Optimizing a mixed allocation
Section 5. Conclusion

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For the stratified designs, we have studied a trade-off allocation situated on a segment between the proportional allocation and the Neyman allocation. A theorem guarantees the existence of a flat region in the vicinity of the optimum and of a particular point that gives an optimal trade-off parameter according to a certain criterion. As part of a survey of businesses in the industry, simulations are conducted showing how the calculation can be done in practice and that the usual choice of a parameter of $1/2$ is not always the most effective. A comparison with other trade-off allocations, such as the classic Bankier allocation, shows that our weight dispersion goal produces more equal weighting at the expense of lower precision for the variable of interest on subdomains of the field. However, it illustrates the variability of the results obtained for the trade-off allocations according to the value of the parameter used; our method for determining parameter $\alpha$ remedies this problem often encountered in the study of these allocation families.

It is possible to replace the Neyman allocation in the trade-off with other specific ad hoc allocations. We postulate that the method remains applicable to obtain the same desirable properties. Different applications of this work were carried out at INSEE with other specific allocations. In the case of the annual Survey on the cost of labour and wage structure (ECMOSS), the specific allocation used for drawing the surveyed businesses is part of a two-stage design where, in each establishment sampled in the first stage, a sample of employees is drawn. The allocation used in the first stage is then optimized to obtain the lowest estimate variance on the estimated total net pay on the final sample of employees, given the dispersion of wages in each establishment. The allocation also integrates precision constraints on certain dissemination domains. Curves of the desired shape are still obtained and the trade-off allocation can be implemented.

Appendix A

Distance term in equation (2.2)

The choice of distance (i.e., a value for $p)$ in the second term of optimization program (1.2) is not crucial in the proposed context, because we will be able to rewrite the second term as follows where $C_p$ is a strictly positive constant dependent only on the choice of $p\text{:}$

$${\Vert n_{\alpha }-n_{\text{Neyman}}\Vert }_p\mathrm{<mo>=</mo>}\alpha C_p.(\text{A}.1)$$

Let us demonstrate this result. By definition (2.1), we have in each stratum $h\text{:}$

$$n_{\alpha \mathrm{,}h}\mathrm{<mo>=</mo>}\alpha n_{\text{prop}\mathrm{,}h}+\left(1-\alpha \right)n_{\text{Neyman}\mathrm{,}h}$$

and therefore,

$$n_{\alpha \mathrm{,}h}-n_{\text{Neyman}\mathrm{,}h}\mathrm{<mo>=</mo>}\alpha \left(n_{\text{prop}\mathrm{,}h}-n_{\text{Neyman}\mathrm{,}h}\right).$$

We therefore have for any choice of $p\text{:}$

$$\begin{array}{ll}{\Vert n_{\alpha }-n_{\text{Neyman}}\Vert }_p\hfill & \mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\left|n_{\alpha \mathrm{,}h}-n_{\text{Neyman}\mathrm{,}h}\right|}^p\right)}^{\frac{1}{p}}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\alpha }^p{\left|\left(n_{\text{prop}\mathrm{,}h}-n_{\text{Neyman}\mathrm{,}h}\right)\right|}^p\right)}^{\frac{1}{p}}\hfill \\ \hfill & \mathrm{<mo>=</mo>}\alpha {\left({\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{\left|n_{\text{prop}\mathrm{,}h}-n_{\text{Neyman}\mathrm{,}h}\right|}^p\right)}^{\frac{1}{p}}\hfill \\ \hfill & \mathrm{<mo>=</mo>}\alpha C_p.\hfill \end{array}$$

We will then integrate $C_p,$ a strictly positive constant, into $\lambda .$

Appendix B

Demonstration of Theorem 1

For a $\lambda \ge 0,$ the minimization function of program (2.2) is written as follows:

$$\begin{array}{ll}f\left(\alpha \right)\hfill & ={\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}{n}_{\alpha \mathrm{,}h}{\left(\frac{N_h}{n_{\alpha \mathrm{,}h}}-\frac{N}{n}\right)}^2+\lambda \alpha \hfill \\ \hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}\left(\frac{N_h^2}{n_{\alpha \mathrm{,}h}}-2\frac{N}{n}{N}_h+\frac{N^2}{n^2}{n}_{\alpha \mathrm{,}h}\right)+\lambda \alpha \hfill \\ \hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}\frac{N_h^2}{n_{\alpha \mathrm{,}h}}-2\frac{N^2}{n}+\frac{N^2}{n}+\lambda \alpha \hfill \\ \hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}\frac{N_h^2}{\alpha \frac{n{N}_h}{N}+\left(1-\alpha \right)n_{\text{Neyman}\mathrm{,}h}}-\frac{N^2}{n}+\lambda \alpha \hfill \\ \hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}\frac{N_h}{\alpha \frac{n}{N}+\left(1-\alpha \right)\frac{n_{\text{Neyman}\mathrm{,}h}}{N_h}}-\frac{N^2}{n}+\lambda \alpha .\hfill \end{array}$$

We now pose for all $h\le H\text{:}$

$${\beta }_h\mathrm{<mo>=</mo>}\frac{n}{N}-\frac{n_{\text{Neyman}\mathrm{,}h}}{N_h}.$$

For each stratum, the ${\beta }_h$ represent the difference between the uniform and Neyman sampling fractions. When ${\beta }_h<0,$ this means that the Neyman allocation is greater than the proportional allocation; the variable of interest is more dispersed in this stratum. Let us now derive $f\text{:}$

$$f^{\prime }\left(\alpha \right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}\frac{-N_h{\beta }_h}{{\left(\alpha {\beta }_h+\frac{n_{\text{Neyman}\mathrm{,}h}}{N_h}\right)}^2}+\lambda .(\text{B}.1)$$

We deduce from equation (B.1) that the derivative cancels out when:

$$\lambda \mathrm{<mo>=</mo>}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H}\frac{N_h{\beta }_h}{{\left(\alpha {\beta }_h+\frac{n_{\text{Neyman}\mathrm{,}h}}{N_h}\right)}^2}\mathrm{<mo>=</mo>\; <mo>:</mo>}g\left(\alpha \right).$$

Now function $g_h$ defined as follows:

$$g_h\text{:}\alpha \to \frac{N_h{\beta }_h}{{\left(\alpha {\beta }_h+\frac{n_{\text{Neyman}\mathrm{,}h}}{N_h}\right)}^2}$$

is decreasing. So:

• If ${\beta }_h$ is negative, the denominator decreases when $\alpha$ increases. In this case, its inverse increases with $\alpha .$ Therefore, we multiply by ${\beta }_h$ to obtain $g_h,$ which implies that $g_h$ is decreasing.
• If ${\beta }_h$ is positive, the denominator increases when $\alpha$ increases. By inverting and then multiplying by ${\beta }_h,$ we find that $g_h$ is decreasing.

So, if $\lambda \in \left[g\left(1\right),g\left(0\right)\right],$ we know that there is an ${\alpha }_0$ that cancels the derivative. As $f^{\prime }$ evolves inversely to $g,f^{\prime }$ is increasing and therefore ${\alpha }_0$ is the minimum of $f$ on $\left[0,1\right].$

Furthermore, as $g\left({\alpha }_0\right)=\lambda$ by definition, the decrease of $g$ implies that when $\lambda$ increases in $\left[g\left(1\right),g\left(0\right)\right],$ then ${\alpha }_0$ decreases. We therefore use the following lemma, admitted because it is relative to a classic property of the Neyman allocation:

Lemma 1. The function that at $\alpha$  associates the variance of the Horvitz-Thompson estimator of the variable of interest $X$  for the allocation $n_{\alpha ,h}$  is increasing.

We deduce that $V\left(\lambda \right)$ is decreasing over $S.$ Finally, by continuity, $V^{″}$ admits a maximum over $S.$

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