= 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 [[Statistics/SurveyFrame|frame]] methods.

Let the set of ''Q'' frames be indexed by ''q'', let ''A,,q,,'' mean the set of cases in frame ''q'', and let ''m,,i,,'' be the number of frames a sample member belongs to. Since

{{attachment:sum1.svg}}

it follows that the population total ''Y'' is given by:

{{attachment:sum2.svg}}

This leads to a multiplicity estimator for ''Y''. Let ''s,,q,,'' be a [[Statistics/SurveySampling|sample]] selected independently from frame ''q''.
 
{{attachment:sum3.svg}}

where ''w,,i,,^(q)^'' is a [[Statistics/SurveyWeights|weight]] accounting for e.g., design.

The variance of ''Ŷ'' is given by:

{{attachment:var1.svg}}

where ''N,,q,,'' and ''n,,q,,'' are the population and sample counts, respectively, for frame ''q''.

The author also notes that under [[Statistics/SimpleRandomSample|SRS]] the sample variance is estimated by:

{{attachment:var2.svg}}

where ''f,,q,,'' is the sampling fraction given by ''n,,q,,/N,,q,,''.

The author then evaluates this estimator using simulations. Importantly it is unbiased to misclassification of cases.



== 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.



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