AI 中文总结
研究非参数统计流形上概率分布距离难计算问题，提出两阶段框架，先投影到有限维参数指数族，再嵌入对称正定矩阵流形，得到可计算伪度量及仿射不变检验统计量，应用于两样本检验，无需连续调整参数。
AI 中文摘要
在非参数统计流形上计算概率分布之间的距离本质上是难以处理的。测地线方程存在于无限维函数空间中，没有一般的封闭形式解。我们开发了一个两阶段框架，在皮斯托内 - 塞姆皮指数流形上产生一个可计算的伪度量，并将其应用于两样本假设检验。第一阶段，通过矩匹配将任意分布投影到选定的有限维参数指数族上。第二阶段，通过增强充分统计量向量的期望外积将参数族嵌入到对称正定矩阵流形中。诱导的拉回度量与费希尔 - 拉奥度量不同，通过涉及充分统计量三阶联合累积量的显式校正；对于高斯族，校正消失，作为特殊情况恢复卡尔沃 - 奥勒嵌入。环境矩阵流形上的仿射不变黎曼度量为伪度量提供了一个封闭形式的下界，可直接从样本矩计算。应用于两样本检验时，该框架产生一个仿射不变的检验统计量，不需要连续调整参数，如带宽。目标指数族的选择决定了比较哪些矩。临界值通过置换获得。
英文摘要
Computing distances between probability distributions on non-parametric statistical manifolds is fundamentally intractable. The geodesicequations live in infinite-dimensional function spaces and admit no general closed-form solution. We develop a two-stage framework that produces a computable pseudo-metric on the Pistone--Sempi exponential manifold and apply it to two-sample hypothesis testing. In the first stage, an arbitrary distribution is projected onto a chosen finite-dimensional parametric exponential family via moment-matching. This projection is many-to-one, so the resulting object is a pseudo-metric rather than a true metric. In the second stage, the parametric family is embedded into the manifold of symmetric positive definite matrices via the expected outer product of the augmented sufficient statistics vector. The embedding is a smooth diffeomorphism. The induced pullback metric differs from the Fisher--Rao metric by an explicit correction involving third-order joint cumulants of the sufficient statistics; the correction vanishes for the Gaussian family, recovering the Calvo--Oller embedding as a special case. The affine-invariant Riemannian metric on the ambient matrix manifold then provides a closed-form lower bound for the pseudo-metric, computable directly from sample moments. Applied to two-sample testing, the framework produces a test statistic that is affine-invariant and requires no continuous tuning parameters such as a bandwidth. The choice of target exponential family determines which moments are compared. Critical values are obtained by permutation.
