MATCH: 非欧几里得数据中条件假设的乘子辅助检验
MATCH: Multiplier-Assisted Tests for Conditional Hypotheses in Non-Euclidean Data
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中文总结 AI 辅助
提出MATCH方法，用于检验非欧几里得数据是否匹配目标模型，涵盖Fréchet回归中的全局显著性、部分显著性和模型充分性检验，通过样本分裂和独立随机乘子解决零假设下损失差的一阶退化问题。
中文摘要 AI 辅助
我们提出了一种新程序MATCH（乘子辅助条件假设检验），用于检验非欧几里得数据是否匹配目标模型，这是Fréchet回归中显著性和设定检验的通用框架。MATCH涵盖了全局显著性、部分显著性以及全局Fréchet回归的充分性，提供了一种统一的方法来比较无限制的条件Fréchet均值与受限备择。关键挑战之一是，在零假设下，普通的留出损失差是一阶退化的：Oracle损失重合，而插件统计量受限于干扰估计误差。MATCH使用样本分裂和留出损失上的独立随机乘子，创建了一个非退化的高斯主导项，无需残差或切空间坐标。为了提高数据利用率和稳定性，我们进一步开发了交叉拟合检验和重复交叉拟合结合p值合并。我们建立了渐近零有效性、固定备择下的一致性以及局部功效保证。针对分布型、对称正定（SPD）矩阵值和球面响应的模拟支持了理论发现，对县级家庭收入分布和北大西洋热带气旋位置的应用展示了所提出检验的实际用途。
英文摘要
We propose a new procedure MATCH (Multiplier-Assisted Tests for Conditional Hypotheses) to test whether the non-Euclidean data match the target model, which is a general framework for significance and specification testing in Fréchet regression. MATCH covers global significance, partial significance, and the adequacy of global Fréchet regression, providing a unified way to compare unrestricted conditional Fréchet means with restricted alternatives. One of the key challenges is that the ordinary held-out loss difference is first-order degenerate under the null: the oracle losses coincide, and plug-in statistics is dominated by nuisance estimation error. MATCH uses sample splitting and independent random multipliers on held-out losses to create a nondegenerate Gaussian leading term without residuals or tangent-space coordinates. To improve data use and stability, we further develop cross-fitted tests and repeated cross-fitting with p-value merging. We establish asymptotic null validity, consistency under fixed alternatives, and local power guarantees. Simulations for distributional, symmetric positive-definite (SPD) matrix-valued, and spherical responses support the theoretical findings, and applications to county-level household income distributions and North Atlantic tropical-cyclone locations demonstrate the practical use of the proposed tests.
