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418 Survey samplingEstimators ̂q are chosen to have good design-based properties, such as(a) Design unbiasedness: E(̂q|Y )=Q, or(b) Design consistency: ̂q → Q as the sample size gets large (Brewer, 1979;Isaki and Fuller, 1982).It is natural to seek an estimate that is design-efficient, in the sense ofhaving minimal variance. However, it became clear that that kind of optimalityis not possible without an assumed model (Horvitz and Thompson, 1952;Godambe, 1955). Design-unbiasedness tends to be too stringent, and designconsistencyis a weak requirement (Firth and Bennett, 1998), leading to manychoices of estimates; in practice, choices are motivated by implicit models, asdiscussed further below. I now give some basic examples of the design-basedapproach.Example 1 (Estimate of a population mean from a simple randomsample): Suppose the target of inference is the population mean Q = Y =(y 1 + ···+ y N )/N and we have a simple random sample of size n, (y 1 ,...,y n ).The usual unbiased estimator is the sample mean ̂q = y =(y 1 + ···+ y n )/n,which has sampling variance V =(1− n/N)S 2 y,whereS 2 y is the populationvariance of Y . The estimated variance ̂v is obtained by replacing S 2 y in V byits sample estimate s 2 y. A 95% confidence interval for Y is y ± 1.96 √̂v.Example 2 (Design weighting): Suppose the target of inference is thepopulation total T =(y 1 + ···+ y N ), and we have a sample (y 1 ,...,y n )wherethe ith unit is selected with probability π i , i ∈{1,...,n}. FollowingHorvitzand Thompson (1952), an unbiased estimate of T is given bŷt HT =N∑w i y i I i ,i=1where w i =1/π i is the sampling weight for unit i, namely the inverse of theprobability of selection. Estimates of variance depend on the specifics of thedesign.Example 3 (Estimating a population mean from a stratified randomsample: For a stratified random sample with selection probability π j = n j /N jin stratum j, theHorvitz–ThompsonestimatorofthepopulationmeanQ =Y =(y 1 + ···+ y N )/N is the stratified mean, viz.y HT = 1 Nn J∑ ∑ jj=1 i=1N jn jy ij = y st =J∑P j y j ,j=1where P j = N j /N and y j is the sample mean in stratum j. Thecorresponding