Horvitz–Thompson estimator

In statistics, the Horvitz–Thompson estimator, named after Daniel G. Horvitz and Donovan J. Thompson,^[1] is a method for estimating the total^[2] and mean of a superpopulation in a stratified sample. Inverse probability weighting is applied to account for different proportions of observations within strata in a target population. The Horvitz–Thompson estimator is frequently applied in survey analyses and can be used to account for missing data.

The method

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

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

and the estimate of the mean is given by:

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

In a Bayesian probabilistic framework $\pi _{i}$ is considered the proportion of individuals in a target population belonging to the ith stratum. Hence, $\pi _{i}^{{-1}}Y_{i}$ 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.^[3]

For post-stratified study designs, estimation of $\pi$ and $\mu$ are done in distinct steps. In such cases, computating the variance of ${\hat {\mu }}_{{HT}}$ 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. The Survey package for R conducts analyses for post-stratified data using the Horvitz–Thompson estimator.

References

1. Horvitz, D. G.; Thompson, D. J. (1952) "A generalization of sampling without replacement from a finite universe", Journal of the American Statistical Association, 47, 663–685, . JSTOR 2280784
2. William G. Cochran (1977), Sampling Techniques, 3rd Edition, Wiley. ISBN 0-471-16240-X
3. Roderick J.A. Little, Donald B. Rubin (2002) Statistical Analysis With Missing Data, 2nd ed., Wiley. ISBN 0-471-18386-5

External links

• Survey Package Website for R