A note on regression estimation with unknown population size 5. ConclusionsA note on regression estimation with unknown population size 5. Conclusions

The regression estimator can be quite efficient if the auxiliary data that it uses are well correlated with the variable of interest. Furthermore, it requires that population totals corresponding to the auxiliary variables are available. In this article, we investigated the behavior of the regression estimator $\left({\stackrel{^}{Y}}_{\text{SREG}}\right)$  proposed by Singh and Raghunath (2011). This estimator uses estimated population count as a control total and the known population totals for the auxiliary variables. We compared it to the Generalized Regression estimator $\left({\stackrel{^}{Y}}_{\text{GREG}}\right),$  its optimal analogue $\left({\stackrel{^}{Y}}_{\text{OPT}}\right),$  and to an alternative estimator $\left({\stackrel{^}{Y}}_{\text{KREG}}\right)$  that uses the first-order inclusion probabilities and auxiliary data for which the population totals are known. As the optimal regression estimator requires the computation of second-order inclusion probabilities, we also included a pseudo-optimal estimator $\left({\stackrel{^}{Y}}_{\text{POPT}}\right)$  that does not require them. We investigated the properties of these estimators in terms of bias and efficiency via a simulation that included various sampling designs, and different values of the intercept in the model for a generated artificial population. We compared the results when the population size was known and unknown.

When the population size is known, the most efficient estimator is the optimal estimator ${\stackrel{^}{Y}}_{\mathrm{OPT}}$ . However, since this estimator can be unstable, the pseudo-optimal estimator ${\stackrel{^}{Y}}_{\text{POPT}}$ is a good alternative to it. This is in line with Rao (1994) who favoured the optimal estimator ${\stackrel{^}{Y}}_{\text{POPT}}$ over the Generalized Regression estimator ${\stackrel{^}{Y}}_{\text{GREG}}.$ The Singh and Raghunath (2011) proposition to use ${\stackrel{^}{Y}}_{\text{SREG}}$ is not viable, as it can be quite inefficient. When the population size is not known, the alternative regression estimator ${\stackrel{^}{Y}}_{\text{KREG}}$ is the best one to use.

Acknowledgements

The authors kindly acknowledge suggestions for improved readability provided by the Associate Editor and the referees.

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