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In [[statistics]], the '''Horvitz–Thompson estimator''', named after Daniel G. Horvitz and Donovan J. Thompson,<ref>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}}</ref> is a method for estimating the total<ref>William G. Cochran (1977), ''Sampling Techniques'', 3rd Edition, Wiley. ISBN 0-471-16240-X</ref> and mean of a [[Sampling (statistics)|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 [[Statistical survey|survey analyses]] and can be used to account for [[missing values|missing data]].
 
==The method==
Formally, let <math>Y_i, i = 1, 2, \ldots, n</math> be an [[independence (probability theory)|independent]] sample from ''n'' of ''N ≥ n'' distinct strata with a common mean&nbsp;''μ''. Suppose further that <math>\pi_i</math> is the [[inclusion probability]] that a randomly sampled individual in a superpopulation belongs to the ''i''th stratum. The Horvitz–Thompson estimate of the total is given by:
: <math>
\hat{Y}_{HT} = \sum_{i=1}^n \pi_i ^{-1} Y_i,
</math>
 
and the estimate of the mean is given by:
: <math>
\hat{\mu}_{HT} = N^{-1}\hat{Y}_{HT} = N^{-1}\sum_{i=1}^n \pi_i ^{-1} Y_i.
</math>
 
In a [[Bayesian probability|Bayesian]] probabilistic framework <math>\pi_i</math> is considered the proportion of individuals in a target population belonging to the ''i''th stratum. Hence, <math>\pi_i^{-1} Y_i</math> could be thought of as an estimate of the complete sample of persons within the ''i''th stratum. The Horvitz–Thompson estimator can also be expressed as the limit of a weighted [[bootstrapping (statistics)|bootstrap]] [[resampling (statistics)|resampling]] estimate of the mean. It can also be viewed as a special case of multiple [[imputation (statistics)|imputation]] approaches.<ref>Roderick J.A. Little, Donald B. Rubin (2002) ''Statistical Analysis With Missing Data'', 2nd ed., Wiley. ISBN 0-471-18386-5</ref>
 
For [[Statistical benchmarking|post-stratified]] study designs, estimation of <math>\pi</math> and <math>\mu</math> are done in distinct steps. In such cases, computating the variance of <math>\hat{\mu}_{HT}</math> 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{{Citation needed|date=July 2013}}. The Survey package for [[R (programming language)|R]] conducts analyses for post-stratified data using the Horvitz–Thompson estimator.
 
== References ==
{{reflist}}
 
==External links==
* [http://faculty.washington.edu/tlumley/survey/ Survey Package Website for R]
 
{{DEFAULTSORT:Horvitz-Thompson estimator}}
[[Category:Sampling (statistics)]]
[[Category:Survey methodology]]
[[Category:Missing data]]

Latest revision as of 06:26, 21 March 2013

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 Yi,i=1,2,…,n be an independent sample from n of N ≥ n distinct strata with a common mean μ. Suppose further that πi 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:

Y^HT=∑i=1nπi−1Yi,

and the estimate of the mean is given by:

μ^HT=N−1Y^HT=N−1∑i=1nπi−1Yi.

In a Bayesian probabilistic framework πi is considered the proportion of individuals in a target population belonging to the ith stratum. Hence, πi−1Yi 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 π and μ are done in distinct steps. In such cases, computating the variance of μ^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 estimatorPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.. The Survey package for R conducts analyses for post-stratified data using the Horvitz–Thompson estimator.

References

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External links

  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, . Template:Jstor
  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
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