Variance estimation in multi-phase calibration
Section 3. Calibration with GLS distance

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Calibration requires the specification of a distance function measuring the distance between the initial weights and the new calibrated weights. Several distance functions have been studied, see a selected summary in Deville and Särndal (1992). We concentrate on the generalized least squares (GLS) distance measure. The conventional form of multi-phase calibration under the GLS distance finds the values ${\tilde{w}}_{ik}$ for the set $k\in s_i$ that minimize the expression

$${\displaystyle \sum _{k\in s_i}}\frac{c_{ik}{\left({\tilde{w}}_{ik}-{\tilde{w}}_{i-\mathrm{1,}k}{w}_{ik}\right)}^2}{{\tilde{w}}_{i-\mathrm{1,}k}{w}_{ik}}(3.1)$$

subject to

$${\displaystyle \sum _{k\in s_i}{\tilde{w}}_{ik}{x}_{ik}}\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in s_{i-1}}{\tilde{w}}_{i-\mathrm{1,}k}{x}_{ik}(3.2)}$$

(alternatively, one can write ${\tilde{w}}_{i-\mathrm{1,}k}{w}_{ik}{g}_{ik}$ instead of ${\tilde{w}}_{ik})$ where $\left\{{\tilde{w}}_{i-\mathrm{1,}k}\text{}\mathrm{<mo>:</mo>}k\in s_i\right\}$ are the initial weights at the beginning of phase $i,$ i.e., the calibrated weights obtained at phase $i-1;$ $\left\{{\tilde{w}}_{ik}\text{}\mathrm{<mo>:</mo>}k\in s_i\right\}$ are the calibrated weights of phase $i$ that we want to obtain; and $\left\{c_{ik}\text{}\mathrm{<mo>:</mo>}k\in s_i\right\}$ are specified positive factors used to control the relative importance that we are willing to assign to each of the elements of the sum on the basis of the auxiliary information available for $k\in s_{i-1}.$ For simplicity of notation assume from now on that $c_{ik}\mathrm{<mo>=</mo>\; 1}$ for all $i\mathrm{,}k.$ The weights resulting from this calibration scheme are ${\tilde{w}}_{ik}\mathrm{<mo>=</mo>}{\tilde{w}}_{i-\mathrm{1,}k}{w}_{ik}{g}_{ik}$ where $g_{ik}\mathrm{<mo>=</mo>\; 1}+{\left({\displaystyle {\sum }_{l\in s_{i-1}}}{\tilde{w}}_{i-\mathrm{1,}l}{x}_{il}-{\displaystyle {\sum }_{l\in s_i}}{\tilde{w}}_{i-\mathrm{1,}l}{w}_{il}{x}_{il}\right)}^{\prime }{T}_i^{-1}{x}_{ik}$ with $T_i\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{l\in s_i}}{w}_{il}^{*}{g}_{i-\mathrm{1,}l}^{*}{x}_{il}{x}_{il}^{\prime }\text{}.$ Hence, the calibration factors in this process operate multiplicatively with an overall calibration factor $g_{ik}^{*}\mathrm{<mo>=</mo>}{\displaystyle {\prod }_{j\mathrm{<mo>=</mo>\; 1}}^i{g}_{jk}}$ for $k\in s_i$ at the end of phase $i.$

Distance measure (3.1) may be criticized, because the factors $1/{\tilde{w}}_{i-\mathrm{1,}k}{w}_{ik}$ for some $i$ may not all necessarily be finite and positive, as the terms $g_{i-\mathrm{1,}k}$ that appear in ${\tilde{w}}_{i-\mathrm{1,}k}$ in the denominator can be zero or negative, contradicting the notion of distance. An alternative choice of distance function, and the one that we shall use in our analysis, is to replace (3.1) with

$${\displaystyle \sum _{k\in s_i}}\frac{{\left({\tilde{w}}_{ik}-{\tilde{w}}_{i-\mathrm{1,}k}{w}_{ik}\right)}^2}{w_{i-\mathrm{1,}k}^{*}{w}_{ik}}(3.3)$$

i.e., with non-calibrated weights in the denominator. It is easy to verify that the overall calibrated weights resulting from minimizing (3.3) subject to (3.2) are (for $p\mathrm{<mo>=</mo>\; 2}$ see Hidiroglou and Särndal 1998)

$${\tilde{w}}_{pk}\mathrm{<mo>=</mo>}{w}_{pk}^{*}\left(g_{1k}+\dots +g_{ik}+\dots +g_{pk}-\left(p-1\right)\right)(3.4)$$

where

$$g_{ik}=1+{\left({\displaystyle \sum _{l\in s_{i-1}}}{\tilde{w}}_{i-\mathrm{1,}l}{x}_{il}-{\displaystyle \sum _{l\in s_i}}{\tilde{w}}_{i-\mathrm{1,}l}{w}_{il}{x}_{il}\right)}^{\prime }{T}_i^{-1}{x}_{ik}(3.5)$$

for $k\in s_p$ with $T_i\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{l\in s_i}{w}_{il}^{*}{x}_{il}{x}_{il}^{{}^{\prime }}}.$ The choice of a distance measure in the construction of calibrated estimators is not critical since the resulting estimators within a wide range of distance measures are asymptotically equivalent to the one that uses the GLS distance measure (3.1), Deville and Särndal (1992). This is the case with distance measure (3.3) as well. Since the Horvitz-Thompson estimator $X_1^{{}^{\prime }}{w}_1^{*}$ is unbiased for $t_1$ with standard deviation of magnitude $N\cdot O\left(n_1^{-1/2}\right)$ then $g_{1k}\mathrm{<mo>=</mo>\; 1}+O\left(n_1^{-1/2}\right)$ for all $k\in s_1$ and hence ${\tilde{w}}_{1k}\mathrm{<mo>=</mo>}{w}_{1k}^{*}\left(1+O\left(n_1^{-1/2}\right)\right).$ Inductively $g_{ik}\mathrm{<mo>=</mo>\; 1}+O\left(n_i^{-1/2}\right)$ for all $i$ and from (3.4) ${\tilde{w}}_{pk}/w_{pk}^{*}\to 1$ in probability with $n_p.$ New techniques to improve estimation were suggested by Farrell and Singh (2002) by proposing other types of penalized chi-square distance function.

3.1 Estimation

The motivation to our next analysis comes from the recursive nature of ${\tilde{w}}_{ik}$ in (3.4), where calibrated weights of previous phases $\mathrm{1,}\dots \mathrm{,}i-1$ are nested in each $g_{ik},$ thus require the computation of the calibrated weights sequentially, i.e., one has to compute all calibrated weights of previous phases in order to obtain those of later ones. Let ${\widehat{B}}_{ij}^{+}\mathrm{<mo>=</mo>}{\left({\displaystyle {\sum }_{k\in s_i}{w}_{ik}^{*}{x}_{ik}{x}_{ik}^{\prime }}\right)}^{-1}{\displaystyle {\sum }_{k\in s_j}{w}_{jk}^{*}{x}_{ik}{x}_{jk}^{\prime }}$ and ${\widehat{B}}_{ij}^{-}\mathrm{<mo>=</mo>}{\left({\displaystyle {\sum }_{k\in s_i}}{w}_{ik}^{*}{x}_{ik}{x}_{ik}^{\prime }\right)}^{-1}{\displaystyle {\sum }_{k\in s_{j-1}}}{w}_{j-\mathrm{1,}k}^{*}{x}_{ik}{x}_{jk}^{\prime }$ be estimators for $B_{ij}\mathrm{<mo>=</mo>}{\left({\displaystyle {\sum }_{k\in U}{x}_{ik}{x}_{ik}^{\prime }}\right)}^{-1}{\displaystyle {\sum }_{k\in U}{x}_{ik}{x}_{jk}^{\prime }\text{},}$ the regression coefficient of $x_j$ on $x_i.$ The difference between the two estimators is that while ${\widehat{B}}_{ij}^{-}$ uses the entire set of units known for $x_j$ which is obtained in $s_{j-1},$ ${\widehat{B}}_{ij}^{+}$ uses only the subset $s_j\subseteq s_{j-1}$ and thus more variables than ${\widehat{B}}_{ij}^{-}.$ Let ${\widehat{Z}}_{ij}\mathrm{<mo>=</mo>}{\widehat{B}}_{ij}^{+}-{\widehat{B}}_{ij}^{-}$ the difference between the two coefficients estimates which is consistent to zero. Denote also ${\widehat{Z}}_{i_1{i}_2\dots i_k}\mathrm{<mo>=</mo>}{\displaystyle {\prod }_{j\mathrm{<mo>=</mo>\; 2}}^k{\widehat{Z}}_{i_{j-1}{i}_j}}$ for $k\ge 2$ and ${\widehat{Z}}_{i_1}\mathrm{<mo>=</mo>\; 1}$ for $k\mathrm{<mo>=</mo>\; 1.}$ Let ${\widehat{t}}_i^{-}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{k\in s_{i-1}}{w}_{i-1k}^{*}{x}_{ik}}$ and ${\widehat{t}}_i^{+}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{k\in s_i}{w}_{ik}^{*}{x}_{ik}}$ be two Horvitz-Thompson estimators for $t_i,$ based on the units obtained in samples $s_i$ and $s_{i-1}$ respectively. Note that all the estimators defined in this paragraph use overall design weights $w^{*}$ and not calibrated weights. In the following lemma we provide a presentation of ${\tilde{w}}_p,$ the vector of calibrated weights after $p$ phases of calibration, that relies solely on the pre-known sampling design weights ${\left\{w_i^{*}\right\}}_{i\mathrm{<mo>=</mo>\; 1}}^p\text{}.$

Lemma 3.1 Consider a multi-phase sampling design with a calibration scheme that produces additive $g-$ factors as defined in (3.3). A presentation of the calibrated weights at phase $p$  that is based entirely on the design weights is

$$\begin{array}{ll}{\tilde{w}}_p\hfill & =D_{p_{}}^{*\prime }{1}_{n_p}+{\displaystyle \sum _{i_1\mathrm{<mo>=</mo>\; 1}}^p{A}_{i_1}}-{\displaystyle \sum _{i_1<i_2}^p{A}_{i_1{i}_2}}\hfill \\ \hfill & +\dots +{\left(-1\right)}^{k+1}{\displaystyle \sum _{i_1\mathrm{<mo><</mo>}\dots \mathrm{<mo><</mo>}{i}_k}^p{A}_{i_1{i}_2\dots i_k}}+\dots +{\left(-1\right)}^{p+1}{A}_{i_1{i}_2\dots i_p}(3.6)\hfill \end{array}$$

where $A_{i_1{i}_2\dots i_k}^{\prime }\mathrm{<mo>=</mo>}{\left({\widehat{t}}_{i_1}^{-}-{\widehat{t}}_{i_1}^{+}\right)}^{\prime }{\widehat{Z}}_{i_1{i}_2\dots i_k}{\left(X_{i_k}^{\prime }{D}_{i_k}^{*}{X}_{i_k}\right)}^{-1}{X}_{i_k}^{\prime }{D}_p^{*}\text{}\mathrm{.}$

Proof. See Appendix A.

Note the “Inclusion-Exclusion” form of ${\tilde{w}}_p$ in Lemma 3.1. The $k^{\prime \text{th}}$ summation involves $\left(\begin{array}{l}p\hfill \\ k\hfill \end{array}\right)$ summands $A_{i_1{i}_2\dots i_k}\text{},$ for which each ${\widehat{Z}}_{i_1{i}_2\dots i_k}\mathrm{<mo>=</mo>}{\displaystyle {\prod }_{j\mathrm{<mo>=</mo>\; 2}}^k\left({\widehat{B}}_{i_{j-1}{i}_j}^{+}-{\widehat{B}}_{i_{j-1}{i}_j}^{-}\right)}$ contains $2^k$ summands. Thus, a total of $\left(\begin{array}{l}p\hfill \\ k\hfill \end{array}\right)2^k$ summands. The overall number of terms in (3.6) is therefore $3^p$ as acknowledged in the proof of the lemma. Note also that the terms $A_{i_1{i}_2\dots i_k}$ involve the product of the components ${\widehat{t}}_{i_1}^{-}-{\widehat{t}}_{i_1}^{+}$ and ${\widehat{Z}}_{i_1{i}_2\dots i_k}\text{},$ both having zero expectation, so the calibrated weight ${\tilde{w}}_p$ therefore equals to $D_p^{*\prime }{1}_{n_p}\text{},$ the overall design weight, plus correction terms of lower orders of magnitude, and maintains the familiar characteristic of calibrated weights. In our discussion so far we have merely provided a presentation to the vector of weights in a multi-phase calibration process which is constituted of design parameters only and does not include $g-$ factors. However, from this presentation of ${\tilde{w}}_p$ it is possible to deduce an innovative estimator for the variance of multi-phase calibrated estimators. Let $y$ be some variable of interest for which we want to estimate the population total $Y.$ Let ${\widehat{\beta }}_j={\left({\displaystyle {\sum }_{k\in s_j}{w}_{jk}^{*}{x}_{jk}{x}_{jk}^{{}^{\prime }}}\right)}^{-1}{\displaystyle {\sum }_{k\in s_p}}{w}_{pk}^{*}{x}_{jk}{y}_k,$ the regression coefficient of $y$ on $x_j,$ and ${\widehat{Y}}_{{\text{HT}}_p}{\mathrm{<mo>=</mo>\; 1}}_{n_p}^{\prime }{D}_p^{*}y$ the non-calibrated Horvitz-Thompson estimator computed over the elements in $s_p.$ Rearranging the terms in (3.6) produces a more conventional presentation of the multi-phased calibrated estimator ${\tilde{w}}_p^{\prime }y$ as a multi-variate regression estimator

$${\tilde{w}}_p^{\prime }y\mathrm{<mo>=</mo>}{\widehat{Y}}_{{\text{HT}}_p}+{\displaystyle \sum _{i_1\mathrm{<mo>=</mo>\; 1}}^p{\left({\widehat{t}}_{i_1}^{-}-{\widehat{t}}_{i_1}^{+}\right)}^{\prime }{\widehat{\gamma }}_{i_1}}(3.7)$$

where

$$\begin{array}{ll}{\widehat{\gamma }}_{i_1}\hfill & \mathrm{<mo>=</mo>}{\widehat{\beta }}_{i_1}-{\displaystyle \sum _{i_1<i_2}^p{\widehat{Z}}_{i_1{i}_2}{\widehat{\beta }}_{i_2}+}\hfill \\ \hfill & \dots +{\left(-1\right)}^{k+1}{\displaystyle \sum _{i_1\mathrm{<mo><</mo>}\dots <i_k}^p{\widehat{Z}}_{i_1{i}_2\dots i_k}{\widehat{\beta }}_{i_k}}+\dots +{\left(-1\right)}^{p-\left(i_1-1\right)+1}{\widehat{Z}}_{i_1\dots p}{\widehat{\beta }}_p\mathrm{.}\hfill \end{array}$$

A derivation of a consistent estimator for the variance of multi-phase calibrated estimators is now straightforward in the sense that it roughly follows the steps used in the derivation of the variance in a one-phase multi-variate calibration scheme.

Theorem 3.1 Let ${\widehat{e}}_{rk}\mathrm{<mo>=</mo>}{x}_{rk}^{\prime }{\widehat{\gamma }}_r-x_{r+\mathrm{1,}k}^{\prime }{\widehat{\gamma }}_{r+1}$  for $r\mathrm{<mo><</mo>}p$  and ${\widehat{e}}_{pk}\mathrm{<mo>=</mo>}{x}_{pk}^{\prime }{\widehat{\gamma }}_p-y_k.$  A consistent estimator for the variance of ${\tilde{w}}_p^{{}^{\prime }}y$  is

$${\displaystyle \sum _{1\le r_1\text{}\mathrm{,}{r}_2\le p}{\displaystyle \sum _{k\in s_{r_m}\text{}\mathrm{,}l\in s_{r_M}}\frac{w_{r_{M}l}^{*}}{w_{r_{m}l}^{*}}}}\left(w_{r_{m}k}^{*}{w}_{r_{m}l}^{*}-w_{r_{m}kl}^{*}\right){\widehat{e}}_{r_{m}k}{\widehat{e}}_{r_{M}l}(3.8)$$

where $r_m\mathrm{<mo>=</mo>}\min \left(r_1\text{}\mathrm{,}{r}_2\right)$  and $r_M\mathrm{<mo>=</mo>}\max \left(r_1\text{}\mathrm{,}{r}_2\right).$

The proof involves evaluation of the highest orders of magnitude and the estimation of their variance. Special attention is given to the evaluation of the joint probability of events $\left\{k\in s_i\text{}\mathrm{,}l\in s_j\right\}$ and estimation of the covariance between units from different phases of sampling.

Proof. In the first step we will show that the substitution of the coefficient estimators ${\widehat{\gamma }}_i;i\mathrm{<mo>=</mo>\; 1}\dots p$ by their true values ${\gamma }_i$ affects the estimation of the variance by a factor of $N^{2}o\left(n_p^{-1}\right)$ and hence not affecting the consistency of the substituted estimator. To this end note that ${\widehat{B}}_{ij}^{+}\mathrm{,}{\widehat{B}}_{ij}^{-}$ are both consistent to $B_{ij}.$ Write ${\widehat{B}}_{ij}^{+}\mathrm{<mo>=</mo>}{B}_{ij}+\left({\widehat{B}}_{ij}^{+}-B_{ij}\right)$ so ${\widehat{B}}_{ij}^{+}\mathrm{<mo>=</mo>}{B}_{ij}+O_p\left(n_j^{-1/2}\right).$ Recall that ${\widehat{Z}}_{ij}\mathrm{<mo>=</mo>}{\widehat{B}}_{ij}^{+}-{\widehat{B}}_{ij}^{-}$ where ${\widehat{B}}_{ij}^{-}$ is based on $s_{j-1}$ while ${\widehat{B}}_{ij}^{+}$ over its subsample $s_j$ and thus ${\widehat{Z}}_{ij}\mathrm{<mo>=</mo>}{O}_p\left(n_j^{-1/2}\right)$ and therefore ${\widehat{Z}}_{i_1{i}_2\dots i_k}$ is bounded by $O_p\left(n_{i_k}^{-1/2}\right).$ Likewise ${\widehat{\beta }}_j$ is ${\beta }_j+O_p\left(n_p^{-1/2}\right)$ because $y$ is observed only at the last phase of sampling $s_p.$ Hence ${\widehat{\gamma }}_i$ is consistent for ${\gamma }_i$ for all $i,$ where ${\widehat{\beta }}_i$ in ${\widehat{\gamma }}_i$ are replaced with ${\beta }_i$ in ${\gamma }_i.$ Consistency does not necessarily imply the convergence of the moments and specifically not of the variance. However, for a finite population, i.e., a finite probability space, the concepts coincide. It follows that for $n_p$ large enough $\text{Var}\left({\widehat{Y}}_{{\text{HT}}_p}+{\displaystyle {\sum }_{i_1\mathrm{<mo>=</mo>\; 1}}^p}{\left({\widehat{t}}_{i_1}^{-}-{\widehat{t}}_{i_1}^{+}\right)}^{\prime }{\widehat{\gamma }}_{i_1}\right)$ and $\text{Var}\left({\widehat{Y}}_{{\text{HT}}_p}+{\displaystyle {\sum }_{i_1\mathrm{<mo>=</mo>\; 1}}^p}{\left({\widehat{t}}_{i_1}^{-}-{\widehat{t}}_{i_1}^{+}\right)}^{\prime }{\gamma }_{i_1}\right)$ are asymptotically equivalent and following the above discussion the difference can be quantified by

$$\text{Var}\left({\tilde{w}}_p^{ \prime }y\right)\mathrm{<mo>=</mo>}\text{Var}\left({\widehat{Y}}_{{\text{HT}}_p}+{\displaystyle \sum _{r\mathrm{<mo>=</mo>\; 1}}^p{\left({\widehat{t}}_r^{-}-{\widehat{t}}_r^{+}\right)}^{\prime }{\gamma }_r}\right)+N^{2}o\left(n_p^{-1}\right)\mathrm{.}$$

The estimator ${\widehat{t}}_r^{+}$ is a summation over units in $s_r$ while ${\widehat{t}}_r^{-}$ is over $s_{r-1}.$ Rearranging the terms, the variance on the right hand side can be written as $\text{Var}\left({\displaystyle {\sum }_{r\mathrm{<mo>=</mo>\; 1}}^p}{\displaystyle {\sum }_{i\in S_r}}{w}_{ri}^{*}{e}_{ri}\right)$ which is equal to

$${\displaystyle \sum _{1\le r_1\text{}\mathrm{,}{r}_2\le p}}{\displaystyle \sum _{k\in U}}{\displaystyle \sum _{l\in U}{w}_{r_{1}k}^{*}{e}_{r_{1}k}{w}_{r_{2}l}^{*}{e}_{r_{2}l}}\text{Cov}\left(I_{k\in s_{r_1}}\text{}\mathrm{,}{I}_{l\in s_{r_2}}\right)$$

so a sample based estimator would be

$${\displaystyle \sum _{1\le r_1\text{}\mathrm{,}{r}_2\le p}}{\displaystyle \sum _{k\in s_{r_1}\text{}\mathrm{,}l\in s_{r_2}}{w}_{r_{1}k}^{*}{\widehat{e}}_{r_{1}k}{w}_{r_{2}l}^{*}{\widehat{e}}_{r_{2}l}}\left[1-\frac{P\left(k\in s_{r_1}\right)P\left(l\in s_{r_2}\right)}{P\left(k\in s_{r_1}\text{}\mathrm{,}l\in s_{r_2}\right)}\right]\mathrm{.}(3.9)$$

To compute the covariance between the indicators $I_{k\in s_{r_1}}$ and $I_{l\in s_{r_2}}$ we need to know the joint probability of events $\left\{k\in s_i\text{}\mathrm{,}l\in s_j\right\}.$ If $s_j\subset s_i,$ then $P\left(k\in s_i\text{}\mathrm{,}l\in s_j\right)$ equals the joint probability that both units $k\text{}\mathrm{,}l$ are in sample $s_i\mathrm{<mo>=</mo>}{s}_{\min \left(i\text{}\mathrm{,}j\right)}\text{},$ multiplied by the conditional probability that unit $l$ is in sample $s_j$ given that it belongs to $s_i.$ Formally, if $s_j\subset s_i$ then $P\left(k\in s_i\text{}\mathrm{,}l\in s_j\right)\mathrm{<mo>=</mo>}\frac{w_{il}^{*}}{w_{jl}^{*}}{w}_{i\text{}\mathrm{,}lk}^{*-1},$ hence eliminating the dependence on $s_{r_2}$ in the brackets in (3.9) and the result follows.

Another way to write (3.8) is

$${\displaystyle \sum _{1\le r\le p}{\displaystyle \sum _{k\text{}\mathrm{,}l\in s_r}\left(w_{rk}^{*}{w}_{rl}^{*}-w_{rkl}^{*}\right){\widehat{e}}_{rk}{\widehat{e}}_{rl}}}+2{\displaystyle \sum _{1\le r_m<r_M\le p}}{\displaystyle \sum _{k\in s_{r_m}}}{\displaystyle \sum _{l\in s_{r_M}}{w}_{r_{m}k}^{*}{\widehat{e}}_{r_{m}k}{w}_{r_{M}l}^{*}{\widehat{e}}_{r_{M}l}}\left(1-\frac{w_{r_{m}kl}^{*}}{w_{r_{m}k}^{*}{w}_{r_{m}l}^{*}}\right)\mathrm{.}$$

When $p\mathrm{<mo>=</mo>\; 2}$ the terms ${\gamma }_i$ coincide with the deviation units obtained from the decomposition of the sampling error of the two-step estimator of Breidt and Fuller (1993). Consistent estimates for the standard deviations of calibrated subpopulation total estimates are derived in the ordinary way by multiplying the target variable by an indicator variable for the specific subpopulation.

In our discussion so far we have provided a presentation of the vector of calibrated weights from which we have derived a new consistent estimator for the variance of multi-phase calibrated estimators. However, under certain cases the estimators can be furthermore simplified without loss of accuracy. Two scenarios will be discussed here briefly and are dependent on whether $n_j$ is significantly smaller than $n_{j-1}$ or not, that is, whether for all $j$ subsample $s_j$ is significantly smaller than $s_{j-1}.$ A typical case of the first scenario is when we have a set of nested administrative files of significantly diminishing sizes. The first set may be, for example, a population registry file that contains a limited number of variables about the whole population, like age, gender, etc. The second set can be a sample data from a wide national survey where comprehensive household data were collected on all sampled units, but with an additional questionnaire for a subgroup of those units (say, every tenth unit). This subgroup of units can now be calibrated to those two former sources of information. An example of the second scenario is when a few phases of calibration are undertaken over the same set of data. In other words, contrary to the customary multi-phase process, the element of sampling is present only in the first phase but not in later phases. Such a scenario may arise if we want to calibrate a survey to many variables for which we don’t have their cross sectional totals but only their marginals. In such cases a sequence of calibrations over the same sample, but with a different set of auxiliary variables on each phase, while usually assigning the last phases for the most important variables, may be a satisfactory compromise. This scenario may better be referred to as sequential. Under these scenarios ${\tilde{w}}_p$ and its variance can be vastly simplified. These scenarios can be stated as corollaries of our analysis but we choose not to consider them here in order to focus on our current results.

3.2 Examples: Two-phase and three-phase calibration

Two-phase calibration. We will use the special case of two-phase calibration $\left(p\mathrm{<mo>=</mo>\; 2}\right)$ to demonstrate the new methodology and its distinction from the alternative estimator commonly used in literature. The calibrated estimator under matrix notation is given according to (3.7) by

$${\tilde{w}}_2^{ \prime }y\mathrm{<mo>=</mo>}{\widehat{Y}}_{{\text{HT}}_2}+{\left({\widehat{t}}_1^{-}-{\widehat{t}}_1^{+}\right)}^{\prime }{\widehat{\gamma }}_1+{\left({\widehat{t}}_2^{-}-{\widehat{t}}_2^{+}\right)}^{\prime }{\widehat{\gamma }}_2$$

where ${\widehat{\gamma }}_1\mathrm{<mo>=</mo>}{\widehat{\beta }}_1-{\widehat{Z}}_{12}{\widehat{\beta }}_2$ and ${\widehat{\gamma }}_2\mathrm{<mo>=</mo>}{\widehat{\beta }}_2.$ Explicitly in non matrix form

$${\tilde{w}}_2^{ \prime }y\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in s_2}{w}_{2k}^{*}{y}_k}+\left({\displaystyle \sum _{k\in U}}{x}_{1k}-{\displaystyle \sum _{k\in s_1}}{w}_{1k}{x}_{1k}\right){\widehat{\gamma }}_1+\left({\displaystyle \sum _{k\in s_1}{w}_{1k}{x}_{2k}}-{\displaystyle \sum _{k\in s_2}{w}_{2k}^{*}{x}_{2k}}\right){\widehat{\gamma }}_2$$

where

$$\begin{array}{ll}{\widehat{\gamma }}_1\hfill & \mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{k\in s_1}{w}_{1k}{x}_{1k}{x}_{1k}^{\prime }}\right)}^{-1}\left[{\displaystyle \sum _{k\in s_2}{w}_{2k}^{*}{x}_{1k}{y}_k}-\left({\displaystyle \sum _{k\in s_2}{w}_{2k}^{*}{x}_{1k}{x}_{2k}^{\prime }}-{\displaystyle \sum _{k\in s_1}{w}_{1k}{x}_{1k}{x}_{2k}^{\prime }}\right){\widehat{\gamma }}_2\right]\hfill \\ {\widehat{\gamma }}_2\hfill & \mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{k\in s_2}{w}_{2k}^{*}{x}_{2k}{x}_{2k}^{\prime }}\right)}^{-1}{\displaystyle \sum _{k\in s_2}{w}_{2k}^{*}{x}_{2k}{y}_k}\mathrm{.}\hfill \end{array}$$

This estimator produces identical estimates to the two-phase calibrated estimator used in Hidiroglou and Särndal (1998) or in Särndal et al. (1992) section 9.7. But once one has computed the estimator of the parameters ${\gamma }_1\text{}\mathrm{,}{\gamma }_2\text{},$ the presentation of ${\tilde{w}}_2^{ \prime }y$ becomes simple and informative, having the structure of a simple multi-variate regression estimator. This linear estimator is based on the coefficients $\gamma$ which encompass the total effect of the variable $x$ they multiply and hence slightly differ from the $\beta$ coefficients. ${\widehat{\gamma }}_i$ encompasses the overall effect of the calibration to variable $x_i$ on the estimation of $Y.$ In the general case it takes into account the projection of $y$ on $x_i,$ the projection of $y$ on $x_{i+1}$ multiplied by the projection of $x_{i+1}$ on $x_i$ and so on. Moreover, as we will now show, the variance estimators differ significantly both in estimates and presentation. Because of the complication in evaluating the variance of estimators that involve $g-$ factors, the common practice used up till now in literature for two phases involved first approximating the $g-$ factors by 1, and then use the law of total variation to obtain two components, one for each phase, according to

$$\begin{array}{ll}{\widehat{V}}_C\left({\tilde{w}}_2^{ \prime }y\right)\hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{k\mathrm{,}l\in s_2}}{w}_{2kl}\left(w_{1k}{w}_{1l}-w_{1kl}\right)\left(g_{1k}{\stackrel{⌣}{e}}_{1k}\right)\left(g_{1l}{\stackrel{⌣}{e}}_{1l}\right)\hfill \\ \hfill & +{\displaystyle \sum _{k\mathrm{,}l\in s_2}}{w}_{1k}{w}_{1l}\left(w_{2k}{w}_{2l}-w_{2kl}\right)\left(g_{2k}{\stackrel{⌣}{e}}_{2k}\right)\left(g_{2l}{\stackrel{⌣}{e}}_{2l}\right)(3.10)\hfill \end{array}$$

where the error terms ${\stackrel{⌣}{e}}_{1k}\mathrm{<mo>=</mo>}{y}_k-x_{1k}^{\prime }{\widehat{\gamma }}_1$ and ${\stackrel{⌣}{e}}_{2k}\mathrm{<mo>=</mo>}{y}_k-x_{2k}^{\prime }{\widehat{\gamma }}_2$ are both defined for $k\in s_2$ because $y$ is observed only at $s_2$ and note the simple presentation of the error terms under the notation that uses the $\gamma$ coefficients. The $g-$ factors are defined as in (3.5). The approximation of the $g-$ factors by 1 in the derivation of (3.10) may undoubtedly lead to unpredictable estimates as those factors depart from unity exactly in those situations where calibration was essential. On the other hand, the variance estimator proposed in (3.8) for a two-phase calibrated estimator is given by

$$\begin{array}{ll}{\widehat{V}}_P\left({\tilde{w}}_2^{ \prime }y\right)\hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{k\text{}\mathrm{,}l\in s_1}}\left(w_{1k}{w}_{1l}-w_{1kl}\right){\widehat{e}}_{1k}{\widehat{e}}_{1l}+{\displaystyle \sum _{k\text{}\mathrm{,}l\in s_2}}\left(w_{2k}^{*}{w}_{2l}^{*}-w_{2kl}^{*}\right){\widehat{e}}_{2k}{\widehat{e}}_{2l}\hfill \\ \hfill & +2{\displaystyle \sum _{k\in s_1\text{}\mathrm{,}l\in s_2}}\frac{w_{2l}^{*}}{w_{1l}}\left(w_{1k}{w}_{1l}-w_{1kl}\right){\widehat{e}}_{1k}{\widehat{e}}_{2l}\mathrm{.}(3.11)\hfill \end{array}$$

The difference in the variance estimator between the two methods represented by equations (3.10) and (3.11) is fundamental. It is expressed in a couple of aspects. While the error term of the second phase in both methods is the same, i.e., ${\widehat{e}}_{2k}\mathrm{<mo>=</mo>}{\stackrel{⌣}{e}}_{2k},$ the error term of the first phase differs. ${\stackrel{⌣}{e}}_{1k}$ is based on the difference between $y_k$ and the regression predictor $x_{1k}^{\prime }{\widehat{\gamma }}_1$ while ${\widehat{e}}_{1k}$ is based on the difference between two predictors of $Y$ from phases one and two $x_{1k}^{\prime }{\widehat{\gamma }}_1-x_{2k}^{\prime }{\widehat{\gamma }}_2.$ This modification causes the first summand in (3.11) to be computed over $s_1$ and not $s_2$ where the sample is larger. Noticeably, the estimator (3.11) has a third summand which involves the product of the two error terms from both phases that has no parallel in (3.10). Although this product will often be close to zero whenever the error terms are not strongly correlated, it may still not be negligible whenever $y$ is strongly correlated with $x_1.$ An evident advantage is the absence of the $g-$ factors which makes the estimator much simpler to compute, i.e., once we have computed the parameters estimates ${\widehat{\gamma }}_i\text{}\mathrm{<mo>;</mo>}i\mathrm{<mo>=</mo>\; 1}\dots p,$ the estimator (3.11) can be computed using design parameters only without carrying the $g-$ factors from all phases of calibration. Last, and maybe from an operational point of view more important, as will be also shown in the simulation study, (3.11) has the advantage that in a wide range of designs the second summand constitutes the absolute majority of the variance while the summands in (3.10) are usually of the same order of magnitude. This characteristic stems from the fact that the term $\left(w_{2k}^{*}{w}_{2l}^{*}-w_{2kl}^{*}\right)$ which involves the total sampling weights is very large in comparison with $w_{2kl}\left(w_{1k}{w}_{1l}-w_{1kl}\right)$ or $w_{1k}{w}_{1l}\left(w_{2k}{w}_{2l}-w_{2kl}\right).$ In the variance estimator the function $f\left(w\right)\mathrm{<mo>=</mo>}{w}_k{w}_l-w_{kl}$ attains its maximum on the diagonal $k\mathrm{<mo>=</mo>}l$ where it is proportional to $w_k^2$ and then it is multiplied by the second power of its remainder ${\widehat{e}}_k$ which is a non-negative term. So when the sampling rate of the second phase is high enough it drastically increases terms which are dependent on total weights of that phase $w_2^{*},$ in comparison with a parallel term from the previous phase. Hence the second summand may therefore be a good estimator of the variance of the calibrated estimator practically on its own.

Three-phase calibration. Multi-phase calibration can be implemented when in a series of samples of diminishing (non-increasing) sizes each pair of consequent phases share some common variables. It can be held whether the samples are nested, i.e., $s_i$ is a subsample of $s_{i-1},$ or not. In practice, the simplest and most common case of course is of two phases when a smaller sample (nested or not) is being calibrated to a much bigger sample such as a Labor Force Survey which in turn is frequently calibrated to an administrative file with demographic variables. However, due to computational feasibility and development of methodology, designs with more phases of calibration are still popular and three-phase designs are second in line in terms of their simplicity and implementation. It is therefore worthwhile to elaborate on the estimator for this case a bit further.

The approximation (3.8) involves six different terms, three for the three phases of sampling and another three for the covariance between phases. We denote these terms by $V_1\text{}\mathrm{,}{V}_2\text{}\mathrm{,}{V}_3$ and $C_{12}\text{}\mathrm{,}{C}_{13}\text{}\mathrm{,}{C}_{23}$ respectively. Each is a multiplication of a term that involves sampling weights multiplied by remainders from the relevant phases. The formulae for three-phase calibration are presented in appendix B. As discussed for the two-phase case, when $w_i\mathrm{<mo>></mo>\; 1}$ the $V_i’\text{s}$ are likely to follow a clear order $V_1\mathrm{<mo><</mo>}{V}_2\mathrm{<mo><</mo>}{V}_3$ and $V_3$ will become more and more dominant the bigger the sampling rates of the third phase will be. This is marked as case 3 in Table 3.1, and in our simulation it is manifested in rows 2 and 6 of Table 4.2 where $w_{3k}$ were 10 and 5 respectively. Clearly, in reality this is many times not the case as the approximation also depends on the sizes of the remainder terms which rely on the choice of the calibrating variables and their specific correlations which may be very strong. In which cases the remainders will be very small and it would be better to use all terms of (3.8). As for the covariance terms, although $C_{13}$ involves overall weights $\left\{w_{3k}^{*}\right\},$ it is unlikely to add any substantial value to the total variance due to the generally weak correlation between the remainders of phases 1 and 3. On the other hand, the term $C_{23},$ although weighted by overall $2^{\text{nd}}$ phase weights only, may be significant due to the strong correlation between the remainders of phases 2 and 3 as they both include the term $x_{3k}^{\prime }{\widehat{\gamma }}_3$ for $k\in s_3.$ The relative importance of the terms for some general designs is specified in Table 3.1. The $\gamma$ coefficients which encompass the total effect of the variables $x$ they multiply now take a more interesting and complicated form. ${\widehat{\gamma }}_1$ for example takes into account the projections of $x_1$ on $x_2$ and of $x_1$ on $x_3,$ but deducted of the projection of $x_1$ over the projection of $x_2$ on $x_3.$

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