= Design-based inference
\b[<Design-based inference>] evaluates an estimator by the random mechanism selecting a sample from a fixed finite population. Population values $y_1,\ldots,y_N$ are held fixed; the selected subset $S$ is random under a known <sampling design> $p(s)$. Thus
$$
\mathbb E_p\widehat T=\sum_s p(s)\widehat T(s),\qquad
\operatorname{Var}_p(\widehat T)=\sum_s p(s)\{\widehat T(s)-\mathbb E_p\widehat T\}^2.
$$
The subscript $p$ identifies the source of randomness. An <unbiased estimator> of a <finite population total> has $\mathbb E_p\widehat T=T$ for each eligible fixed population. The unequal-probability total developed by <D. G. Horvitz> and <D. J. Thompson> is a central example.<A Generalization of Sampling Without Replacement From a Finite Universe by D. G. Horvitz and D. J. Thompson>[1952, unequal-probability sampling and unbiased finite-universe estimation]{ref}
For an original example, let the fixed population values be $(1,1,10)$ and sample one label uniformly. Every <inclusion probability> is $1/3$, so the <Horvitz-Thompson estimator> is $3y_i$. Its three equally probable values are $3,3,30$, while the actual total is $12$. Consequently,
$$
\mathbb E_p\widehat T=12,\qquad
\operatorname{Var}_p(\widehat T)=\frac{(3-12)^2+(3-12)^2+(30-12)^2}{3}=162.
$$
The estimator is unbiased even though none of its possible estimates equals the target. Unbiasedness describes averaging across the <sampling design>, rather than guaranteed accuracy in an individual sample.
A <confidence interval> with design-based coverage must likewise succeed at its stated rate across repeated selections under the specified design. That property requires a justified construction; unbiasedness of a point estimate alone supplies no coverage guarantee. Repeating a convenient calculation on the same selected units does not reproduce the selection process. Assumptions about measurement error or missing responses must also be stated separately, since knowledge of the original selection probabilities alone does not repair those later stages.
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