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| == Reading Notes == As [[AlternativeSurveySampleDesigns|Lohr]] points out, the claim that this estimator is unbiased assumes a multi frame design where the misclassification is between e.g. domains ''ab'' and ''bc''. The more likely case of misclassification between ''a'' and ''ab'' does in face lead to bias. |
A single frame multiplicity estimator for multiple frame surveys
A single frame multiplicity estimator for multiple frame surveys was written by Fulvia Mecatti in 2007. It was published in Survey Methodology (vol. 33).
The author proposes a multiplicity adjustment approach to multi frame methods.
Let the set of Q frames be indexed by q, let Aq mean the set of cases in frame q, and let mi be the number of frames a sample member belongs to. Since
it follows that the population total Y is given by:
This leads to a multiplicity estimator for Y. Let sq be a sample selected independently from frame q.
where wi(q) is a weight accounting for e.g., design.
The variance of Ŷ is given by:
where Nq and nq are the population and sample counts, respectively, for frame q.
The author also notes that under SRS the sample variance is estimated by:
where fq is the sampling fraction given by nq/Nq.
The author then evaluates this estimator using simulations. Importantly it is unbiased to misclassification of cases.
Reading Notes
As Lohr points out, the claim that this estimator is unbiased assumes a multi frame design where the misclassification is between e.g. domains ab and bc. The more likely case of misclassification between a and ab does in face lead to bias.