Optimal linear estimation in two-phase sampling
Section 2. Two-phase sampling design: Structure and notation

• Previous
• Next

Let $U=\left\{\mathrm{1,}\dots ,k\mathrm{,}\dots ,N\right\}$ denote a finite population of $N$ units. A first-phase sample $s_1$ of size $n_1$ is drawn from the population $U,$ using a sampling design that defines inclusion probability ${\pi }_{1k}=P\mathrm{<mo\; stretchy="false">(</mo>}k\in s_1\mathrm{<mo\; stretchy="false">)</mo>}$ for unit $k\in U,$ and joint inclusion probability ${\pi }_{1kl}=P\mathrm{<mo\; stretchy="false">(</mo>}k\mathrm{,}l\in s_1\mathrm{<mo\; stretchy="false">)</mo>}$ for units $k\mathrm{,}l\in U.$ Then, a second-phase sample $s_2$ of size $n_2$ is drawn from $s_1$ using a sampling design that defines conditional inclusion probability ${\pi }_{2k}=P\mathrm{<mo\; stretchy="false">(</mo>}k\in s_2|s_1\mathrm{<mo\; stretchy="false">)</mo>}$ for $k\in s_1,$ and joint conditional inclusion probability ${\pi }_{2kl}=P\mathrm{<mo\; stretchy="false">(</mo>}k\mathrm{,}l\in s_2|s_1\mathrm{<mo\; stretchy="false">)</mo>}$ for units $k\mathrm{,}l\in s_1.$ Assuming that ${\pi }_{1k}>0$ for all $k\in U$ and ${\pi }_{2k}>0$ for all $k\in s_1,$ the first-phase design weight for $k\in s_1$ is $w_{1k}=1/{\pi }_{1k},$ the conditional second-phase design weight for $k\in s_2$ is $w_{2k}=1/{\pi }_{2k},$ and the overall design weight for $k\in s_2$ is $w_k=w_{1k}{w}_{2k}.$

The standard type of auxiliary variables in two-phase sampling (see, for example, Särndal et al. (1992)) involves a vector of auxiliary variables $x,$ partitioned as $x=\mathrm{<mo\; stretchy="false">(</mo>}{x}_1^{\prime }\mathrm{,}{x}_2^{\prime }{)}^{\prime }$  by $p$ and $q$ components of it, with population total $t_x={\displaystyle {\sum }_U}{x}_k$ and known total $t_{x_1}={\displaystyle {\sum }_U}{x}_{1k}$ of $x_1.$ The value $x_k$ is observed for every unit $k\in s_1,$ whereas for a $d$ -dimensional vector of target variables $y,$ with total $t_y={\displaystyle {\sum }_U}{y}_k,$ the value $y_k$ is observed only for the units $k\in s_2.$ In some surveys, components of the vector $x_2$ are also target variables. An unbiased estimator of the total $t_y,$ the common Horvitz-Thompson (HT) estimator, given by ${\tilde{t}}_y={\displaystyle {\sum }_{s_2}}{w}_k{y}_k,$ is obtained using the second-phase sample $s_2,$ while two HT estimators of the total $t_x,$ given by ${\widehat{t}}_x={\displaystyle {\sum }_{s_1}}{w}_{1k}{x}_k$ and ${\tilde{t}}_x={\displaystyle {\sum }_{s_2}}{w}_k{x}_k,$ are obtained using the samples $s_1$ and $s_2,$ respectively. In the derivation of results involving these estimators we will use the vector notation ${\tilde{t}}_y=Y_2^{\prime }w,$ ${\widehat{t}}_x=X_1^{\prime }{w}_1,$ ${\tilde{t}}_x=X_2^{\prime }w,$  ${\widehat{t}}_{x_1}=X_{11}^{\prime }{w}_1,$ where $w_1$ and $w$ denote the vectors of design weights for samples $s_1$ and $s_2,$ respectively, $X_1$ and $X_{11}$ denote the sample $s_1$ matrices of $x$ and $x_1$ of dimensions $n_1\times \mathrm{<mo\; stretchy="false">(</mo>}p+q\mathrm{<mo\; stretchy="false">)</mo>}$ and $n_1\times p,$ respectively, and $Y_2,$ $X_2$ denote the sample $s_2$ matrices of $y$ and $x$ of dimensions $n_2\times d$ and $n_2\times \mathrm{<mo\; stretchy="false">(</mo>}p+q\mathrm{<mo\; stretchy="false">)</mo>.}$

The primary target of estimation is the total $t_y.$ However, for better understanding of the construction of the proposed estimators, and because components of the vector $x_2$ may also be target variables, a unified approach to the estimation of both $t_y$ and $t_x$ will be taken.

• Previous
• Article main page
• Next