Finite populations and sampling designs

Let $N\ge 1$ be an integer, let $U=\{1,\dots ,N\}$, and let $y:U\to \mathbb{R}$ be a fixed function. The pair $(U,y)$ is a finite population; its total and mean are $T(y):={\sum }_{i\in U}{y}_i,{\bar{y}}_U:=T(y)\mathrm{/}N\mathrm{.}$

A sampling design on $U$ is a nonempty finite set ${\mathcal{S}}_d\subseteq \mathcal{P}(U)$ and a function $p_d:{\mathcal{S}}_d\to (0,1]$ with ${\sum }_{s\in {\mathcal{S}}_d}{p}_d(s)=1$. Here ${\mathcal{S}}_d$ is the support of the design law: zero-weight subsets are omitted. Thus $({\mathcal{S}}_d,p_d)$ is a finite probability space in the sense of Finite probability spaces, outcome weights, events, and event probabilities. For $A\subseteq {\mathcal{S}}_d$ and $g:{\mathcal{S}}_d\to \mathbb{R}$, put $P_d(A):={\sum }_{s\in A}{p}_d(s),E_d[g]:={\sum }_{s\in {\mathcal{S}}_d}{p}_d(s)g(s)\mathrm{.}$ The random sample is the subset $S\in {\mathcal{S}}_d$; probability and expectation here average over the design while holding every $y_i$ fixed.

For $i,j\in U$, define the inclusion indicator $I_i(s):=\{\begin{array}{ll}1,& i\in s,\\ 0,& i\mathrm{\notin }s,\end{array}$ the first-order inclusion probability ${\pi }_i:=P_d(\{s\in {\mathcal{S}}_d:i\in s\}),$ and the second-order inclusion probability ${\pi }_{ij}:=P_d(\{s\in {\mathcal{S}}_d:i,j\in s\})\mathrm{.}$ In particular, ${\pi }_{ii}={\pi }_i$.

The observed record includes the selected unit labels and their values. For a common record space set ${\mathcal{X}}_d:=\{(s,z):s\in {\mathcal{S}}_d,z:s\to \mathbb{R}\},{\mathcal{A}}_d:=\mathcal{P}({\mathcal{X}}_d),$ and for each $y\in {\mathbb{R}}^U$ define $O_y(s):=(s,y{\mathrm{\mid }}_s),P_y^d(B):={\sum }_{\begin{array}{c}s\in {\mathcal{S}}_d\\ O_y(s)\in B\end{array}}{p}_d(s)(B\in {\mathcal{A}}_d)\mathrm{.}$ For a real-valued statistic $h$ on ${\mathcal{X}}_d$, write $E_y^d[h]:={\sum }_{s\in {\mathcal{S}}_d}{p}_d(s)h(O_y(s))=E_d[h\circ O_y];$ this is expectation under the induced observed-data law. The local well-definedness result A finite-population design induces probability laws on observed records ↗ shows that each $P_y^d$ is a probability measure on $({\mathcal{X}}_d,{\mathcal{A}}_d)$. Hence ${\mathsf{E}}_d:=(({\mathcal{X}}_d,{\mathcal{A}}_d),{\mathbb{R}}^U,(P_y^d{)}_{y\in {\mathbb{R}}^U})$ is a statistical experiment (Statistical experiments), with parametric model ${\mathsf{P}}_d(y)=P_y^d$ (Parametric statistical models). The total $y\mapsto T(y)$ and mean $y\mapsto {\bar{y}}_U$ are estimands (Estimands on a statistical model).

This is design probability: the $y_i$ are fixed and only $S$ is random. A superpopulation model instead assigns a probability law to the values $y_i$; that additional law is separate from $p_d$. The finite-population target and the sampling unit must be recorded in the scientific model contract (Scientific statistical model contracts). In fixed-design regression, the matrix $X$ is held fixed and has no selection law; that convention alone does not specify a sampling design on $U$ (Fixed-design linear regression model).