Neyman Allocation Derivation

PUBHBIO 7225

Stratified sampling: \(\displaystyle V(\bar{y}_{str})= \sum_{h=1}^H \left(\frac{N_h}{N}\right)^2 \left(1-\frac{n_h}{N_h}\right)\frac{S_h^2}{n_h}\)

Goal: Find \(n_h, h=1,\dots,H\) such that:

  1. \(\displaystyle \sum_{h=1}^H n_h = n\)

  2. \(V(\bar{y}_{str})\) is minimized

Solution: Use Lagrange multipliers

Minimize:
\(\displaystyle V(\bar{y}_{str}) = f(\underline{S}_h, \underline{n}_h, n) = \sum_{h=1}^H \left(\frac{N_h}{N}\right)^2 \left(1-\frac{n_h}{N_h}\right)\frac{S_h^2}{n_h} = \sum_{h=1}^H \left(\frac{N_h}{N}\right)^2 \frac{S_h^2}{n_h} - \sum_{h=1}^H \left(\frac{N_h}{N}\right)^2 \frac{S_h^2}{N_h}\)

Subject to constraint:
\(\displaystyle g(\underline{S}_h, \underline{n}_h, n) = \sum_{h=1}^H n_h = n \rightarrow \sum_{h=1}^H n_h - n = 0\)

Using Lagrange multipliers:
\(\displaystyle \mathcal{L}(\underline{n}_h, \lambda) = \sum_{h=1}^H \left(\frac{N_h}{N}\right)^2 \frac{S_h^2}{n_h} - \sum_{h=1}^H \left(\frac{N_h}{N}\right)^2 \frac{S_h^2}{N_h} + \lambda \left( \sum_{h=1}^H n_h - n \right)\)

\(\displaystyle \frac{\partial \mathcal{L}}{\partial n_h} = - \left(\frac{N_h}{N}\right)^2 \frac{S_h^2}{n_h^2} + \lambda = 0\)

\(\displaystyle \frac{\partial \mathcal{L}}{\partial \lambda} = \sum_{h=1}^H n_h - n = 0\)

\[\begin{aligned} \text{Thus, from } \frac{\partial \mathcal{L}}{\partial n_h} \rightarrow n_h^2 &= \left(\frac{N_h}{N}\right)^2 \frac{S_h^2}{\lambda}& \\ n_h &= \frac{N_h}{N} \frac{S_h}{\sqrt{\lambda}}\\ \text{From } \frac{\partial \mathcal{L}}{\partial \lambda} \rightarrow \sum_{h=1}^H n_h &= \sum_{h=1}^H \frac{N_h}{N} \frac{S_h}{\sqrt{\lambda}} = n \\ \sqrt{\lambda} &= \sum_{h=1}^H \frac{N_h}{N} \frac{S_h}{n} \\ \text{Plug into above: }n_h &= \frac{N_h}{N} \frac{S_h}{\sqrt{\lambda}} = \frac{N_h}{N} \frac{S_h}{\sum_{h=1}^H \frac{N_h}{N} \frac{S_h}{n}} = n \left(\frac{N_h S_h}{\sum_{l=1}^H N_l S_l} \right) \end{aligned}\]