COMPARISON OF VARIANCE ESTIMATORS OF THE HORVITZ-THOMPSON ESTIMATOR FOR RANDOMIZED VARIABLE PROBABILITY SYSTEMATIC-SAMPLING

The National Stream Survey (NSS) and Environmental Monitoring and Asse ssment Program (EMAP) use variable probability, systematic sampling, a nd the Horvitz-Thompson estimator to estimate population parameters of ecological interest. A common strategy of variance estimation for sys tematic sampling is to assume that the population order had been rando mized prior to sampling and to estimate variance under this randomized population model. The Yates-Grundy variance estimator is generally re commended for estimating the variance of the Horvitz-Thompson estimato r under this model. But design features of NSS and EMAP preclude appli cation of the Yates-Grundy estimator, so use of the Horvitz-Thompson v ariance estimator is required. Further, because the first-order inclus ion probabilities are known only for the sample units and not the enti re population, neither the actual pairwise inclusion probabilities (pi (uv)'s) nor the Hartley-Rao approximation of the pi(uv)'s can be compu ted. Thus the variance estimator proposed for use in these surveys was the Horvitz-Thompson variance estimator computed with a new approxima tion to the pi(uv)'s. Having to use this estimator, denoted upsilon(HT )o, 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 upsilo n(HT)o. To permit comparison of variance estimators, we restricted att ention to fixed sample size, variable probability systematic sampling, from a randomly sorted list. Properties of upsilon(HT)o were compared to those of three other variance estimators: the Yates-Grundy estimat or calculated with both the new pi(uv) approximation and the Hartley-R ao approximation, and the Horvitz-Thompson variance estimator calculat ed with the Hartley-Rao approximation. An empirical study, designed to permit generalization beyond a few special case populations, demonstr ated that superiority of the Yates-Grundy variance estimator was restr icted to populations having both high correlation between the response variable, y, and the selection variable, x, and approximately equal c oefficients of variation for the x and y populations. With the excepti on of these populations, upsilon(HT)o performed nearly the same as the Yates-Grundy estimators studied and performed better than the Horvitz -Thompson variance estimator computed with the Hartley-Rao approximati on. In NSS and EMAP most response variables are not expected to be hig hly correlated with the selection variable, so upsilon(HT)o should fur nish an adequate variance approximation when the randomized population model holds.