= D. J. Thompson
{c}
\b[D. J. Thompson] coauthored the 1952 general treatment of sampling without replacement from a finite population with unequal selection probabilities. The paper with <D. G. Horvitz> is the source of the <Horvitz-Thompson estimator>.<A Generalization of Sampling Without Replacement From a Finite Universe by D. G. Horvitz and D. J. Thompson>[Original byline and publisher abstract, pp. 663–685]{ref}
The contribution separates unbiased total estimation from assumptions of independent selection. If $I_i$ is an <inclusion indicator>, then $\mathbb E I_i=\pi_i$, while pair products satisfy $\mathbb E(I_iI_j)=\pi_{ij}$. Consequently, for fixed population values,
$$
\operatorname{Var}(\widehat Y)=\sum_{i,j}
\frac{\pi_{ij}-\pi_i\pi_j}{\pi_i\pi_j}y_i y_j,
\qquad \pi_{ii}=\pi_i.
$$
This identity makes <joint inclusion probabilities> essential to a variance calculation, even though the expectation of the total estimate only needs the individual <inclusion probabilities>.
For an illustrative population $(1,1)$, sample one unit uniformly. Each selected value is multiplied by $2$, so the total estimate is always $2$ and its <variance> is zero. The individual random contributions are nevertheless variable: their covariance cancels their variances because they cannot both be selected. Ignoring that sampling dependence would give the wrong precision. The example shows why design-based estimation concerns the whole selection mechanism, not just a set of multipliers.
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