🤖 AI Summary
This paper addresses the problem of testing parameter homogeneity across multiple high-dimensional populations (with large $k$) under small per-group sample sizes. We propose an asymptotically distribution-free nonparametric test that relaxes the classical assumptions of fixed $k$ and large overall sample size. The test statistic is constructed via asymptotic distribution theory and enhanced by linear bootstrap calibration to improve finite-sample accuracy. We establish its asymptotic exactness under the null hypothesis and consistency under alternatives. Simulation studies demonstrate superior finite-sample performance compared to existing methods, and real-data analyses confirm its robustness and practical utility in high-dimensional, multi-group settings. The key contribution is the first nonparametric homogeneity test framework specifically designed for large-$k$, small-$n$ scenarios—offering both rigorous theoretical guarantees (asymptotic validity and consistency) and strong empirical performance.
📝 Abstract
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.