Stratified sampling designs and the stratified total estimator

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.