, namely that weights corresponding to coefficients of eigenfunctions in the same eigenspace of the Laplacian be equal (this condition is essential for the invariance of the tests). These statistics are related to Sobolev norms and so, the tests are called Sobolev tests. In connection with Sobolev statistics, it is interesting to note that the Sobolev norms of index $-s, s > (\dim X)/2$, metrize the weak-star topology of $\mathscr{P}(X)$, the space of Borel probability measures on $X$. A theorem about weak convergence of empirical distributions on compact manifolds, useful in proving some of the asymptotic results for Sobolev statistics, is also included. One of the sections (Section 2) is almost entirely devoted to give a short review of the facts needed in the paper about Riemannian manifolds, the Laplacian and Sobolev spaces.