A degenerated U-statistic with rank 2 is known to converge in distribution to a weighted sum of independent chi-square random variables. However, a result for the asymptotic distribution of a degenerated U-statistic with rank higher than 2 is not available in the literature. We derive explicitly the asymptotic distribution of a degenerated U-statistic with any rank, which is useful for hypothesis testing when the test statistic is a degenerated U-statistic under null hypothesis. Interestingly, the limit for a degenerated U-statistic with rank higher than 2 is a weighted sum of independent polynomials of the standard normal random variables.

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To cite this article: Jun Shao (2025) Asymptotic distributions of degenerated U-statistics, Statistical Theory and Related Fields, 9:4, 434-443, DOI: 10.1080/24754269.2025.2579414

To link to this article: https://doi.org/10.1080/24754269.2025.2579414