PROBABILITY SAMPLE U-STATISTICS: THEORY AND APPLICATIONS FOR COMPLEX SAMPLE DESIGNS (VARIANCE COMPONENTS, ROBUST, INFERENCE)

The classical theory of U-statistics is extended to the realm of multistage unequal probability sample designs. As in the classical domain, probability sample U-statistic theory provides robust nonparametric inference for generalized symmetric means. In the unequal probability sampling context, unbiased variance and variance component estimators are identified as degree 2 probability sample U-statistics. Considering the central role that variance and variance component estimates play in probability sample design and inference, the associated U-statistic theory provides a valuable new research and analysis tool for survey statistics.;Recognizing that most variance and variance component estimation problems involve nonlinear functions of U-statistics, extensions of the Taylor series linearization (delta method), balanced repeated replication (BRR), and the Jackknife are developed for probability sample U-statistics. Previous sample design limitations on the BRR and Jackknife methods relating to with replacement primary unit selections or uniform finite population correction factors across strata are removed. These developments also provide BRR analogs that are no longer constrained to designs with equal stratum sample sizes.;Three areas of application are illustrated. The first application explores the small sample properties, bias, and mean-squared-error, of a new class of ratio variance estimators. The second application estimates the variance of a probability sample t-type statistic and approximates the associated degrees of freedom by equating moments to the non-central t distribution. The third application develops a new variance component model and associated component estimators for a complex two stage unequal probability sample.;Strictly unbiased covariance estimators are developed for probability sample U-statistics. The deeply stratified nature of most probability sample designs leads to stratum specific sample sizes that are too small to justify large sample variance approximations. The unbiased U-statistic covariance estimator is analogous to the Yates-Grundy-Sen variance estimator for degree 1 Horvitz-Thompson statistics. Durbin's theorem for unbiased multistage variance estimation is extended to multistage U-statistics.