Abstract:In this paper, we study the problem of testing whether or not a given probability measure $\mu$ on $\mathbb{R}^{d}$ can be decomposed as a mixture of two probability measures whose second order statistics are significantly different. We call this the problem of testing the mixture model hypothesis. To tackle it, we introduce a new set of computable orthogonal invariants of $\mu$, namely, the eigenvalues of the 4th moment operator $T_{\mu}$ associated with the measure. We prove that the largest eigenvalue is always an outlier eigenvalue. Further, we show how the first and second largest eigenvalues of $T_{\mu}$ give nonasymptotic bounds for this problem and give a complete resolution of the asymptotic version of the problem under the $L^{8}$-$L^{2}$ equivalence assumption.
| Subjects: | Probability (math.PR); Statistics Theory (math.ST) |
| Cite as: | arXiv:2603.03245 [math.PR] |
| (or arXiv:2603.03245v2 [math.PR] for this version) | |
| https://doi.org/10.48550/arXiv.2603.03245 arXiv-issued DOI via DataCite |
From: March Boedihardjo [view email]
[v1]
Tue, 3 Mar 2026 18:37:53 UTC (19 KB)
[v2]
Fri, 22 May 2026 18:34:21 UTC (23 KB)
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