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 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

it follows that the population total Y is given by:

This leads to a multiplicity estimator for Y. Let s_q be a sample selected independently from frame q.

where w_i^(q) is a weight accounting for e.g., design.

The variance of Ŷ is given by:

where N_q and n_q are the population and sample counts, respectively, for frame q.

The author also notes that under SRS the sample variance is estimated by:

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 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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