Based on a decomposition of a $U$-statistic, Nobre, Singer and Silvapulle (In Beyond Parametrics in Interdisciplinary Research, Festschrift to P.K. Sen (2008) 197–210 Institute of Mathematical Statistics) proposed a test for the hypothesis that the within-treatment variance component in a one-way random effects model is null, specially useful when very mild assumptions are imposed on the underlying distributions. We consider a bootstrap version of that $U$-test and evaluate its performance via simulation studies in different scenarios. The bootstrap $U$-test has better statistical properties than the original test even in small samples. Furthermore, it is easy to implement and has a low computational cost. We consider two examples with unbalanced small sample datasets, for illustrative purposes.
"Improved $U$-tests for variance components in one-way random effects models." Braz. J. Probab. Stat. 34 (3) 464 - 477, August 2020. https://doi.org/10.1214/19-BJPS436