1 Introduction
Peter M. Aronow and Cyrus Samii
Sampling designs sometimes result in pairs of units having zero probability of being jointly included in the sample. Horvitz and Thompson (1952)'s statement of the properties of the finite population total makes clear that general, unbiased variance estimation for estimators of population totals is impossible for such non-measurable designs (Särndal, Swensson and Wretman 1992, page 33). Optimal methods for variance estimation in these cases remains an open problem. This paper analyzes the nature of the biases that non-measurability introduces for the standard Horvitz-Thompson estimator and studies an approach to correct for this bias in a conservative manner. While our results cannot offer a solution to the non-measurability problem for all practical applications, we do clarify conditions under which the standard estimator performs well and where the conservative bias correction outperforms commonly-used approximations.
Despite their theoretical drawbacks, sampling designs with zero pairwise inclusion probabilities are quite common. A common non-measurable design is one that draws only single units or clusters from a set of strata. This may occur if the population or a subpopulation of interest is incidentally sparse over stratification cells. Another common non-measurable design is a systematic sample in which unit indices are sampled from a list in multiples from a random starting value. In these designs, units whose indices are multiples from different starting values have zero joint probability of inclusion.
Approximate methods have been proposed for special cases, as discussed in Hansen, Hurwitz and Madow (1953, Section 9.15), Särndal et al. (1992, Chapter 3), and Wolter (2007, Chapters 2 and 8). In the single-unit per stratum case, a common approach is to collapse strata and assume units were drawn via a simple random sample from the larger, collapsed stratum. For systematic samples, the standard approach is to use an approximation based on an assumption of simple random sampling with replacement. These approximate methods are generally biased to a degree that cannot be determined from the data. In some cases, it can be shown that the bias will tend to be positive, but such is not the case generally, and especially so when the zero pairwise inclusion probabilities occur in a haphazard manner.
This paper begins in Section 2 by decomposing the bias of the Horvitz-Thompson variance estimator under non-measurability. This exposes precisely how conditions on the underlying data result in more or less bias. In Section 3, we also show how a simple application of Young's inequality yields a bias correction and a class of estimators guaranteed to have weakly positive bias as well as no bias under special conditions. We discuss implications for applied work in Section 4.
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