Stratification and allocation

Let $(U,y)$ be a finite population as in Finite populations and sampling designs, and let $U_1,\dots ,U_H$ be a finite partition of $U$ into nonempty strata. Write $N_h=\mathrm{\mid }{U}_h\mathrm{\mid }$ and $N={\sum }_{h=1}^H{N}_h$. Choose integers $1\le n_h\le N_h$. In stratum $h$, draw $R_h$ by SRSWOR of size $n_h$ as in Simple random sampling without replacement, independently across strata. Thus the joint probability of a tuple $(r_1,\dots ,r_H)$ is ${\prod }_{h=1}^H{\left(\genfrac{}{}{0px}{}{N_h}{n_h}\right)}^{-1},$ and its sample in $U$ is $R={\bigcup }_{h=1}^H{r}_h$. This is the stratified SRSWOR design. The union map is one-to-one because the strata are disjoint; the support has ${\prod }_h\left(\genfrac{}{}{0px}{}{N_h}{n_h}\right)$ elements, so the displayed mass function sums to one.

The fixed mean in stratum $h$ is ${\bar{y}}_h=\frac{1}{N_h}{\sum }_{i\in U_h}{y}_i\mathrm{.}$ For a noncertainty stratum ($n_h<N_h$), necessarily $N_h>1$, and its finite- population variance is $S_h^2=\frac{1}{N_h-1}{\sum }_{i\in U_h}(y_i-{\bar{y}}_h{)}^2\mathrm{.}$ For every noncertainty stratum write $S_h=\sqrt{S_h^2}$ for its nonnegative population standard deviation. For a realized sample, set ${\bar{y}}_{h,R}=\frac{1}{n_h}{\sum }_{i\in R_h}{y}_i,{\widehat{T}}_{st}={\sum }_{h=1}^H{N}_h{\bar{y}}_{h,R}\mathrm{.}$ The quantity ${\widehat{T}}_{st}$ is the stratified total estimator. When $n_h=N_h$, the stratum is a certainty stratum and its contribution is the fixed total $N_h{\bar{y}}_h$; no $S_h^2$ is needed for that term. The population values $y_i$ remain fixed: only the sample selections are random.

Let $(U,y)$ be a finite population with a partition and stratified SRSWOR design as in Stratified sampling designs and the stratified total estimator. Under the design probability $P_d$, the stratified total estimator satisfies $E_d[{\widehat{T}}_{st}]={\sum }_{i\in U}{y}_i\mathrm{.}$ Its design variance is ${\mathrm{Var}}_d({\widehat{T}}_{st})={\sum }_{h:n_h<N_h}{N}_h^2(1-\frac{n_h}{N_h})\frac{S_h^2}{n_h}\mathrm{.}$ Thus certainty strata contribute zero and require no within-stratum variance term.

Use the notation of Stratified sampling designs and the stratified total estimator and Unbiasedness and variance of the stratified total estimator. Let $J$ be a nonempty set of noncertainty strata whose allocations vary, and suppose $S_h>0$ for each $h\in J$, where $S_h:=\sqrt{S_h^2}$ is the positive population standard deviation. Hold all other stratum allocations fixed. For real $x_h>0$ with ${\sum }_{h\in J}{x}_h=n^{\star }>0$, consider the uncapped continuous relaxation of the variance expression $V_0+{\sum }_{h\in J}{N}_h^2(1-\frac{x_h}{N_h})\frac{S_h^2}{x_h},$ where $V_0$ collects the fixed strata's variance terms. Its unique minimum over this positive simplex is attained at $x_h=n^{\star }\frac{N_h{S}_h}{{\sum }_{k\in J}{N}_k{S}_k}(h\in J)\mathrm{.}$ This is the continuous Neyman allocation. The relaxation imposes no integer, capacity, lower-bound, or minimum-sample constraints; if the displayed allocation violates a design constraint, it is not itself an admissible stratified SRSWOR design.

Use the notation of Stratified sampling designs and the stratified total estimator and Unbiasedness and variance of the stratified total estimator. Let $J$ be a nonempty set of variable noncertainty strata with $S_h>0$, and hold all other allocations fixed. Assign each $h\in J$ a positive per-unit cost $c_h$. For a fixed positive residual cost budget $C^{\star }$, the uncapped continuous relaxation of the stratified variance expression, minimized over positive real allocations $x_h>0$ for every $h\in J$ subject to ${\sum }_{h\in J}{c}_h{x}_h=C^{\star }$, has the unique minimizer $x_h=C^{\star }\frac{N_h{S}_h\mathrm{/}\sqrt{c_h}}{{\sum }_{k\in J}{N}_k{S}_k\sqrt{c_k}}(h\in J)\mathrm{.}$ Thus $x_h\propto N_h{S}_h\mathrm{/}\sqrt{c_h}$. As in the fixed-size result, this continuous formula does not enforce integer, capacity, lower-bound, or minimum-sample constraints.