Horvitz–Thompson estimator

Formally, let $Y\_i,\; i\; =\; 1,\; 2,\; \backslash ldots,\; n$ be an independent sample from $n$ of $N\; \backslash ge\; n$ distinct strata with an overall mean $\backslash mu$. Suppose further that $\backslash pi\_i$ is the inclusion probability that a randomly sampled individual in a superpopulation belongs to the $i$th stratum. The Horvitz–Thompson estimator of the total is given by:{{cite book

|title = Model Assisted Survey Sampling

|isbn= 9780387975283

|year = 1992

|last1= Särndal

|first1= Carl-Erik

|last2= Swensson

|first2= Bengt

|last3= Wretman

|first3= Jan Hȧkan

}}{{rp|51}}

: $$

\hat{Y}_\mathrm{HT} = \sum_{i=1}^n \frac{Y_i}{\pi_i},

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

: $$

\hat{\mu}_\mathrm{HT} = \frac1N\hat{Y}_{HT} = \frac1N\sum_{i=1}^n \frac{Y_i}{\pi_i}.

In a Bayesian probabilistic framework $\backslash pi\_i$ is considered the proportion of individuals in a target population belonging to the $i$th stratum. Hence, $Y\_i/\backslash pi\_i$ 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 bootstrap resampling estimate of the mean. It can also be viewed as a special case of multiple imputation approaches.Roderick J.A. Little, Donald B. Rubin (2002) Statistical Analysis With Missing Data, 2nd ed., Wiley. {{ISBN|0-471-18386-5}}

For post-stratified study designs, estimation of $\backslash pi$ and $\backslash mu$ are done in distinct steps. In such cases, computating the variance of $\backslash hat\{\backslash mu\}\_\{HT\}$ 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.{{cite journal |last=Quatember |first=A. |title=The Finite Population Bootstrap - from the Maximum Likelihood to the Horvitz-Thompson Approach |journal=Austrian Journal of Statistics |date=2014 |volume=43 |issue=2 |pages=93–102 |doi=10.17713/ajs.v43i2.10 |doi-access=free }} The "survey" package for R conducts analyses for post-stratified data using the Horvitz–Thompson estimator.{{cite web |url=https://cran.r-project.org/web/packages/survey/ |title = CRAN - Package survey| date=19 July 2021 }}

For this proof it will be useful to represent the sample as a random subset $S\backslash subseteq\backslash \{1,\backslash ldots,N\backslash \}$ of size $n$. We can then define indicator random variables $I\_j\; =\; \backslash mathbf\{1\}[j\; \backslash in\; S]$ representing whether for each $j$ in $\backslash \{1,\backslash ldots,N\backslash \}$ whether it is present in the sample. Note that for any observation in the sample, the expectation is the definition of the inclusion probability:

\pi_i = \operatorname{\mathbb E}\left(I_i\right) = \Pr(i\in S)

.

{{efn|Technically, the indexing scheme in the proof is different from the indexing in the description of the estimator. In the proof, $Y\_j$ is the $j$th value in a global ordering out of $N$ strata. In the description, $Y\_i$ is the $i$th value in the sample, out of $n$. To unify these two, we could explicitly define a function mapping sample-indices to global indices.}}

Taking the expectation of the estimator we can prove it is unbiased as follows:

:$$

\begin{align}

\operatorname{\mathbb E}\left(\hat{\mu}_\mathrm{HT}\right) &= \operatorname{\mathbb E}\left(\frac{1}{N}\sum_{i\in S} \frac{Y_i}{\pi_i}\right)\\[6pt]

&=\operatorname{\mathbb E}\left(\frac{1}{N} \sum_{j=1}^N \frac{Y_j}{\pi_j}I_j\right) \\[6pt]

&= \frac{1}{N} \sum_{j=1}^N \frac{Y_j}{\pi_j}\operatorname{\mathbb E}\left(I_j \right)\\&= \frac{1}{N}\sum_{j=1}^N \frac{Y_j}{\pi_j}\pi_j \\[6pt]

&= \frac{1}{N}\sum_{j=1}^N Y_i

\end{align}

The Hansen–Hurwitz (1943) is known to be inferior to the Horvitz–Thompson (1952) strategy, associated with a number of Inclusion Probabilities Proportional to Size (IPPS) sampling procedures.PRABHU-AJGAONKAR, S. G. "Comparison of the Horvitz–Thompson Strategy with the Hansen–Hurwitz Strategy." Survey Methodology (1987): 221. [https://www150.statcan.gc.ca/n1/en/pub/12-001-x/1987002/article/14609-eng.pdf?st=mgQEBG-Z (pdf)]

{{notelist}}

{{reflist}}

• [https://cran.r-project.org/web/packages/survey/ Survey Package Website for R]

{{DEFAULTSORT:Horvitz-Thompson estimator}}

Category:Sampling (statistics)

Category:Survey methodology

Category:Missing data