2DkcTXceO

R.J. Little 419estimate of variance iŝv st =J∑j=1(1 − n jN j) s2jn j,where s 2 j is the sample variance of Y in stratum j. Acorresponding95%confidence interval for Y is y st ± 1.96 √̂v st .Example 4 (Estimating a population mean from a PPS sample):In applications such as establishment surveys or auditing, it is common tohave measure of size X available for all units in the population. Since largeunits often contribute more to summaries of interest, it is efficient to samplethem with higher probability. In particular, for probability proportional tosize (PPS) sampling, unit i with size X = x i is sampled with probabilitycx i ,wherec is chosen to yield the desired sample size; units that come inwith certainty are sampled and removed from the pool. Simple methods ofimplementation are available from lists of population units, with cumulatedranges of size. The Horvitz–Thompson estimator̂t HT = cN∑i=1y ix iI iis the standard estimator of the population total in this setting.The Horvitz–Thompson estimator often works well in the context of PPSsampling, but it is dangerous to apply it to all situations. A useful guide is toask when it yields sensible predictions of nonsampled values from a modelingperspective. A model corresponding to the HT estimator is the HT modely iiid∼N(βxi ,σ 2 x 2 i ), (37.1)where N (µ, τ 2 )denotestheNormaldistributionwithmeanµ and varianceτ 2 .Thisleadstopredictionŝβx i ,wherêβ = n −1N ∑i=1y ix iI i ,so ̂t HT = ̂β(x 1 + ···+ x N )istheresultofusingthismodeltopredictthesampled and nonsampled values. If the HT model makes very little sense,the HT estimator and associated estimates of variance can perform poorly.The famous elephant example of Basu (1971) provides an extreme and comicillustration.Models like the HT estimator often motivate the choice of estimator inthe design-based approach. Another, more modern use of models is in modelassistedinference, where predictions from a model are adjusted to protect