发表机构
Yale University（耶鲁大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明高斯位置混合模型下NPMLE对几乎所有数据集唯一，基于有限支撑定理和最优对数似然可微性，并推广至一般协方差。
AI 中文摘要
对于具有单位协方差的高斯位置混合模型，我们证明了对于Lebesgue几乎所有的数据集，混合分布的非参数最大似然估计是唯一的，这一结论对所有样本量和维度均成立。特别地，当观测值的联合分布绝对连续时，唯一性几乎必然成立。证明使用了高斯位置NPMLE的有限支撑定理以及最优对数似然的几乎处处可微性。在该最优值可微的任意位置，所有NPMLE在观测点处具有相同的密度导数。随后，高斯评估向量之间的最小线性相关性排除了不同解的存在。附录将该结果推广到任意固定的、已知的、观测特定的正定协方差情形。
英文摘要
For Gaussian location mixtures with identity covariance, we prove that the nonparametric maximum likelihood estimator of the mixing distribution is unique for Lebesgue-almost every dataset, for every sample size and dimension. In particular, uniqueness holds almost surely whenever the joint distribution of the observations is absolutely continuous. The proof uses the finite-support theorem for Gaussian location NPMLEs and almost-everywhere differentiability of the optimal log-likelihood. Wherever this optimal value is differentiable, all NPMLEs have the same density derivatives at the observations. A minimal linear dependence among Gaussian evaluation vectors then rules out distinct solutions. An appendix extends the result to arbitrary fixed, known, observation-specific positive-definite covariances.
Comments7 pages
