The effect of cluster sampling on the covariance and correlation matrices of sample distribution functions

In the analysis of continuous variables collected through a complex sample design, the sparseness of data in the tail region may lead to relatively poor performance for design-based estimation of distribution functions and also to potential instability of direct design-based estimators of their covariance matrices. Consequently, it is of interest to consider approximation methods that may lead to more stable covariance matrix estimators. Accordingly, one may seek to obtain better inference methods by fitting an appropriate parametric model to data from these tail regions. This paper develops one such approximation method by examining the effect of cluster sampling on the covariance and correlation matrices of sample distribution functions based on a superpopulation model. The results are applied to data from a stratified multistage sampling design. It is then compared with the empirical result from medical examination data from the US Third National Health and Nutrition Examination Survey.