Horvitz–Thompson estimator

In statistics, the Horvitz–Thompson estimator, named after Daniel G. Horvitz and Donovan J. Thompson, is a method for estimating the total and mean of a pseudo-population in a stratified sample by applying inverse probability weighting to account for the difference in the sampling distribution between the collected data and the target population. The Horvitz–Thompson estimator is frequently applied in survey analyses and can be used to account for missing data, as well as many sources of unequal selection probabilities.

The method

Formally, let

Y

i

,

i

=

1

,

2

,

…

,

n

{\displaystyle Y_{i},i=1,2,\ldots ,n}

be an independent sample from

n

{\displaystyle n}

of

N

≥

n

{\displaystyle N\geq n}

distinct strata with an overall mean

μ

{\displaystyle \mu }

. Suppose further that

π

i

{\displaystyle \pi _{i}}

is the inclusion probability that a randomly sampled individual in a superpopulation belongs to the

i

{\displaystyle i}

th stratum. The Horvitz–Thompson estimator of the total is given by:

Y

^

H

T

=

∑

i

=

1

n

Y

i

π

i

,

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

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

μ

^

H

T

=

1

N

Y

^

H

T

=

1

N

∑

i

=

1

n

Y

i

π

i

.

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

In a Bayesian probabilistic framework

π

i

{\displaystyle \pi _{i}}

is considered the proportion of individuals in a target population belonging to the

i

{\displaystyle i}

th stratum. Hence,

Y

i

/

π

i

{\displaystyle Y_{i}/\pi _{i}}

could be thought of as an estimate of the complete sample of persons within the

i

{\displaystyle i}

th 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.

For post-stratified study designs, estimation of

π

{\displaystyle \pi }

and

μ

{\displaystyle \mu }

are done in distinct steps.