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The cross-classified sampling design consists in drawing samples from atwo-dimension population, independently in each dimension. Such design iscommonly used in consumer price index surveys and has been recently applied todraw a sample of babies in the French ELFE survey, by crossing a sample ofmaternity units and a sample of days. We propose to derive a general theory ofestimation for this sampling design. We consider the Horvitz-Thompson estimatorfor a total, and show that the cross-classified design will usually result in aloss of efficiency as compared to the widespread two-stage design. We obtainthe asymptotic distribution of the Horvitz-Thompson estimator, and severalunbiased variance estimators. Facing the problem of possibly negative values,we propose simplified non-negative variance estimators and study their biasunder a super-population model. The proposed estimators are compared for totalsand ratios on simulated data. An application on real data from the ELFE surveyis also presented, and we make some recommendations. Supplementary materialsare available online.

Estimation under cross-classified sampling with application to a childhood survey