Multivariate Mean Estimation Under Efficient Sampling Designs

In many studies, the researchers attempt to describe the relationship between more than two outcome (or response) variables with its determinants (covariates). In this paper, we present an efficient procedure based on ranked set sampling to estimate and perform the hypothesis testing on a multivariate outcome mean. The method is based on ranking on an auxiliary covariate, which is assumed to be correlated with the multivariate response, in order to improve the efficiency of the estimation. We show that the proposed estimator developed under this sampling scheme is unbiased, has smaller variance in the multivariate sense, and is asymptotically Gaussian. A bootstrap routine is developed in the statistical software R to perform the inference when the sample size is small. We also extend the regression estimators based on ranked set sampling to multivariate regression. We use a simulation study to investigate the performance of the method under known conditions and apply the method to the biomarker data collected in China Health and Nutrition Survey (CHNS 2009) data.