Horvitz–Thompson estimator

Formally, let ${\displaystyle Y_{i},i=1,2,\ldots ,n}$ be an independent sample from n of N ≥ n distinct strata with a common mean μ. Suppose further that ${\displaystyle \pi _{i}}$ is the inclusion probability that a randomly sampled individual in a superpopulation belongs to the ith stratum. The Horvitz–Thompson estimator of the total is given by:[3]^: 51

${\displaystyle {\hat {Y}}_{HT}=\sum _{i=1}^{n}\pi _{i}^{-1}Y_{i},}$

and the Horvitz–Thompson estimate of the mean is given by:

${\displaystyle {\hat {\mu }}_{HT}=N^{-1}{\hat {Y}}_{HT}=N^{-1}\sum _{i=1}^{n}\pi _{i}^{-1}Y_{i}.}$

In a Bayesian probabilistic framework ${\displaystyle \pi _{i}}$ is considered the proportion of individuals in a target population belonging to the ith stratum. Hence, ${\displaystyle \pi _{i}^{-1}Y_{i}}$ could be thought of as an estimate of the complete sample of persons within the ith stratum. The Horvitz–Thompson estimator can also be expressed as the limit of a weighted bootstrap resampling estimate of the mean. It can also be viewed as a special case of multiple imputation approaches.[4]

For post-stratified study designs, estimation of ${\displaystyle \pi }$ and ${\displaystyle \mu }$ are done in distinct steps. In such cases, computating the variance of ${\displaystyle {\hat {\mu }}_{HT}}$ is not straightforward. Resampling techniques such as the bootstrap or the jackknife can be applied to gain consistent estimates of the variance of the Horvitz–Thompson estimator.[5] The "survey" package for R conducts analyses for post-stratified data using the Horvitz–Thompson estimator.[6]

The Horvitz–Thompson estimator can be shown to be unbiased when evaluating the expectation of the Horvitz–Thompson estimator, ${\displaystyle \mathbf {E} {\bar {X}}_{n}^{HT}}$, as follows:

${\displaystyle \mathbf {E} {\bar {X}}_{n}^{HT}=\mathbf {E} {\frac {1}{N}}\sum _{i=1}^{n}{\frac {\mathbf {X} _{I_{i}}}{\pi _{I_{i}}}}}$

${\displaystyle =\mathbf {E} {\frac {1}{N}}\sum _{i=1}^{N}{\frac {X_{i}}{\pi _{i}}}1_{i\in D_{n}}}$

${\displaystyle =\sum _{b=1}^{B}P(D_{n}^{(b)})\left[{\frac {1}{N}}\sum _{i=1}^{N}{\frac {X_{i}}{\pi _{i}}}1_{i\in D_{n}^{(b)}}\right]}$

${\displaystyle ={\frac {1}{N}}\sum _{i=1}^{N}{\frac {X_{i}}{\pi _{i}}}\sum _{b=1}^{B}1_{i\in D_{n}^{(b)}}P(D_{n}^{(b)})}$

${\displaystyle ={\frac {1}{N}}\sum _{i=1}^{N}\left({\frac {X_{i}}{\pi _{i}}}\right)\pi _{i}}$

${\displaystyle ={\frac {1}{N}}\sum _{i=1}^{N}X_{i}}$

${\displaystyle {\text{where}}~D_{n}=\{x_{1},x_{2},...,x_{n}\}}$

The Hansen-Hurwitz (1943) is known to be inferior to the Horvitz–Thompson (1952) strategy, associated with a number of Inclusion Probabilities Proportional to Size (IPPS) sampling procedures.[7]

• Survey Package Website for R