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arXiv · 2609.23903

Almost-sure uniqueness of the Gaussian location NPMLE

Haiyang Wang
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Abstract

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.

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Haiyang Wang. 2026-09-20. Almost-sure uniqueness of the Gaussian location NPMLE. https://arxiv.org/abs/2609.23903

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