= D. G. Horvitz
{c}
\b[D. G. Horvitz] coauthored the 1952 paper that generalized finite-population sampling without replacement to unequal selection probabilities. The joint work with <D. J. Thompson> provides the foundation for 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, Journal of the American Statistical Association 47, pp. 663–685]{ref}
The central idea is to compensate for unequal chances of inclusion. For fixed population values $y_i$ and positive <inclusion probabilities> $\pi_i$, write a total estimate as
$$
\widehat Y=\sum_{i\in S}\frac{y_i}{\pi_i}.
$$
Using an <inclusion indicator> $I_i$ gives $\mathbb E(I_i/\pi_i)=1$, so $\mathbb E\widehat Y=\sum_i y_i$. Unbiasedness follows from the design probabilities rather than from assuming that the sampled values are an independent realization of a population model.<Unequal Probability Sampling in Penn State STAT 506>[Lesson 3, inverse-inclusion-probability total estimation]{ref}
For an illustrative population $(4,6)$, select exactly one unit with probabilities $(1/4,3/4)$. The total estimate is $16$ or $8$, respectively, with <expected value> $(1/4)16+(3/4)8=10$, the true total. The less frequently selected unit receives the larger multiplier. This example explains how a sampling design can deliberately favor some units without forcing an unweighted analysis to be correct.
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