Comparison of Variance Estimators of the Horvitz-Thompson Estimator for Randomized Variable Probability Systematic Sampling

Abstract

Abstract The National Stream Survey (NSS) and Environmental Monitoring and Assessment Program (EMAP) use variable probability, systematic sampling, and the Horvitz-Thompson estimator to estimate population parameters of ecological interest. A common strategy of variance estimation for systematic sampling is to assume that the population order had been randomized prior to sampling and to estimate variance under this randomized population model. The Yates-Grundy variance estimator is generally recommended for estimating the variance of the Horvitz-Thompson estimator under this model. But design features of NSS and EMAP preclude application of the Yates-Grundy estimator, so use of the Horvitz-Thompson variance estimator is required. Further, because the first-order inclusion probabilities are known only for the sample units and not the entire population, neither the actual pairwise inclusion probabilities (πuv's) nor the Hartley-Rao approximation of the πuv's can be computed. Thus the variance estimator proposed for use in these surveys was the Horvitz-Thompson variance estimator computed with a new approximation to the πuv's. Having to use this estimator, denoted v o HT, motivated exploration of the general question of when behaviors of the Horvitz-Thompson and Yates-Grundy variance estimators differ and also investigation of the specific performance of the estimator v o HT.To permit comparison of variance estimators, we restricted attention to fixed sample size, variable probability systematic sampling, from a randomly sorted list. Properties of v o HT were compared to those of three other variance estimators; the Yates-Grundy estimator calculated with both the new πuv approximation and the Hartley-Rao approximation, and the Horvitz-Thompson variance estimator calculated with the Hartley-Rao approximation. An empirical study, designed to permit generalization beyond a few special case populations, demonstrated that superiority of the Yates-Grundy variance estimator was restricted to populations having both high correlation between the response variable, y, and the selection variable, x, and approximately equal coefficients of variation for the x and y populations. With the exception of these populations, v o HT performed nearly the same as the Yates-Grundy estimators studied and performed better than the Horvitz-Thompson variance estimator computed with the Hartley-Rao approximation. In NSS and EMAP most response variables are not expected to be highly correlated with the selection variable, so v o HT should furnish an adequate variance approximation when the randomized population model holds.