Model-assisted optimal allocation for planned domains using composite estimation 3. Optimizing the designModel-assisted optimal allocation for planned domains using composite estimation 3. Optimizing the design

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3.1 Optimal design for $F$

One way of measuring the performance of designs for small area estimation is with a linear combination of the anticipated MSEs of the small area mean and overall mean estimators. Following Longford (2006), but using anticipated MSEs instead of design-based MSEs, we define the criterion

$$\begin{array}{ll}F\hfill & ={\displaystyle \sum _{h\in U^1}}{N}_h^q{\text{AMSE}}_h+{\text{GN}}_{+}^{\left(q\right)}{E}_{\xi }{\text{var}}_p\left[{\widehat{\overline{Y}}}_r\right]\hfill \\ \hfill & ={\displaystyle \sum _{h\in U^1}}{N}_h^q{\text{AMSE}}_h+{\text{GN}}_{+}^{\left(q\right)}{E}_{\xi }{\text{var}}_p\left[{\displaystyle \sum _{h\in U^1}}{P}_h{\overline{y}}_{hr}\right]\hfill \\ \hfill & \approx {\displaystyle \sum _{h\in U^1}}{N}_h^q{\text{AMSE}}_h+{\text{GN}}_{+}^{\left(q\right)}{E}_{\xi }{\displaystyle \sum _{h\in U^1}}{P}_h^2{n}_h^{-1}{S}_{hw}^2\hfill \\ \hfill & ={\displaystyle \sum _{h\in U^1}}{N}_h^q{\sigma }_h^2\rho \left(1-\rho \right){\left[1+\left(n_h-1\right)\rho \right]}^{-1}+{\text{GN}}_{+}^{\left(q\right)}{\displaystyle \sum _{h\in U^1}}{\sigma }_h^2{P}_h^2{n}_h^{-1}\left(1-\rho \right)(3.1)\hfill \end{array}$$

where the weights $N_h^q$ reflect the inferential priorities for area $h,$ with $0\le q\le 2,$ and $N_{+}^{\left(q\right)}={\displaystyle {\sum }_{h\in U^1}}{N}_h^q,$ and ${\overline{y}}_{hr}$ is the grand mean estimator defined in Section 2. This objective reflects the fact that surveys have many stakeholders, some of whom will be only concerned with one specific small area, while others will place priority only on national estimators. Estimators for small regions are often considered a priority, particularly if they correspond to administrative or governmental jurisdictions, although smaller areas may be assigned less priority than larger regions. The quantity $G$ is a relative priority coefficient. Ignoring the goal of national estimation corresponds to $G=0$ and ignoring the goal of small area estimation corresponds to large values of $G,$ since when $G$ is very large the second component in (3.1) dominates. The factor $N_{+}^{\left(q\right)}$ is introduced to appropriately scale for the effect of the absolute sizes of $N_h^q$ and the number of areas on the relative priority $G.$ The criterion in (3.1) is algebraically similar to the criterion in Longford (2006). Here, however, we adopt the model-assisted approach which treats the design-based inference as the real goal of survey sampling, but employs models to choose between valid randomization-based alternatives (e.g., Chapter 6 of Särndal, Swensson and Wretman 1992).

Suppose that national estimation has no priority $\left(G=0\right),$ and the aim is to minimize (3.1) subject to a fixed total sampling cost function $C_f={\displaystyle {\sum }_h}{C}_h{n}_h,$ where $C_h$ is the unit cost of surveying a unit in stratum $h.$ The unique stationary point for this optimization is

$$\begin{array}{ll}{n}_{h\mathrm{,}\text{opt}\mathrm{.}}\hfill & =\frac{C_f\sqrt{N_h^q{\sigma }_h^2{C}_h^{-1}}}{{\displaystyle \sum _{h\in U^1}}\sqrt{N_h^q{\sigma }_h^2{C}_h}}+\frac{1-\rho }{\rho }\left(\frac{\overline{C}\sqrt{N_h^q{\sigma }_h^2{C}_h^{-1}}}{H^{-1}{\displaystyle \sum _{h\in U^1}}\sqrt{N_h^q{\sigma }_h^2{C}_h}}-1\right)\hfill \end{array}(3.2)$$

where $\overline{C}=H^{-1}{\displaystyle {\sum }_h}{C}_h.$ We will concentrate on the case when unit costs are equal across strata, so that the constraint becomes $n={\displaystyle {\sum }_h{n}_h}$ and (3.2) simplifies to

$$\begin{array}{ll}{n}_{h\mathrm{,}\text{opt}\mathrm{.}}\hfill & =\frac{n\sqrt{{\sigma }_h^2{N}_h^q}}{{\displaystyle \sum _{h\in U^1}}\sqrt{{\sigma }_h^2{N}_h^q}}+\frac{1-\rho }{\rho }\left(\frac{\sqrt{{\sigma }_h^2{N}_h^q}}{H^{-1}{\displaystyle \sum _{h\in U^1}}\sqrt{{\sigma }_h^2{N}_h^q}}-1\right)\hfill \end{array}.(3.3)$$

If there are other active constraints (e.g., minimum stratum sample sizes or maximum stratum MSEs), or if $G\text{>}0,$ then (3.2) and (3.3) do not apply and $F$ must be minimized numerically, for example by NLP as in Choudhry et al. (2012).

In practice it would almost always be appropriate to set $0\le q\le 2,$ with $q=0$ corresponding to all areas being equally important regardless of size, and $q=2$ giving much greater weight to larger areas. (The value of $q=2$ would lead to proportional allocation if direct estimators were used rather than composite - see for example Bankier 1988.) In many cases $q=1$ would be a sensible compromise. For example, this has been used to motivate power allocations (Bankier 1988) for master household samples in Vietnam and South Africa (Kalton, Brick and Lê 2005, paragraph 76, page 89).

The first term in (3.3) is the optimal allocation for the direct estimator and corresponds to power allocation (Bankier 1988). The second term will be positive for more populous areas (large $N_h)$ and negative for less populous areas. Therefore, the allocation optimal for model-assisted composite estimation has more dispersed subsample sizes $n_{h\mathrm{,}\text{opt}\mathrm{.}}$ than the allocation that is optimal for direct estimators.

To understand the properties of the optimal allocation when $G\text{>}0,$ and to provide a non-iterative method, Molefe (2011, Chapter 3) derived Taylor Series approximations to the optimal $n_h,$ based on small $\rho .$ However, the resulting approximation tended to result in very large negative and very large positive values of $n_{h\mathrm{,}\text{opt}\mathrm{.}}$ unless $\rho$ is very small. (In practice, these would be truncated to either 0 or the population size, respectively.) Mathematically, the issue is apparently that the optimal $n_h$ are quite nonlinear in $\rho$ at $\rho =0,$ so that Taylor Series approximations are only a good approximation in a small neighbourhood of $\rho =0.$ Taylor Series based on small values of a function of both $G$ and $\rho$ were also considered but had similar difficulties, and so these approaches are not further discussed here.

3.2 Power allocation

Power allocations (Bankier 1988) are defined by

$$n_h=\frac{n{N}_h^p}{{\displaystyle \sum _{h\in U^1}}{N}_h^p}(3.4)$$

for $h\in U^1,$ where $0\le p\le 1.$ A special case is the square root allocation when $p=1/2.$ The exponent $p$ is called the power of the allocation. Setting $p\mathrm{<mo>=</mo>\; 1}$ results in proportional allocation and $p=0$ results in equal allocation.

Bankier (1988) proposed choosing $p$ based on perceived relative priorities. However, this was based on direct estimators being used in each stratum. We are interested in the case where composite estimation is to be used, and the objective is to obtain a low value for $F$ in (3.1). We obtain numerically the value of $p$ which minimizes $F$ by one-dimensional optimization. We further consider imposing minimum stratum sample sizes, with $p$ re-optimized accordingly. (Alternatively, maximum stratum MSE constraints could be imposed.)

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