Suppose
$$X\_{ij} = \mu\_j + \varepsilon\_{ij}, \quad j = 1, \cdots, J, \quad i = 1, \cdots, N\_j$$
ANOVA can allow us to test whether $\mu\_1 = \cdots = \mu\_J$.


In traditional ANOVA, however, the number of groups $J$ is fixed. But I want to know the proof technique in the case that $J$ goes to infinity.


Does anyone know some papers containing this question with detailed proof? Thanks so much!



Asymptotic properties of ANOVA when the number of groups goes to infinity
The following article is what you are looking for:



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> Akritas, M., Arnold, S. (2000). Asymptotics for Analysis of Variance When the Number of Levels is Large, Journal of the American Statistical Association, 95:449, 212-226.
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