= Horvitz-Thompson estimator
{c}
{title2=$\widehat T_{\mathrm{HT}}$}
{wiki=Horvitz–Thompson_estimator}
The \b[Horvitz-Thompson estimator] estimates a <finite population total> by expanding each observed unit's contribution by the reciprocal of its <inclusion probability>. <D. G. Horvitz> and <D. J. Thompson> introduced the general unequal-probability construction in 1952.<A Generalization of Sampling Without Replacement From a Finite Universe by D. G. Horvitz and D. J. Thompson>[Original publication, Journal of the American Statistical Association 47(260), pp. 663–685, December 1952]{ref}
Let $y_1,\ldots,y_N$ be fixed values attached to labelled population units. A <sampling design> selects a random subset $S$, with known <inclusion probabilities> $\pi_i=\mathbb P(i\in S)>0$. Then
$$
\boxed{\widehat T_{\mathrm{HT}}=\sum_{i\in S}\frac{y_i}{\pi_i}
=\sum_{i=1}^N\frac{I_i y_i}{\pi_i},\qquad T=\sum_{i=1}^Ny_i,}
$$
where $I_i$ is the <inclusion indicator>. This is <inverse probability weighting> for a finite population.<Unequal Probability Sampling in Penn State STAT 506>[Lesson 3, Definition 3.2, Horvitz–Thompson estimator]{ref} Its <expected value> is immediately
$$
\mathbb E_p\widehat T_{\mathrm{HT}}
=\sum_i\frac{y_i}{\pi_i}\mathbb E_pI_i
=\sum_i y_i=T.
$$
The subscript $p$ emphasizes <design-based inference>: the fixed values are not assumed to follow a probability model; the sample-selection mechanism supplies the randomness. The <inclusion indicators> need not be independent. Dividing by known $N$ gives an <unbiased estimator> of the <finite population mean>.
For an original exact example, use values $(y_1,y_2,y_3)=(2,4,6)$ and choose samples $\{1,2\}$, $\{1,3\}$, $\{2,3\}$ with <probabilities> $1/2,1/3,1/6$, respectively. The <inclusion probabilities> are $(5/6,2/3,1/2)$. The corresponding estimates of the total are $8.4,14.4,18$, and
$$
\mathbb E_p\widehat T_{\mathrm{HT}}
=\tfrac12(8.4)+\tfrac13(14.4)+\tfrac16(18)=12=T.
$$
Simply multiplying each <sample mean> by three instead gives totals $9,12,15$, with <expected value> $11$. Equal treatment of the observed values ignores the unequal chances of observing them.
\Image[/horvitz-thompson-three-unit-design.png]
{title=Inverse-inclusion weighting corrects the expected total in an unequal-probability sample}
{description=Original exact enumeration of three possible two-unit samples from values 2, 4 and 6. The left panel shows their selection probabilities; the right compares the Horvitz-Thompson total with three times the sample mean. The population total is 12.}
The <sampling variance> depends on pair information. Put $\pi_{ij}=\mathbb P(i,j\in S)$, including $\pi_{ii}=\pi_i$. Bilinearity of <covariance> gives
$$
\operatorname{Var}_p(\widehat T_{\mathrm{HT}})
=\sum_{i=1}^N\sum_{j=1}^N
\frac{\pi_{ij}-\pi_i\pi_j}{\pi_i\pi_j}y_i y_j.
$$
For the three-unit design it is $14.4$, found directly from $\tfrac12(8.4-12)^2+\tfrac13(14.4-12)^2+\tfrac16(18-12)^2$. Thus an <unbiased estimator> need not give a nearly correct total in each sample.
With <pairwise inclusion positivity>, the same expansion device applied to pairs constructs the <Horvitz-Thompson variance estimator>:
$$
\widehat V_{\mathrm{HT}}
=\sum_{i\in S}\sum_{j\in S}
\frac{\pi_{ij}-\pi_i\pi_j}{\pi_{ij}}\frac{y_i}{\pi_i}\frac{y_j}{\pi_j}.
$$
Its <expected value> equals the preceding population sum because $\mathbb E_p(I_iI_j/\pi_{ij})=1$. In the worked design its three possible values are $9.76,58.56,-60$, whose probability-weighted average is $14.4$. This particular <unbiased estimator> of <variance> can therefore be negative for an individual sample. Its <expected value> identity is a statement about repeated selection.
If $\pi_i=0$, an arbitrary unobserved $y_i$ cannot contribute to an unbiased total estimate. If a distinct pair has $\pi_{ij}=0$, its product cannot be recovered by the pair-expansion argument. Finally, dividing the weighted sum by the random sum of its inverse-probability weights changes the estimator into a ratio; the indicator proof above no longer establishes that this ratio is an <unbiased estimator>. The known population size and the actual <sampling design> are part of the result.
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