= 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 $\widehat T=2y_I$. Its expectation is $y_1+y_2$, and its design variance is
$$
\operatorname{Var}_p(\widehat T)=\tfrac12(y_1-y_2)^2+\tfrac12(y_2-y_1)^2=(y_1-y_2)^2.
$$
Suppose 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 require
$$
\tfrac12g_1(y_1)+\tfrac12g_2(y_2)=(y_1-y_2)^2.
$$
The 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.<Conservative variance estimation for sampling designs with zero pairwise inclusion probabilities by Peter M. Aronow and Cyrus Samii>[§2.1, pair-expansion variance estimator; §2.2 and Proposition1, missing products under zero pairwise probabilities]{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.
Back to article page
