= Pairwise inclusion positivity
{title2=$\pi_{ij}>0$}
\b[<Pairwise inclusion positivity>] requires every distinct pair of population units to have positive <joint inclusion probability> under the <sampling design>:
$$
\pi_{ij}=\mathbb P(i,j\in S)>0\quad(i\ne j).
$$
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 $\pi_i\pi_j$.<Conservative variance estimation for sampling designs with zero pairwise inclusion probabilities by Peter M. Aronow and Cyrus Samii>[Section 2, conditions for measurability; section 2.1, pair expansion]{ref}
For <simple random sampling> of $n$ distinct labels from $N\geq2$, $\pi_{ij}=n(n-1)/\{N(N-1)\}$. Consequently, $n\geq2$ 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> is
$$
V_p=(y_1-y_2)^2=y_1^2+y_2^2-2y_1y_2.
$$
Any 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 instead
$$
V_p(1,1)-V_p(1,0)-V_p(0,1)+V_p(0,0)=0-1-1+0=-2.
$$
No 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.<Conservative variance estimation for sampling designs with zero pairwise inclusion probabilities by Peter M. Aronow and Cyrus Samii>[Sections 1 and 2.2, impossible unbiased recovery of jointly unobservable cross-products]{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>.
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