In many applications, data cluster. Failing to take the cluster structure into consideration generally leads to underestimated variances of point estimators and inflated type I errors in hypothesis tests. Many circumstance-dependent approaches have been developed to handle clustered data. A working covariance matrix may be used in generalised estimating equations. One may throw out the cluster structure and use only the cluster means, or explicitly model the cluster structure. Our interest is the case where multiple samples of clustered data are collected, and the population quantiles are particularly important. We develop a composite empirical likelihood for clustered data under a density ratio model. This approach avoids parametric assumptions on the population distributions or the cluster structure. It efficiently utilises the common features of the multiple populations and the exchangeability of the cluster members. We also develop a cluster-based bootstrap method to provide valid variance estimation and to control the type I errors. We examine the performance of the proposed method through simulation experiments and illustrate its usage via a real-world example.

在许多应用程序中，数据集群。如果不考虑聚类结构，通常会导致点估计量的方差被低估，并且在假设检验中会导致I型错误膨胀。已经开发了许多与环境有关的方法来处理聚类数据。可以在广义估计方程中使用工作协方差矩阵。一个人可以扔掉集群结构，只使用集群方法，或者显式地对集群结构建模。我们感兴趣的是收集多个聚类数据样本的情况，而总体分位数尤为重要。我们在密度比模型下为聚类数据开发了一个综合的经验似然性。这种方法避免了关于人口分布或集群结构的参数假设。它有效地利用了多个种群的共同特征和集群成员的可交换性。我们还开发了基于群集的引导程序方法，以提供有效的方差估计并控制I型错误。我们通过仿真实验检查了该方法的性能，并通过一个实际示例说明了其用法。