Two-stage sampling laws and the expansion estimator

Use the PSU partition and fixed values from Cluster sampling units, cluster totals, and one-stage SRSWOR. A first-stage design has a nonempty finite support ${\mathcal{S}}_1\subseteq 2^{[M]}$ and probability mass function $p_1$, positive on that support and summing to $1$. For each PSU $i$, fix a second-stage sampling design with nonempty finite support ${\mathcal{S}}_{2i}\subseteq 2^{U_i}$ and mass function $p_{2i}$, positive on that support and summing to $1$. Given the selected PSU set $S_1$, the second-stage samples $(S_i{)}_{i\in S_1}$ are independent, with each $S_i$ drawn from $p_{2i}$, independently of the identities of other selected PSUs. The joint law is $P_d(S_1,(S_i{)}_{i\in S_1})=p_1(S_1){\prod }_{i\in S_1}{p}_{2i}(S_i)\mathrm{.}$ It is a probability mass function: summing over the second-stage supports first gives $p_1(S_1)$ for each $S_1$, and summing these values gives $1$. For a function $g$ of the joint outcome, write $E_1[h]={\sum }_{S_1\in {\mathcal{S}}_1}{p}_1(S_1)h(S_1),E_2[g\mid S_1=s]={\sum }_{(S_i{)}_{i\in s}}({\prod }_{i\in s}{p}_{2i}(S_i))g(s,(S_i{)}_{i\in s})\mathrm{.}$ For a first-stage function $h$, define ${\mathrm{Var}}_1(h)=E_1[(h-E_{1}h{)}^2]$. For a function $g$ of the full second-stage outcome and fixed $S_1=s$, define ${\mathrm{Var}}_2(g\mid S_1=s)=E_2[(g-E_2[g\mid S_1=s]{)}^2\mid S_1=s]\mathrm{.}$ The symbol ${\mathrm{Var}}_{2i}$ denotes centered-square variance under $p_{2i}$, and ${\mathrm{Var}}_d$ denotes centered-square variance under the joint law. These use the variance definition in Moments, variance, and covariance on a probability space.

For the first stage, put ${\pi }_i^{(1)}=P_d(i\in S_1),{\pi }_{ir}^{(1)}=P_d(i,r\in S_1)\mathrm{.}$ For the second-stage design in PSU $i$, put ${\pi }_{j\mid i}^{(2)}=P_{2i}(j\in S_i),{\pi }_{j\mathrm{\ell }\mid i}^{(2)}=P_{2i}(j,\mathrm{\ell }\in S_i)\mathrm{.}$ The stage-specific covariance differences are ${\mathrm{\Delta }}_{ir}^{(1)}={\pi }_{ir}^{(1)}-{\pi }_i^{(1)}{\pi }_r^{(1)},{\mathrm{\Delta }}_{j\mathrm{\ell }\mid i}^{(2)}={\pi }_{j\mathrm{\ell }\mid i}^{(2)}-{\pi }_{j\mid i}^{(2)}{\pi }_{\mathrm{\ell }\mid i}^{(2)}\mathrm{.}$

If ${\pi }_i^{(1)}>0$ for every PSU and ${\pi }_{j\mid i}^{(2)}>0$ for every SSU, the within-PSU expansion total and the two-stage expansion total are defined by ${\widehat{T}}_i={\sum }_{j\in S_i}\frac{y_{ij}}{{\pi }_{j\mid i}^{(2)}},\widehat{T}={\sum }_{i\in S_1}\frac{{\widehat{T}}_i}{{\pi }_i^{(1)}}\mathrm{.}$ No positivity assumption on the pair-inclusion probabilities is part of this definition.

The product in the joint law is an explicit conditional-independence assumption across selected PSUs. A design with dependent second-stage samples across PSUs, or a second-stage law depending jointly on the first-stage set, has a different variance formula. The expansion estimator's divisions use only first-order inclusion probabilities.