In Lohr's Sampling and Design Textbook, for cluster sampling, it says that the model based variance of $\hat{T}-T$ is:
The textbook says that this equation is minimized for $\hat{T}_r$ when $m_i$ is proportional to $M_i$ using Lagrange multipiers, with constraint $\sum{m_i} = L$
It doesn't really give any further explanations other than that, and I don't fully understand this statement.
My attempt:
$g(m_1,...,m_i, L) = L - \sum_{i\in S}{m_i}$
$F(b_{ij},m_i,M_i,M_0,\lambda) = V_{M1}[\hat{T}-T] - \lambda g$
I'm not sure my input variables for function F $b_{ij},m_i,M_i,M_0,\lambda$ are correct though.