Optimal linear estimation in two-phase sampling
Section 4. Optimal linear estimation in two-phase sampling

4.1 The two-phase optimal estimator

The matrix $B$ in (3.7) comprises variances and covariances which need to be estimated. In view of $Var ({\hat{t}}_{X}) = X_{U}^{'} Δ X_{U}$ and $Cov ({\hat{t}}_{Ψ}, {\hat{t}}_{X}) = Ψ_{U}^{'} Δ X_{U},$ and recalling (3.8), the obvious unbiased estimates are $\hat{Var} ({\hat{t}}_{X}) = X^{'} \hat{Δ} X$ and $\hat{Cov} ({\hat{t}}_{Ψ}, {\hat{t}}_{X}) = Ψ^{'} \hat{Δ} X,$ where the $(n_{1} + n_{2}) \times (n_{1} + n_{2})$ matrix $\hat{Δ} =$ $\hat{Var} (w_{U}^{*})$ has diagonal blocks ${\hat{Δ}}_{1} = {(π_{1 k l} - π_{1 k} π_{1 l}) / π_{1 k} π_{1 l} π_{1 k l}}$ , ${\hat{Δ}}_{2} = {(π_{1 k l} π_{2 k l} - π_{1 k} π_{2 k} π_{1 l} π_{2 l}) / π_{1 k} π_{2 k} π_{1 l} π_{2 l} π_{1 k l} π_{2 k l}},$ and off-diagonal blocks ${\hat{Δ}}_{12},$ ${\hat{Δ}}_{21} = {\hat{Δ}}_{12}^{'}$ with ${\hat{Δ}}_{12} = {(π_{1 k l} - π_{1 k} π_{1 l}) / π_{1 k} π_{1 l} π_{1 k l} π_{2 l}},$ and $X,$ $Ψ$ are the sample matrices in (3.5).

We now obtain, as elements of the matrices $\hat{Var} ({\hat{t}}_{X})$ and $\hat{Cov} ({\hat{t}}_{Ψ}, {\hat{t}}_{X}),$ the unbiased estimates of all variances and covariances in (3.9), i.e, $\hat{Var} ({\hat{t}}_{x}) = X_{1}^{'} {\hat{Δ}}_{1} X_{1},$ $\hat{Var} ({\tilde{t}}_{x}) = X_{2}^{'} {\hat{Δ}}_{2} X_{2},$ $\hat{Var} ({\hat{t}}_{x_{1}}) = X_{11}^{'} {\hat{Δ}}_{1} X_{11},$ $\hat{Cov} ({\tilde{t}}_{x}, {\hat{t}}_{x_{1}}) = X_{2}^{'} {\hat{Δ}}_{21} X_{11},$ $\hat{Cov} ({\tilde{t}}_{y}, {\hat{t}}_{x}) = Y_{2}^{'} {\hat{Δ}}_{21} X_{1},$ $\hat{Cov} ({\tilde{t}}_{y}, {\hat{t}}_{x_{1}}) = Y_{2}^{'} {\hat{Δ}}_{21} X_{11},$ $\hat{Cov} ({\tilde{t}}_{y}, {\tilde{t}}_{x}) = Y_{2}^{'} {\hat{Δ}}_{2} X_{2} .$ However, the matrix $\hat{Var} ({\hat{t}}_{X})$ includes also the elements $\hat{Cov} ({\hat{t}}_{x}, {\tilde{t}}_{x}) = X_{1}^{'} {\hat{Δ}}_{12} X_{2}$ and $\hat{Cov} ({\hat{t}}_{x_{1}}, {\hat{t}}_{x} - {\tilde{t}}_{x}) = X_{11}^{'} {\hat{Δ}}_{1} X_{1} - X_{11}^{'} {\hat{Δ}}_{12} X_{2},$ which clearly do not retain the properties $Cov ({\hat{t}}_{x}, {\tilde{t}}_{x}) = Var ({\hat{t}}_{x})$ and $Cov ({\hat{t}}_{x_{1}}, {\hat{t}}_{x} - {\tilde{t}}_{x}) = 0,$ respectively. Unbiased estimates for the variances and covariances in (3.9) could be directly used, but then the estimate of the simple form $B$ in (3.9) could not be expressed as $Ψ^{'} \hat{Δ} X {(X^{'} \hat{Δ} X)}^{- 1},$ and thus the resulting estimator would not retain the calibration form of the BLUE in (3.12). This complication is circumvented using the following reformulation. Reset $w^{*},$ $X$ and $Ψ$ as

$w^{*} = (\begin{matrix} w_{1} \\ w \\ w_{1} \end{matrix}), X = (\begin{matrix} - X_{1} & 0 \\ X_{2} & 0 \\ 0 & X_{11} \end{matrix}), Ψ = (\begin{matrix} - Y_{1} & - X_{1} \\ Y_{2} & X_{2} \\ Y_{1} & X_{1} \end{matrix}), (4.1)$

where the sample matrices $X_{1},$ $X_{2},$ $X_{11}$ and $Y_{2}$ are as before, and $Y_{1}$ is the matrix of $y$ for sample $s_{1}$ with dummy values $y_{k}$ for $k \notin s_{2} .$ Clearly, $X^{'} w^{*}$ and $Ψ^{'} w^{*}$ are exactly as in (3.6). Then, having as before ${\hat{t}}_{X} = X^{'} w^{*}$ and ${\hat{t}}_{Ψ} = Ψ^{'} w^{*},$ we obtain again $B = Cov ({\hat{t}}_{Ψ}, {\hat{t}}_{X}) {[Var ({\hat{t}}_{X})]}^{- 1},$ where $Var ({\hat{t}}_{X}) = X_{U}^{'} Var (w_{U}^{*}) X_{U}$ and $Cov ({\hat{t}}_{Ψ}, {\hat{t}}_{X}) = Ψ_{U}^{'} Var (w_{U}^{*}) X,$ as in (3.7) but with $w_{U}^{*},$ $X_{U}$ and $Ψ_{U}$ being the population counterparts of the redefined $w^{*},$ $X,$ $Ψ .$ An extension of Lemma 1 to the redefined $w^{*}$ gives

$Var (w_{U}^{*}) = (\begin{matrix} Var (w_{1 U}) & Var (w_{1 U}) & Var (w_{1 U}) \\ Var (w_{1 U}) & Var (w_{U}) & Var (w_{1 U}) \\ Var (w_{1 U}) & Var (w_{1 U}) & Var (w_{1 U}) \end{matrix}),$

where $Var (w_{1 U})$ and $Var (w_{U})$ are the same as in Lemma 1. It is easy now to verify that again $B$ may be expressed analytically as in (3.9), and the two components of the BLUE are identical to those given by (3.10). More importantly, it follows from this special form of $Var (w_{U}^{*})$ that we have again $Var ({\hat{t}}_{X}) = X_{U}^{'} Δ X_{U}$ and $Cov ({\hat{t}}_{Ψ}, {\hat{t}}_{X}) = Ψ_{U}^{'} Δ X_{U},$ where now $Δ = diag (- Δ_{1}, Δ_{2}, Δ_{1})$ and $Δ_{1},$ $Δ_{2}$ as already defined. Thus we obtain again the BLUE in the calibration form of (3.12), and the retained orthogonal decomposition of the vector of calibrated weights in (3.13) leads readily to the expression (3.14). Now the orthogonality property $X_{12 U}^{'} Δ X_{1 U} = 0$ is induced by the block-diagonal structure of the redefined $X_{U},$ rather than by the special structure of the initial matrix $Δ$ used in (3.12).

For the reconstructed BLUE we now have the unbiased estimates $\hat{Var} ({\hat{t}}_{X}) = X^{'} \hat{Δ} X$ and $\hat{Cov} ({\hat{t}}_{Ψ}, {\hat{t}}_{X}) = Ψ^{'} \hat{Δ} X,$ where $X,$ $Ψ$ are the sample matrices in (4.1), and $\hat{Δ} = diag (- {\hat{Δ}}_{1}, {\hat{Δ}}_{2}, {\hat{Δ}}_{1})$ with ${\hat{Δ}}_{1},$ ${\hat{Δ}}_{2}$ as defined at the beginning of the section. From these we rederive easily the unbiased estimates of the variances and covariances in (3.9), but two of the elements of the sample matrix $Ψ^{'} \hat{Δ} X$ which involve $Y_{1},$ namely $Y_{1}^{'} {\hat{Δ}}_{1} X_{1}$ and $Y_{1}^{'} {\hat{Δ}}_{1} X_{11},$ require special consideration. The dummy (unobserved) values $y_{k}$ for $k \notin s_{2},$ necessary for expanding $Y_{1}$ to the population matrix $Y_{U}$ in the reconstructed BLUE, are set equal to zero, and the values $y_{k}$ for $k \in s_{2}$ are then necessarily weighted by $1 / π_{2 k} .$ Then $Y_{1}^{'} {\hat{Δ}}_{1} X_{1}$ and $Y_{1}^{'} {\hat{Δ}}_{1} X_{11}$ reduce to $Y_{2}^{'} {\hat{Δ}}_{21} X_{1}$ and $Y_{2}^{'} {\hat{Δ}}_{21} X_{11},$ which are the unbiased estimates $\hat{Cov} ({\tilde{t}}_{y}, {\hat{t}}_{x})$ and $\hat{Cov} ({\tilde{t}}_{y}, {\hat{t}}_{x_{1}}),$ respectively. The estimated $B$ in (3.9) is now given by

$\hat{B} = [\begin{matrix} [Y_{2}^{'} {\hat{Δ}}_{2} X_{2} - Y_{2}^{'} {\hat{Δ}}_{21} X_{1}] {[X_{2}^{'} {\hat{Δ}}_{2} X_{2} - X_{1}^{'} {\hat{Δ}}_{1} X_{1}]}^{- 1} & Y_{2}^{'} {\hat{Δ}}_{21} X_{11} {[X_{11}^{'} {\hat{Δ}}_{1} X_{11}]}^{- 1} \\ I & X_{1}^{'} {\hat{Δ}}_{1} X_{11} {[X_{11}^{'} {\hat{Δ}}_{1} X_{11}]}^{- 1} \end{matrix}] .$

The BLUE ${\hat{t}}_{Ψ}^{B} = {\hat{t}}_{Ψ} + B (t_{X} - {\hat{t}}_{X})$ with estimated $B$ will be called optimal linear unbiased estimator, optimal estimator in short, denoted by ${\hat{t}}_{Ψ}^{O} = {\hat{t}}_{Ψ} + \hat{B} (t_{X} - {\hat{t}}_{X}),$ with its two components given by

$\begin{array}{l} {\hat{t}}_{y}^{O} & = {\tilde{t}}_{y} + [Y_{2}^{'} {\hat{Δ}}_{2} X_{2} - Y_{2}^{'} {\hat{Δ}}_{21} X_{1}] {[X_{2}^{'} {\hat{Δ}}_{2} X_{2} - X_{1}^{'} {\hat{Δ}}_{1} X_{1}]}^{- 1} ({\hat{t}}_{x} - {\tilde{t}}_{x}) \\ + Y_{2}^{'} {\hat{Δ}}_{21} X_{11} {[X_{11}^{'} {\hat{Δ}}_{1} X_{11}]}^{- 1} (t_{x_{1}} - {\hat{t}}_{x_{1}}) \\ {\hat{t}}_{x}^{O} & = {\hat{t}}_{x} + X_{1}^{'} {\hat{Δ}}_{1} X_{11} {[X_{11}^{'} {\hat{Δ}}_{1} X_{11}]}^{- 1} (t_{x_{1}} - {\hat{t}}_{x_{1}}). \end{array} (4.2)$

This is the sample version of the BLUEs in (3.14), with estimated coefficients. In particular, ${\hat{t}}_{x}^{O}$ is the customary single-phase optimal estimator of $t_{x}$ using $x_{1}$ as auxiliary variable, and data from the full first-phase sample $s_{1};$ see Montanari (1987) and Rao (1994).

Remark 4.1. When $n_{2}$ is very close to $n_{1},$ the optimal estimator ${\hat{t}}_{y}^{O}$ can be quite unstable because of the near singularity of the inverted matrix in the coefficient of ${\hat{t}}_{x} - {\tilde{t}}_{x},$ and thus can become very inefficient; see, though, later Remark 6.1 on two-phase designs in which this is not an issue. Generally this is not a realistic setting in two-phase sampling, where $n_{2}$ is typically much smaller than $n_{1} .$

Following the construction of $Y_{2}^{'} {\hat{Δ}}_{21} X_{1}$ and $Y_{2}^{'} {\hat{Δ}}_{21} X_{11}$ as two of the estimates in $\hat{B},$ it transpires that these two bilinear forms can be written alternatively as ${\overset{⌣}{Y}}_{1}^{'} {\hat{Δ}}_{1} X_{1}$ and ${\overset{⌣}{Y}}_{1}^{'} {\hat{Δ}}_{1} X_{11},$ respectively, where ${\overset{⌣}{Y}}_{1}$ is a weighted version of $Y_{1}$ in which ${\overset{⌣}{y}}_{k} = y_{k} / π_{2 k}$ if $k \in s_{2}$ and ${\overset{⌣}{y}}_{k} = 0$ if $k \notin s_{2} .$ Then ${\hat{t}}_{Ψ} = Ψ^{'} w^{*} = {\overset{⌣}{Ψ}}^{'} w^{*},$ where $\overset{⌣}{Ψ}$ is $Ψ$ in (4.1) with ${\overset{⌣}{Y}}_{1}$ in place of $Y_{1},$ and $\hat{B}$ can be written compactly as $\hat{B} = {\overset{⌣}{Ψ}}^{'} \hat{Δ} X {(X^{'} \hat{Δ} X)}^{- 1},$ where $\hat{Δ} = diag (- {\hat{Δ}}_{1}, {\hat{Δ}}_{2}, {\hat{Δ}}_{1}).$ Henceforth, $\hat{Δ}$ will be meant to be the matrix $diag (- {\hat{Δ}}_{1}, {\hat{Δ}}_{2}, {\hat{Δ}}_{1}).$

As in Montanari (1987) and Rao (1994) for the single-phase optimal estimator, for large samples $s_{1}$ and $s_{2}$ the optimal estimator ${\hat{t}}_{Ψ}^{O} = {\hat{t}}_{Ψ} + \hat{B} (t_{X} - {\hat{t}}_{X})$ approximates the BLUE ${\hat{t}}_{Ψ}^{B},$ and thus it is approximately unbiased. Furthermore, the variance of ${\hat{t}}_{Ψ}^{O}$ approximates that of ${\hat{t}}_{Ψ}^{B},$ which works out easily to be $Var ({\hat{t}}_{Ψ}^{B}) = Var ({\hat{t}}_{Ψ}) - Cov ({\hat{t}}_{Ψ}, {\hat{t}}_{X}) {[Var ({\hat{t}}_{X})]}^{- 1} {Cov}^{'} ({\hat{t}}_{Ψ}, {\hat{t}}_{X}),$ i.e., the compact form of (3.11). Then, using the estimates $\hat{Var} ({\hat{t}}_{Ψ})$ and $\hat{Cov} ({\hat{t}}_{Ψ}, {\hat{t}}_{X}),$ derived earlier, we obtain the estimated approximate variance of ${\hat{t}}_{Ψ}^{O}$ as $\hat{AV} ({\hat{t}}_{Ψ}^{O}) = \hat{Var} ({\hat{t}}_{Ψ}) - \hat{Cov} ({\hat{t}}_{Ψ}, {\hat{t}}_{X}) {[\hat{Var} ({\hat{t}}_{X})]}^{- 1} {\hat{Cov}}^{'} ({\hat{t}}_{Ψ}, {\hat{t}}_{X}).$ From this we derive the computationally convenient expressions $\hat{AV} ({\hat{t}}_{y}^{O}) = Y_{2}^{'} {\hat{Δ}}_{2} Y_{2} - {\overset{⌣}{Ψ}}_{1}^{'} \hat{Δ} X {(X^{'} \hat{Δ} X)}^{- 1} X^{'} \hat{Δ} {\overset{⌣}{Ψ}}_{1},$ where ${\overset{⌣}{Ψ}}_{1}$ is the first column submatrix of $\overset{⌣}{Ψ},$ and $\hat{AV} ({\hat{t}}_{x}^{O}) = X_{1}^{'} {\hat{Δ}}_{1} X_{1} - X_{1}^{'} {\hat{Δ}}_{1} X_{11} {[X_{11}^{'} {\hat{Δ}}_{1} X_{11}]}^{- 1} X_{11}^{'} {\hat{Δ}}_{1} X_{1} .$

4.2 The two-phase optimal estimator as calibration estimator

The optimal estimator ${\hat{t}}_{Ψ}^{O} = {\hat{t}}_{Ψ} + \hat{B} (t_{X} - {\hat{t}}_{X}),$ with $\hat{B} = {\overset{⌣}{Ψ}}^{'} \hat{Δ} X {(X^{'} \hat{Δ} X)}^{- 1}$ , takes the form

${\hat{t}}_{Ψ}^{O} = {\overset{⌣}{Ψ}}^{'} [w^{*} + \hat{Δ} X {(X^{'} \hat{Δ} X)}^{- 1} (t_{X} - X^{'} w^{*})],$

of a calibration estimator, with vector of calibration totals $t_{X}$ and sample vector of calibrated weights $c^{*} = w^{*} + \hat{Δ} X {(X^{'} \hat{Δ} X)}^{- 1} (t_{X} - X^{'} w^{*})$ satisfying $X^{'} c^{*} = t_{X} .$ This is established formally by the following theorem; the proof is similar to that of Theorem 1, and is omitted.

Theorem 2. The vector $c^{*} = w^{*} + \hat{Δ} X {(X^{'} \hat{Δ} X)}^{- 1} (t_{X} - X^{'} w^{*})$ minimizes the generalized least-squares distance $(c^{*} - w^{*})^{'} {\hat{Δ}}^{- 1} (c^{*} - w^{*})$ subject to the constraints $X^{'} c^{*} = t_{X},$ i.e., $X_{1}^{'} c_{1} = X_{2}^{'} c$ and $X_{11}^{'} c_{1} = t_{x_{1}},$ where $(c_{1}, c)$ corresponds to $(w_{1}, w).$

The sample vector $c^{*}$ admits the same orthogonal decomposition as its population counterpart $c_{U}^{*}$ in (3.13). We may now write formally the optimal estimator ${\hat{t}}_{Ψ}^{O}$ as a calibration estimator ${\hat{t}}_{Ψ}^{O} = {\overset{⌣}{Ψ}}^{'} c^{*},$ which in view of $X^{'} c^{*} = t_{X}$ is generated by the simultaneous calibration of the two estimates ${\hat{t}}_{x}$ and ${\tilde{t}}_{x}$ of $t_{x}$ to each other, and of the estimate ${\hat{t}}_{x_{1}}$ to the total $t_{x_{1}} .$

Now, in expanded form the vector $c^{*}$ is

$c^{*} = (\begin{array}{l} c_{1} \\ c_{2} \\ c_{3} \end{array}) = (\begin{array}{l} w_{1} + {\hat{Δ}}_{1} X_{1} {[X_{2}^{'} {\hat{Δ}}_{2} X_{2} - X_{1}^{'} {\hat{Δ}}_{1} X_{1}]}^{- 1} ({\hat{t}}_{x} - {\tilde{t}}_{x}) \\ w + {\hat{Δ}}_{2} X_{2} {[X_{2}^{'} {\hat{Δ}}_{2} X_{2} - X_{1}^{'} {\hat{Δ}}_{1} X_{1}]}^{- 1} ({\hat{t}}_{x} - {\tilde{t}}_{x}) \\ w_{1} + {\hat{Δ}}_{1} X_{11} {(X_{11}^{'} {\hat{Δ}}_{1} X_{11})}^{- 1} (t_{x_{1}} - {\hat{t}}_{x_{1}}) \end{array}) . (4.3)$

Then, using the partition $X = (X_{12}, X_{1}),$ where $X_{12}$ and $X_{1}$ are the two orthogonal column submatrices of $X$ shown in (4.1), the two constraints are written as $X_{12}^{'} c^{*} = X_{2}^{'} c_{2} - X_{1}^{'} c_{1} = 0$ and $X_{1}^{'} c^{*} = X_{11}^{'} c_{3} = t_{x_{1}} .$ It also follows from (4.3) that ${\hat{t}}_{Ψ}^{O} = {\overset{⌣}{Ψ}}^{'} c^{*}$ implies (4.2). Regarding the two components of ${\hat{t}}_{Ψ}^{O}$ we observe that ${\hat{t}}_{x}^{O} = - X_{1}^{'} c_{1} + X_{2}^{'} c_{2} + X_{1}^{'} c_{3} = X_{1}^{'} c_{3}$ and that

${\hat{t}}_{y}^{O} = {\overset{⌣}{Y}}_{1}^{'} (c_{3} - c_{1}) + Y_{2}^{'} c_{2} = \sum_{s_{2}} [(c_{3 k} - c_{1 k}) / π_{2 k} + c_{2 k}] y_{k} .$

The explicit expression of ${\hat{t}}_{y}^{O},$ in terms of sample units, is

$\begin{array}{l} {\hat{t}}_{y}^{O} & = {\tilde{t}}_{y} + [\sum_{s_{2}} \sum_{s_{2}} {\hat{Δ}}_{2 k l} y_{k} x_{l}^{'} - \sum_{s_{2}} \sum_{s_{1}} {\hat{Δ}}_{1 k l} {\overset{⌣}{y}}_{k} x_{l}^{'}] \times \\ {[\sum_{s_{2}} \sum_{s_{2}} {\hat{Δ}}_{2 k l} x_{k} x_{l}^{'} - \sum_{s_{1}} \sum_{s_{1}} {\hat{Δ}}_{1 k l} x_{k} x_{l}^{'}]}^{- 1} ({\hat{t}}_{x} - {\tilde{t}}_{x}) \\ + (\sum_{s_{2}} \sum_{s_{1}} {\hat{Δ}}_{1 k l} {\overset{⌣}{y}}_{k} x_{1 l}^{'}) {(\sum_{s_{1}} \sum_{s_{1}} {\hat{Δ}}_{1 k l} x_{1 k} x_{1 l}^{'})}^{- 1} (t_{x_{1}} - {\hat{t}}_{x_{1}}), \end{array} (4.4)$

where ${\hat{Δ}}_{1 k l}$ and ${\hat{Δ}}_{2 k l}$ are the $k l^{th}$ elements of ${\hat{Δ}}_{1}$ and ${\hat{Δ}}_{2},$ respectively. Formula (4.4) is simplified in certain two-phase designs employed in important large scale surveys; examples of such surveys are presented in Hidiroglou and Särndal (1998) and Turmelle and Beaucage (2013). Specifically, this is the case when independent sampling (Poisson, or stratified Poisson) is used in one of the two phases, that is, when $π_{1 k l} = π_{1 k} π_{1 l}$ or $π_{2 k l} = π_{2 k} π_{2 l} .$ The simplification is considerable in the case of independent sampling in both phases. Then, both ${\hat{Δ}}_{1}$ and ${\hat{Δ}}_{2}$ are diagonal, with diagonal elements ${\hat{Δ}}_{1 k k} = (1 / π_{1 k}) ((1 / π_{1 k}) - 1)$ and ${\hat{Δ}}_{2 k k} = (1 / π_{1 k} π_{2 k}) ((1 / π_{1 k} π_{2 k}) - 1),$ respectively, and (4.4) involves only single summations. Other two-phase designs in which (4.4) involves single summations only, although ${\hat{Δ}}_{1}$ and ${\hat{Δ}}_{2}$ are not diagonal, involve simple random sampling or stratified simple random sampling in either phase; for an example of a survey with such two-phase design see Hidiroglou (2001). In general, however, the optimal estimator may not be practical because it requires the use of first-phase and second-phase joint inclusion probabilities $π_{1 k l}$ and $π_{2 k l},$ which are not known for some complex sampling designs. Even when these joint probabilities are known, but the matrices ${\hat{Δ}}_{1}$ and ${\hat{Δ}}_{2}$ are not diagonal, the estimated coefficient $\hat{B}$ and, hence, the optimal estimator may be unstable in very small samples ‒ especially if the dimension of the auxiliary vector $x$ is large. These difficulties may be overcome, at some loss of optimality, by employing simple approximations of the variances and covariances in $\hat{B};$ for approximate variance estimates based only on first order inclusion probabilities see, for example, Haziza, Mecatti and Rao (2008) and references therein. A computationally very convenient approximation of $\hat{B}$ leading to a two-phase estimator that belongs to the class of generalized regression estimators is described in the next section.

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Optimal linear estimation in two-phase sampling
Section 4. Optimal linear estimation in two-phase sampling

4.1 The two-phase optimal estimator

4.2 The two-phase optimal estimator as calibration estimator

Optimal linear estimation in two-phase sampling Section 4. Optimal linear estimation in two-phase sampling

4.1 The two-phase optimal estimator

4.2 The two-phase optimal estimator as calibration estimator

Editorial policy

Submission of Manuscripts

Note of appreciation

Standards of service to the public

Copyright

Optimal linear estimation in two-phase sampling
Section 4. Optimal linear estimation in two-phase sampling