= Horvitz-Thompson Estimator =

The '''Horvitz-Thompson estimator''' is an estimator for population totals and means.

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== Description ==

While a population total ''Y'' can be determined by measuring all members of the population, an estimate can be calculated using a smaller sample ''s''. For each unit ''i'' of the sample, there is a corresponding measurement ''y,,i,,'', and the population total is estimated as:

{{attachment:total.svg}}

where ''π,,i,,'' is the probability of inclusion in the sample for unit ''i''. Equivalently, let ''w,,i,,'' be a weight equal to the inverse of that probability, i.e. ''w,,i,, = π,,i,,^-1^''.

It should be clear that the population mean is then estimated as:

{{attachment:mean.svg}}

where ''n(s)'' is the sample size.



=== Domain Estimator ===

Let ''1,,d,,'' be an [[Analysis/IndicatorFunctions|indicator function]] for a domain ''d''.

A direct estimator for a domain mean is:

{{attachment:domain1.svg}}

The numerator can be considered an estimator for the population total within the domain (''Y,,d,,''), denoted ''Ŷ,,d,,''. And the denominator can be considered an estimator for the domain total (''t,,d,,''), denoted ''t̂,,d,,''. Therefore this estimator is a function of two estimators.

Sometimes this estimator is written as a summation over ''s,,d,,'', i.e. the subset of ''s'' that is in ''d'', instead. That is:

{{attachment:domain2.svg}}

This formulation ignores measurements from the rest of the sample. Importantly it hides the fact that the estimator is a ratio (i.e. is not linear). As such, the variance is generally approximated using [[Statistics/LinearizedVariance|linearization]].



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