 Pairwise inclusion positivity

Pairwise inclusion positivity requires every distinct pair of population units to have positive joint inclusion probability under the sampling design:For Horvitz-Thompson variance estimator, the probabilities must also be known, and individual inclusion probabilities must be positive. Each pair can then, at least in principle, contribute an observed product to a pair-expansion estimate. Positivity is not independence: ${\pi }_{ij}$ need not equal ${\pi }_i{\pi }_j$.^[ref]

For simple random sampling of $n$ distinct labels from $N\ge 2$, ${\pi }_{ij}=n(n-1)/\{N(N-1)\}$. Consequently, $n\ge 2$ gives pairwise positivity, whereas selecting exactly one label gives positive individual inclusion probabilities but zero probability of ever observing two distinct values together. This distinction separates unbiased total estimation from unbiased estimation of its sampling variance.

An original two-unit proof shows what information is missing. Sample one of two fixed values $y_1,y_2$ uniformly. The Horvitz-Thompson estimator is $2y_1$ or $2y_2$, and its variance isAny estimator based only on the observed label and value has expected value of the form $\{g_1(y_1)+g_2(y_2)\}/2$. Its mixed difference over $(y_1,y_2)\in \{0,1{\}}^2$ is zero. The target's mixed difference is insteadNo such estimator can therefore be unbiased for this variance for every possible pair of values. Additional randomization internal to the estimator does not help: conditioning on the observed data still produces functions of this same separated form. This is an information limitation, not division by zero that can be repaired by assigning a convenient numerical convention.^[ref]

Strict positivity alone also does not guarantee useful precision. A very small ${\pi }_{ij}$ produces a large inverse-probability multiplier on a rare observed pair. Designing a usable survey therefore requires attention to both observability and the resulting sampling variance.