On a surprising behavior of the likelihood ratio test in non-parametric mixture models

Yan Zhang; Stanislav Volgushev

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arXiv · 2509.05610

On a surprising behavior of the likelihood ratio test in non-parametric mixture models

Yan Zhang·Stanislav Volgushev

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Abstract

We study the likelihood ratio test in general mixture models where the base density is parametric, the null is a known fixed mixing distribution, and the alternative is a general mixing distribution supported on a bounded parameter space. For Gaussian location mixtures and Poisson mixtures, we show a surprising result: the non-parametric likelihood ratio test statistic converges to a tight limit if and only if the null distribution is a finite mixture, and diverges to infinity otherwise. We further demonstrate that the likelihood ratio test diverges for a fairly general class of distributions when the null mixing distribution is not finitely discrete.

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Yan Zhang, Stanislav Volgushev. 2025-09-06. On a surprising behavior of the likelihood ratio test in non-parametric mixture models. https://arxiv.org/abs/2509.05610

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