A Bayesian Approach for the Variance of Fine Stratification

Fine stratification is a popular design as it permits the stratification to be carried out to the fullest possible extent. Some examples include the Current Population Survey and National Crime Victimization Survey both conducted by the U.S. Census Bureau, and the National Survey of Family Growth conducted by the University of Michigan's Institute for Social Research. Clearly, the fine stratification survey has proved useful in many applications as its point estimator is unbiased and efficient. A common practice to estimate the variance in this context is collapsing the adjacent strata to create pseudo-strata and then estimating the variance, but the attained estimator of variance is not design-unbiased, and the bias increases as the population means of the pseudo-strata become more variant. Additionally, the estimator may suffer from a large mean squared error (MSE). In this paper, we propose a hierarchical Bayesian estimator for the variance of collapsed strata and compare the results with a nonparametric Bayes variance estimator. Additionally, we make comparisons with a kernel-based variance estimator recently proposed by Breidt et al. (2016). We show our proposed estimator is superior compared to the alternatives given in the literature such that it has a smaller frequentist MSE and bias. We verify this throughout multiple simulation studies and data analysis from the 2007-8 National Health and Nutrition Examination Survey and the 1998 Survey of Mental Health Organizations.