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

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 ( ruv's) nor the Hartley-Rao approximation of the r,'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 r,'s. Having to use this estimator, denoted motivated exploration of the general question of when behaviors of the Horvitz-Thompson andYates-Grundy variance estimators differ and also investigation ofthe specific performance of the estimator t)gT. To permit comparison of variance estimators, we restricted attention to fixed sample size, variable probability systematic sampling, from a randomly sorted list. Properties of uGT were compared to those of three other variance estimators: the Yates-Grundy estimator calculated with both the new ruu 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, uGT 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 wGT should furnish an adequate variance approximation when the randomized population model holds.

Use of Multiple Methods to Estimate Walrus (<i>Odobenus rosmarus rosmarus</i>) Abundance in the Penny Strait-Lancaster Sound and West Jones Sound Stocks, Canada

“…Although it is preferred to obtain tag data concurrent with the survey period (Thompson et al 1997, Gilbert et al 2005 and over the whole survey area (Huber et al 2001, Sharples et al 2009), there were no tag data for the years of maximum counts. Counts are samples with unequal probabilities and, to adjust for varying haulout behaviour recorded by the tags, we used the Horvitz-Thompson estimator (Cochran 1977, Udevitz et al 2009, with a variance estimator to allow for variable sample size (Stehman and Overton 1994). Ultimately, we demonstrated that the Horvitz-Thompson estimator required more days of data from more tags to avoid spurious, negative variance estimators (C. Schwarz pers.…”

A Sample-Based Forest Monitoring Strategy Using Landsat, AVHRR and MODIS Data to Estimate Gross Forest Cover Loss in Malaysia between 1990 and 2005

“…This variance estimator uses an approximation to the pairwise inclusion probabilities for the randomized systematic implementation of the πpx sampling design [36], where a pairwise inclusion probability is defined as the probability that a particular pair of blocks would be jointly included in the sample.…”

Using Linear Programming to Design Samples for a Complex Survey

“…The Horvitz-Thompson formula contains the selection probabilities for pairs of individuals as well as the individual selection probabilities, and the pairwise probabilities are often difficult to work out. Some approximations are compared in Stehman and Overton (1994).…”