Optimal cost-constrained stratified sampling allocation ($x_i^{\ast }=\frac{b}{A}\sqrt{v_i/a_i},A={\sum }_i\sqrt{a_i{v}_i}$)

Suppose the variance is ${\sum }_i{v}_i/x_i$ and the sampling cost is ${\sum }_i{a}_i{x}_i\le b$, with positive costs, variance coefficients and budget. The Cauchy-Schwarz inequality gives $A^2\le ({\sum }_i{v}_i/x_i)({\sum }_i{a}_i{x}_i)\le b{\sum }_i{v}_i/x_i$. Equality forces $x_i$ proportional to $\sqrt{v_i/a_i}$ and a tight budget, proving the displayed unique optimum. The minimal variance is $A^2/b$. For the optimization Lagrangian $f-\lambda ({\sum }_i{a}_i{x}_i-b)$, the multiplier and budget derivative both equal $-A^2/b^2$. Therefore a small resource change $\delta b$ changes the optimum by $\lambda \delta b+O((\delta b{)}^2)$. This is a continuous allocation; integer sample-size restrictions define a separate optimization problem.