Horvitz–Thompson estimator
Encyclopedia
In statistics
, the Horvitz–Thompson estimator, named after Daniel G. Horvitz and Donovan J. Thompson, is a method for estimating the 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. It is frequently applied in survey analyses
and can be used to account for missing data
.
Formally, let
be an independent sample from n of N ≥ n distinct strata with a common mean μ. Suppose further that 
is the inclusion probability
that a randomly sampled individual in a superpopulation belongs to the
-th stratum. The Horvitz–Thompson estimate of the mean is given by:
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, the Horvitz–Thompson estimator, named after Daniel G. Horvitz and Donovan J. Thompson, is a method for estimating the mean of a superpopulation
Sampling (statistics)
In statistics and survey methodology, sampling is concerned with the selection of a subset of individuals from within a population to estimate characteristics of the whole population....
in a stratified sample. Inverse probability weighting
Inverse probability weighting
Inverse probability weighting is a statistical technique for calculating statistics standardized to a population different from that in which the data was collected. Study designs with a disparate sampling population and population of target inference are common in application...
is applied to account for different proportions of observations within strata in a target population. It is frequently applied in survey analyses
Statistical survey
Survey methodology is the field that studies surveys, that is, the sample of individuals from a population with a view towards making statistical inferences about the population using the sample. Polls about public opinion, such as political beliefs, are reported in the news media in democracies....
and can be used to account for missing data
Missing values
In statistics, missing data, or missing values, occur when no data value is stored for the variable in the current observation. Missing data are a common occurrence and can have a significant effect on the conclusions that can be drawn from the data....
.
Formally, let
be an independent sample from n of N ≥ n distinct strata with a common mean μ. Suppose further that is the inclusion probabilityInclusion probability
In statistics, in the theory relating to sampling from finite populations, the inclusion probability of an element or member of the population is its probability of becoming part of the sample during the drawing of a single sample....
that a randomly sampled individual in a superpopulation belongs to the
-th stratum. The Horvitz–Thompson estimate of the mean is given by:-
In a BayesianBayesianBayesian refers to methods in probability and statistics named after the Reverend Thomas Bayes , in particular methods related to statistical inference:...
probabilistic framework
is considered the proportion of individuals in a target population belonging to the ith stratum. Hence, 
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 bootstrapBootstrapping (statistics)In statistics, bootstrapping is a computer-based method for assigning measures of accuracy to sample estimates . This technique allows estimation of the sample distribution of almost any statistic using only very simple methods...
resamplingResampling (statistics)In statistics, resampling is any of a variety of methods for doing one of the following:# Estimating the precision of sample statistics by using subsets of available data or drawing randomly with replacement from a set of data points # Exchanging labels on data points when performing significance...
estimate of the mean. It can also be viewed as a special case of multiple imputationImputation (statistics)In statistics, imputation is the substitution of some value for a missing data point or a missing component of a data point. Once all missing values have been imputed, the dataset can then be analysed using standard techniques for complete data...
approaches.
For post-stratifiedStatistical benchmarkingIn statistics, benchmarking is a method of using auxiliary information to adjust the sampling weights used in an estimation process, in order to yield more accurate estimates of totals....
study designs, estimation of
and 
are done in distinct steps. In such cases, computating the variance of 
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 RR (programming language)R is a programming language and software environment for statistical computing and graphics. The R language is widely used among statisticians for developing statistical software, and R is widely used for statistical software development and data analysis....
conducts analyses for post-stratified data using the Horvitz–Thompson estimator.
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