Heteroscedastic one-way ANOVA and lack-of-fit tests

Recent articles have considered the asymptotic behavior of the one-way analysis of variance (ANOVA) F statistic when the number of levels or groups is large. In these articles, the results were obtained under the assumption of homoscedasticity and for the case when the sample or group sizes n _i remain fixed as the number of groups, a, tends to infinity. In this article, we study both weighted and unweighted test statistics in the heteroscedastic case. The unweighted statistic is new and can be used even with small group sizes. We demonstrate that an asymptotic approximation to the distribution of the weighted statistic is possible only if the group sizes tend to infinity suitably fast in relation to a. Our investigation of local alternatives reveals a similarity between lack-of-fit tests for constant regression in the present case of replicated observations and the case of no replications, which uses smoothing techniques. The asymptotic theory uses a novel application of the projection principle to obtain the asymptotic distribution of quadratic forms.