The comparison of a parameter in $k$ populations is a classical problem in statistics. Testing for the equality of means or variances are typical examples. Most procedures designed to deal with this problem assume that $k$ is fixed and that samples with increasing sample sizes are available from each population. This paper introduces and studies a test for the comparison of an estimable parameter across $k$ populations, when $k$ is large and the sample sizes from each population are small when compared with $k$. The proposed test statistic is asymptotically distribution-free under the null hypothesis of parameter homogeneity, enabling asymptotically exact inference without parametric assumptions. Additionally, the behaviour of the proposal is studied under alternatives. Simulations are conducted to evaluate its finite-sample performance, and a linear bootstrap method is implemented to improve its behaviour for small $k$. Finally, an application to a real dataset is presented.

翻译：在统计学中，比较 $k$ 个总体中的某个参数是一个经典问题。检验均值或方差的相等性即为典型示例。大多数处理此问题的程序均假设 $k$ 固定，且可从每个总体中获得样本量递增的样本。本文针对 $k$ 较大、且各总体样本量相对于 $k$ 较小时的情形，提出并研究了一种用于比较 $k$ 个总体中某个可估参数的检验方法。所提出的检验统计量在原假设（参数同质性）下是渐近分布自由的，从而能够在非参数假设下实现渐近精确推断。此外，本文还研究了该检验在备择假设下的表现。通过模拟实验评估了其有限样本性能，并采用线性自助法以改善其在 $k$ 较小时的表现。最后，本文展示了该方法在一个真实数据集上的应用。