Small area estimation methods under cut-off sampling
Section 10. Conclusions

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Cut-off sampling is frequently used in business surveys, when drawing a representative sample from the whole population entails a cost that does not really compensate the subsequent gain in accuracy. On the other hand, in some surveys, part of the target population may not be actually available for sampling; that is, there may be population sectors that cannot be represented in the sample. These situations appear more often than desired, providing biased direct estimates as we have seen along this work.

We have studied the theoretical design properties of basic direct, calibration and model-based estimators under cut-off sampling in small areas. Our results show that EBLUP for a linear parameter, similarly as calibration estimators, reduce considerably the bias due to cut-off sampling if the models for the included and excluded individuals are reasonably similar. In terms of MSE, EBLUP performs significantly better than calibration estimators, specially for domains with small sample size.

In our simulation studies and in the application, we compared the proposed methods by assuming that the model is the same for all units in the population (included or excluded). The model assumption could be arguable because there is no way of checking the model for the excluded units. In the case that estimation for the overall domain (and not only for $U_{iI})$ is required as is the case in this work, one will need to rely on subjective prior information concerning the validity of the assumed model for the excluded units. In any case, estimates can be considered just as indicatives of what could be the true values in the case that the same model holds for all the domain units. In fact, the case of different models for included and excluded units was also analyzed in simulations. In this case, model-based estimators remained to be the most efficient, with not much larger bias than that of calibration estimators.

MSEs of calibration and model-based estimators are obtained under the model. Design MSEs are preferred by National Statistical Institutes because they do not assume that a model is correct and therefore account for model failures. However, finding design-unbiased estimators for the design MSE under cut-off sampling encounters the same problems as finding design-unbiased estimators of the target domain indicators $H_i.$ We plan to use the ideas of Strzalkowska-Kominiak and Molina (2019), based on borrowing strength from the other domains also for estimating the design MSE in a given domain, to find design MSE estimators with reduced cut-off sampling bias.

Finally, we have considered that the domains act as sampling strata and cut-off sampling is applied within each domain. Considering that the strata are different from the domains (typically cutting-across the domains) and applying cut-off sampling within each strata yields random domain sample sizes. Small area estimation is seldom studied under this case in the literature. Nevertheless, putting together the subsamples from the different strata corresponding to the same domain we get a sample from each domain. Inference could then be done conditionally on the observed domain sample sizes Rao (1985), which would reduce to the same problem considered here.

Acknowledgements

The work of M. Guadarrama and I. Molina is supported by the Spanish Ministerio de Economía y Competitividad, grants MTM2015-69638-R (MINECO/FEDER, UE) and MTM2015-72907-EXP.

Appendix

Estimates of total sales by provinces

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