Optimizing a mixed allocation
Section 2. Optimization program

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The program (1.2) is difficult to resolve and analyze, which is why we will simply look for a solution on a segment between the proportional allocation and a given specific allocation, the Neyman allocation, the one most frequently used. Often, the choice of an $\alpha =1/2$ is a good trade-off. For example, this is proposed in Chiodini et al. (2010a), or in some INSEE business survey designs.

This method combines the benefits of both methods at a low cost. However, we can question the arbitrary choice of the factor $1/2.$ In this paragraph, we will present a method based on a minimization program involving the dispersion of weights as well as the distance to the Neyman allocation to choose a parameter $\alpha$ such as the “optimal” mixed allocation between proportional allocation and the Neyman allocation:

$$n_{\alpha }^{\text{opt}}=\alpha n_{\text{prop}}+\left(1-\alpha \right)n_{\text{Neyman}}.(2.1)$$

We situate ourselves here in the context of stratified sampling with $H$ strata, ignoring the influence of non-response. This could be integrated by considering anticipated response rates or a second Poisson phase, but this unnecessarily complicates the form of the results. We will focus here on a set of allocations $\left(n_{\alpha }\right)$ that go through a segment between the proportional allocation $\left(n_{\text{prop}}\right)$ and the Neyman allocation $\left(n_{\text{Neyman}}\right),$ as indicated in equation (2.1). We therefore limit ourselves to achieving the following minimization program, a simplified form of that in equation (1.2):

$$\underset{\alpha \in \left[0,1\right]}{\min }{\displaystyle \sum _{h=1}^H{n}_{\alpha ,h}}{\left(\frac{N_h}{n_{\alpha ,h}}-\frac{N}{n}\right)}^2+\lambda \alpha .(2.2)$$

The term on the right corresponds to the distance between the desired allocation and the Neyman allocation, up to a constant, integrated in $\lambda \text{:}$ this result is shown in Appendix A.

This minimization program depends on the chosen constant $\lambda \ge 0.$ It is clear that when $\lambda$ is large enough, the term of distance becomes preponderant and we obtain $\alpha =0$ and therefore $n_{\alpha }=n_{\text{Neyman}}.$ Similarly, when $\lambda$ tends toward 0, the factor representing the dispersion of weights becomes preponderant and the allocation tends toward the proportional allocation.

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