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  "body": "\n## Definition\n\nLet $U = \\set{1, \\dots, N}$ be a finite population, and let $y_1, \\dots, y_N$ be fixed real values attached to the units.\nLet $S \\subseteq U$ be a random sample drawn by a sampling design, and let $\\pi_i = P(i \\in S)$ be the inclusion probability of unit $i$.\nAssume $\\pi_i \u003e 0$ for every $i \\in U$.\nThe *Horvitz–Thompson estimator* of the population total $Y = \\sum_{i=1}^N y_i$ is the [[estimator|0TTNRP5PFQ]]\n$$\\hat{Y}_{HT} = \\sum_{i \\in S} \\frac{y_i}{\\pi_i}.$$\nThe Horvitz–Thompson estimator of the population mean $Y/N$ is $\\hat{Y}_{HT}/N$.\n\n## Remarks\n\nThe estimator is unbiased: write $\\hat{Y}_{HT} = \\sum_{i=1}^N \\frac{y_i}{\\pi_i} \\mathbf{1}[i \\in S]$, and take the [[expected value|0DCT0QZVQ1]], using $E(\\mathbf{1}[i \\in S]) = \\pi_i$, to get $E(\\hat{Y}_{HT}) = Y$.\n\nExample: in a [[simple random sample|Y0PTCJBPEA]] of size $n$ drawn without replacement, every unit has $\\pi_i = n/N$.\nThen $\\hat{Y}_{HT} = \\frac{N}{n}\\sum_{i \\in S} y_i = N \\bar{y}$, where $\\bar{y}$ is the sample mean.\n\nNon-example: let some unit $j$ have $\\pi_j = 0$.\nThen it never appears in the sample, so no estimator built from the sample can be unbiased for every choice of $y_j$.\nThe condition $\\pi_i \u003e 0$ fails, and the definition does not apply.\n\nWith the joint inclusion probabilities $\\pi_{ij} = P(i \\in S, j \\in S)$ and $\\pi_{ii} = \\pi_i$, the variance is\n$$\\operatorname{Var}(\\hat{Y}_{HT}) = \\sum_{i=1}^N \\sum_{j=1}^N (\\pi_{ij} - \\pi_i \\pi_j) \\frac{y_i}{\\pi_i} \\frac{y_j}{\\pi_j}.$$\n",
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