Model-assisted optimal allocation for planned domains using composite estimation 5. ConclusionsModel-assisted optimal allocation for planned domains using composite estimation 5. Conclusions

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The anticipated MSE is a sensible objective criterion for sample design, because the particular sample which will be selected is not available in advance of the survey. Hence a criterion which averages over all possible samples is appropriate. Särndal et al. (1992, Chapter 12) base their optimal designs on the anticipated variance, which similarly averages over both model realizations and sample selection, although they consider only approximately design-unbiased estimators.

When both strata composite estimators and overall estimators are a priority, it makes sense to optimise an objective criterion which is a linear combination of the relevant anticipated MSEs. Allocations which are optimal in this sense give lower values of the objective function than either proportional or equal allocation. An optimal power allocation, $n_h\propto N_h^p$ where $p$ is obtained numerically to minimize the objective function, is simpler and avoids the possibility of negative sample sizes which need to be truncated. Under conditions, it is very nearly as efficient as the optimal allocation. When there is no priority on national estimation $\left(G=0\right),$ the optimal exponent turns out to be close to $p=q/2,$ where $q$ is the exponent applied to stratum population sizes in the objective criterion. This removes the need to perform an optimization. Thus, we recommend an objective criterion very similar to that of Longford (2006), but we suggest a simple power allocation with $p=q/2$ when $G=0,$ rather than the optimal allocation for $F.$ This extends the the domain of application of power allocation to surveys using stratum composite estimators.

Rather than just relying on the overall objective criterion to appropriately balance resources across strata, it may often be desirable to also impose minimum stratum sample sizes or maximum stratum RRMSEs. These were successfully implemented using NLP. In the Swiss canton example in Section 4, an upper limit of 8% for stratum RRMSEs significantly reduced the highest RRMSE with little loss in the objective criterion. More complex constraints, for example on cross-strata domains or for multiple variables of interest, could also be implemented using NLP.

Acknowledgements

The authors wish to thank Professors Raymond Chambers and David Steel for their helpful suggestions on this paper.

Appendix

Derivation of (3.2)

The steps of this derivation are similar to Longford (2006) although $F$ is defined differently and unequal costs are allowed. A stationary point of (3.1) subject to $C_f={\displaystyle \sum C_h{n}_h}$ is given by

$$\begin{array}{ll}0\hfill & =\frac{\partial F}{\partial n_h}+\lambda C_h\hfill \\ \hfill & =-N_h^q{\sigma }_h^2{\rho }^2\left(1-\rho \right){\left(1+\left(n_h-1\right)\rho \right)}^{-2}+\lambda C_h.\hfill \end{array}$$

Writing $\gamma =\lambda {\rho }^{-2}{\left(1-\rho \right)}^{-1}$ and rearranging gives

$$\begin{array}{ll}\hfill {\left(1+\left(n_h-1\right)\rho \right)}^{-2}& =\gamma C_h{N}_h^{-q}{\sigma }_h^{-2}\hfill \\ \hfill 1+\left(n_h-1\right)\rho & ={\gamma }^{-1/2}\sqrt{C_h^{-1}{N}_h^q{\sigma }_h^2}\hfill \\ \hfill n_h& ={\gamma }^{-1/2}{\rho }^{-1}\sqrt{C_h^{-1}{N}_h^q{\sigma }_h^2}-\frac{1-\rho }{\rho }.(\text{A}\text{.1})\hfill \end{array}$$

Substituting into the constraint $C_f={\displaystyle \sum C_h{n}_h}$ and solving for $\gamma$ gives

$${\gamma }^{-1/2}=\frac{C_f\rho +\left(1-\rho \right)H\overline{C}}{{\displaystyle \sum _h}\sqrt{{\sigma }_h^2{N}_h^q{C}_h^{-1}}}$$

where $\overline{C}=H^{-1}{\displaystyle {\sum }_h{C}_h}.$ Substituting back into (A.1) and rearranging gives the result.

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