Horvitz–Thompson estimator

<h2 id="the-method">The method</h2>
<p>Formally, let 
  
    
      
        
          Y
          
            i
          
        
        ,
        i
        =
        1
        ,
        2
        ,
        …
        ,
        n
      
    
    {\displaystyle Y_{i},i=1,2,\ldots ,n}
  
 be an <a href="/facts/Independence_(probability_theory)/NUzQtnUL">independent</a> sample from 
  
    
      
        n
      
    
    {\displaystyle n}
  
 of 
  
    
      
        N
        ≥
        n
      
    
    {\displaystyle N\geq n}
  
 distinct <a href="/facts/Stratum_(statistics)/CvAUCegT">strata</a> with an overall mean 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
. Suppose further that 
  
    
      
        
          π
          
            i
          
        
      
    
    {\displaystyle \pi _{i}}
  
 is the <a href="/facts/Inclusion_probability/Vob3umvl">inclusion probability</a> 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:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>: 51 
</p>

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

<p>and the Horvitz–Thompson estimate of the mean is given by:
</p>

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}}}.}

<p>In a <a href="/facts/Bayesian_probability/Cvn7WWfG">Bayesian</a> 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 <a href="/facts/Bootstrapping_(statistics)/zCHuBeIz">bootstrap</a> <a href="/facts/Resampling_(statistics)/yq6LXqnX">resampling</a> estimate of the mean. It can also be viewed as a special case of multiple <a href="/facts/Imputation_(statistics)/UXBwWUKJ">imputation</a> approaches.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p><p>For <a href="/facts/Statistical_benchmarking/z8Pk7HQ4">post-stratified</a> study designs, estimation of 
  
    
      
        π
      
    
    {\displaystyle \pi }
  
 and 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 are done in distinct steps. In such cases, computating the variance of 
  
    
      
        
          
            
              
                μ
                ^
              
            
          
          
            H
            T
          
        
      
    
    {\displaystyle {\hat {\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.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> The "survey" package for <a href="/facts/R_(programming_language)/LSrkr8K8">R</a> conducts analyses for post-stratified data using the Horvitz–Thompson estimator.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
</p>
<h2 id="proof-of-horvitzthompson-unbiased-estimation-of-the-mean">Proof of Horvitz–Thompson unbiased estimation of the mean</h2>
<p>For this proof it will be useful to represent the sample as a random subset 
  
    
      
        S
        ⊆
        {
        1
        ,
        …
        ,
        N
        }
      
    
    {\displaystyle S\subseteq \{1,\ldots ,N\}}
  
 of size 
  
    
      
        n
      
    
    {\displaystyle n}
  
. We can then define indicator random variables 
  
    
      
        
          I
          
            j
          
        
        =
        
          1
        
        [
        j
        ∈
        S
        ]
      
    
    {\displaystyle I_{j}=\mathbf {1} [j\in S]}
  
 representing whether  for each 
  
    
      
        j
      
    
    {\displaystyle j}
  
 in 
  
    
      
        {
        1
        ,
        …
        ,
        N
        }
      
    
    {\displaystyle \{1,\ldots ,N\}}
  
 whether it is present in the sample. Note that for any observation in the sample, the expectation is the definition of the inclusion probability:

π
          
            i
          
        
        =
        
          
            E
          
        
        ⁡
        
          (
          
            I
            
              i
            
          
          )
        
        =
        Pr
        (
        i
        ∈
        S
        )
      
    
    {\displaystyle \pi _{i}=\operatorname {\mathbb {E} } \left(I_{i}\right)=\Pr(i\in S)}
  
.
<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>
</p><p>
Taking the expectation of the estimator we can prove it is unbiased as follows:
</p>

E
                  
                
                ⁡
                
                  (
                  
                    
                      
                        
                          μ
                          ^
                        
                      
                    
                    
                      
                        H
                        T
                      
                    
                  
                  )
                
              
              
                
                =
                
                  
                    E
                  
                
                ⁡
                
                  (
                  
                    
                      
                        1
                        N
                      
                    
                    
                      ∑
                      
                        i
                        ∈
                        S
                      
                    
                    
                      
                        
                          Y
                          
                            i
                          
                        
                        
                          π
                          
                            i
                          
                        
                      
                    
                  
                  )
                
              
            
            
              
              
                
                =
                
                  
                    E
                  
                
                ⁡
                
                  (
                  
                    
                      
                        1
                        N
                      
                    
                    
                      ∑
                      
                        j
                        =
                        1
                      
                      
                        N
                      
                    
                    
                      
                        
                          Y
                          
                            j
                          
                        
                        
                          π
                          
                            j
                          
                        
                      
                    
                    
                      I
                      
                        j
                      
                    
                  
                  )
                
              
            
            
              
              
                
                =
                
                  
                    1
                    N
                  
                
                
                  ∑
                  
                    j
                    =
                    1
                  
                  
                    N
                  
                
                
                  
                    
                      Y
                      
                        j
                      
                    
                    
                      π
                      
                        j
                      
                    
                  
                
                
                  
                    E
                  
                
                ⁡
                
                  (
                  
                    I
                    
                      j
                    
                  
                  )
                
              
            
            
              
              
                
                =
                
                  
                    1
                    N
                  
                
                
                  ∑
                  
                    j
                    =
                    1
                  
                  
                    N
                  
                
                
                  
                    
                      Y
                      
                        j
                      
                    
                    
                      π
                      
                        j
                      
                    
                  
                
                
                  π
                  
                    j
                  
                
              
            
            
              
              
                
                =
                
                  
                    1
                    N
                  
                
                
                  ∑
                  
                    j
                    =
                    1
                  
                  
                    N
                  
                
                
                  Y
                  
                    i
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\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{aligned}}}

<p>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.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>
</p>
<h2 id="notes">Notes</h2>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://cran.r-project.org/web/packages/survey/">Survey Package Website for R</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>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 <a href="/wiki/Journal_of_the_American_Statistical_Association" target="_blank">/wiki/Journal_of_the_American_Statistical_Association</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>William G. Cochran (1977), Sampling Techniques, 3rd Edition, Wiley. ISBN 0-471-16240-X <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Särndal, Carl-Erik; Swensson, Bengt; Wretman, Jan Hȧkan (1992). Model Assisted Survey Sampling. ISBN 9780387975283. <a href="9780387975283" target="_blank">9780387975283</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Roderick J.A. Little, Donald B. Rubin (2002) Statistical Analysis With Missing Data, 2nd ed., Wiley. ISBN 0-471-18386-5 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Quatember, A. (2014). "The Finite Population Bootstrap - from the Maximum Likelihood to the Horvitz-Thompson Approach". Austrian Journal of Statistics. 43 (2): 93–102. doi:10.17713/ajs.v43i2.10. <a href="https://doi.org/10.17713%2Fajs.v43i2.10" target="_blank">https://doi.org/10.17713%2Fajs.v43i2.10</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>"CRAN - Package survey". 19 July 2021. <a href="https://cran.r-project.org/web/packages/survey/" target="_blank">https://cran.r-project.org/web/packages/survey/</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Technically, the indexing scheme in the proof is different from the indexing in the description of the estimator. In the proof, 
  
    
      
        
          Y
          
            j
          
        
      
    
    {\displaystyle Y_{j}}
  
 is the 
  
    
      
        j
      
    
    {\displaystyle j}
  
th value in a global ordering out of 
  
    
      
        N
      
    
    {\displaystyle N}
  
 strata. In the description, 
  
    
      
        
          Y
          
            i
          
        
      
    
    {\displaystyle Y_{i}}
  
 is the 
  
    
      
        i
      
    
    {\displaystyle i}
  
th value in the sample, out of 
  
    
      
        n
      
    
    {\displaystyle n}
  
. To unify these two, we could explicitly define a function mapping sample-indices to global indices. <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>PRABHU-AJGAONKAR, S. G. "Comparison of the Horvitz–Thompson Strategy with the Hansen–Hurwitz Strategy." Survey Methodology (1987): 221. (pdf) <a href="https://www150.statcan.gc.ca/n1/en/pub/12-001-x/1987002/article/14609-eng.pdf?st=mgQEBG-Z" target="_blank">https://www150.statcan.gc.ca/n1/en/pub/12-001-x/1987002/article/14609-eng.pdf?st=mgQEBG-Z</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
</ol>

Horvitz–Thompson estimator open-in-new

Horvitz–Thompson estimator