Single-draw variance identifiability obstruction

The single-draw variance identifiability obstruction is the inability, for an unrestricted finite population, to construct a universally design-unbiased estimate of a total estimator's variance from just one observed unit. The estimator can estimate the total unbiasedly while the required cross-products for estimating its sampling variance are never jointly observed. This is an obstruction to the stated one-draw estimation problem, not a claim that a parameter can never be identified from repeated independent samples.

An original two-unit proof makes the distinction exact. Sample label $I=1$ or2 with equal probability and observe only $y_I$. The Horvitz-Thompson estimator of the finite population total is $\hat{T}=2y_I$. Its expectation is $y_1+y_2$, and its design variance isSuppose a proposed variance estimate returns $g_1(y_1)$ after observing unit1 and $g_2(y_2)$ after observing unit2. Unbiasedness for every pair would requireThe left side is additive in the two unknown values, while the right side contains an unobserved product.

To prove impossibility without smoothness assumptions, evaluate both sides at $(y_1,y_2)=(0,0),(1,0),(0,1),(1,1)$. Every additive function $H$ has rectangular contrast $H(1,1)-H(1,0)-H(0,1)+H(0,0)=0$. The required variance has contrast $0-1-1+0=-2$. The contradiction rules out such an unbiased estimator. Independent auxiliary randomization does not help: averaging it out merely replaces each $g_i$ by another function of the same single observed value.

Here the individual inclusion probabilities are positive, but the joint inclusion probability for distinct units is zero. The usual Horvitz-Thompson variance estimator needs pairwise inclusion positivity to turn sample cross-products into unbiased population cross-product estimates. The primary analysis of zero pairwise probabilities identifies precisely these missing terms.^[ref]

Extra information can change the problem: observing suitable pairs, imposing a justified population relation, or accepting a model-based or conservative estimate uses assumptions absent from the proof. Observing one value also does not prevent writing a biased variance estimate. What fails is a single-draw unbiased estimator valid for every unrestricted population, even though the total estimator itself remains unbiased.