Given $k$ independent samples of functional data, this paper deals with the problem of testing for the equality of their mean functions. In contrast to the classical setting, where $k$ is kept fixed and the sample size from each population increases without bound, here $k$ is assumed to be large and the size of each sample is either bounded or small in comparison to $k$. A new test is proposed. The asymptotic distribution of the test statistic is stated under the null hypothesis of equality of the $k$ mean functions as well as under alternatives, which allows us to study the consistency of the test. Specifically, it is shown that the test statistic is asymptotically free distributed under the null hypothesis. The finite sample performance of the test based on the asymptotic null distribution is studied via simulation. Although we start by assuming that the data are functions, the proposed test can also be applied to finite dimensional data. The practical behavior of the test for one dimensional data is numerically studied and compared with other tests.

给定的 $克$函数数据的独立样本，本文处理测试它们的平均函数是否相等的问题。与经典环境相比，$克$ 保持固定，每个群体的样本量无限制地增加，这里 $克$ 被假定为很大，并且每个样本的大小与 $克$. 建议进行新的测试。检验统计量的渐近分布是在等式的零假设下陈述的$克$均值函数以及替代方案，这使我们能够研究测试的一致性。具体来说，它表明检验统计量在零假设下是渐近自由分布的。通过模拟研究了基于渐近零分布的测试的有限样本性能。尽管我们首先假设数据是函数，但所提出的测试也可以应用于有限维数据。对一维数据测试的实际行为进行了数值研究，并与其他测试进行了比较。