Variance estimation in multi-phase calibration
Section 2. Notation

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We use a similar notation to the one used by Särndal et al. (1992) and Hidiroglou and Särndal (1998). Consider a finite population $U\mathrm{<mo>=</mo>}\left\{\mathrm{1,}\dots \mathrm{,}k\mathrm{,}\dots \mathrm{,}N\right\}.$ A first phase probability sample $s_1\left(s_1\subseteq U\right)$ is drawn from the population $U$ using a sampling design that generates the selection probability ${\pi }_{1k}$ for the $k^{\text{th}}$ unit in the population. Given that $s_{i-1}$ has been drawn, the $i^{\text{th}}$ phase sample $s_i\left(s_i\subseteq s_{i-1}\right)$ is selected from $s_{i-1}$ through a sampling design with the selection probabilities ${\pi }_{ik|s_{i-1}}\equiv \text{Pr}\left(k\in s_i|k\in s_{i-1}\right).$ Note the conditional nature of the consequent phase selection probabilities. From this point on we work only with weights in the estimation process. The conditioned $i^{\text{th}}$ phase sampling weight of unit $k\in s_i$ and its overall sampling weight will be denoted by $w_{ik}\mathrm{<mo>=</mo>}1/{\pi }_{ik|s_{i-1}}$ and $w_{ik}^{*}={\displaystyle {\prod }_{j\mathrm{<mo>=</mo>\; 1}}^i{w}_{jk}}$ respectively.

Let $y_k$ be the value of the target variable for the $k^{\text{th}}$ population unit with which an auxiliary vector $x_k=\left(x_{1k}\text{}\mathrm{,}\dots \mathrm{,}{x}_{jk}\text{}\mathrm{,}\dots \mathrm{,}{x}_{Jk}\right)$ is associated. Denote by $y$ the vector of elements of the target variable obtained at the last phase of sampling, $p.$ As outlined in Särndal et al. (1992, chapter 9), we partition the vector $x$ as $x\mathrm{<mo>=</mo>}{\left(x_1^{\prime }\mathrm{,}{x}_2^{\prime }\mathrm{,}\dots \mathrm{,}{x}_p^{\prime }\right)}^{\prime }$ with $p\le J$ so that at certain phases maybe more than one auxiliary variable is obtained. The population total of $x,$ $t_x\mathrm{<mo>=</mo>}{\displaystyle {\sum }_U{x}_k}$ is assumed to be unknown. However, some demographic totals may be known from relatively accurate sources such as census data or other types of administrative files. Without loss of generality let $x_1$ be the vector of variables known for all units in the population $U.$ Let $x_2$ be the vector of variables obtained in the first phase sample $s_1,$ and so on. For elements in $s_r\mathrm{,}r\le p$ the complete information is then summarized in the vector $x\mathrm{<mo>=</mo>}{\left(x_1^{\prime }\mathrm{,}{x}_2^{\prime }\mathrm{,}\dots \mathrm{,}{x}_r^{\prime }\right)}^{\prime }.$ Denote also $t_i\mathrm{<mo>=</mo>}{t}_{x_i}.$

Let $X_r$ be the design matrix with $n_r$ rows representing $n_r$ sampled units, and a number of columns as the number of auxiliary variables in the vector $x_r.$ Note that $X_r$ is obtained in sample $s_{r-1}$ at the $r-1^{\text{th}}$ phase of sampling so we may think of $U$ as sample $s_0.$ In the setting that appears for example in Särndal et al. (1992) and Hidiroglou and Särndal (1998), the design matrix $X_r$ includes all auxiliary variables $x_1\mathrm{,}\dots \mathrm{,}{x}_r,$ and not just $x_r,$ and is referred to as the full vector. The analysis however is the same in both cases.

The auxiliary information available at each phase of sampling can be used to obtain improved weights through the process of calibration which produces calibration factors to be used in the estimation process. We use the superscript “*” to denote overall weights, i.e., weights taking all phases into account. The super-imposed symbol “ $\sim$ ” denotes calibrated weights. The $i^{\text{th}}$ phase $g-$ factors are denoted by $g_{ik},$ resulting with $i^{\text{th}}$ phase calibrated weights ${\tilde{w}}_{ik}\mathrm{<mo>=</mo>}{\tilde{w}}_{i-\mathrm{1,}k}{w}_{ik}{g}_{ik}$ for $k\in s_i,$ where ${\tilde{w}}_{i-\mathrm{1,}k}$ are the calibrated weights of the $i-1^{\text{th}}$ phase and ${\tilde{w}}_{0k}\mathrm{<mo>=</mo>\; 1.}$ For $k\in s_i$ the calibration with respect to all phases produces overall calibration factors denoted as $g_{ik}^{*}\text{}.$ As a result we will have overall calibrated weights ${\tilde{w}}_{ik}\mathrm{<mo>=</mo>}{w}_{ik}^{*}{g}_{ik}^{*}$ where $w_{ik}^{*}$ is the overall sampling weight. Denote by $w_i$ the vector with components $w_{ik}\text{}\mathrm{<mo>;</mo>}k\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}{n}_i\text{},$ and $D_i$ a diagonal matrix of size $n_i$ with $w_i$ on its diagonal. The same notation will be used with the vectors $w_i^{*}\text{}\mathrm{,}{\tilde{w}}_i$ and $g_i.$

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