Optimal linear estimation in two-phase sampling
Section 3. Best linear unbiased estimation in two-phase
sampling

3.1 An analytic form of the best linear unbiased estimator

For more efficient estimation of the totals $t_{y}$ and $t_{x},$ incorporating all the available information from both phases through the correlation of $y$ and $x,$ we consider the best linear unbiased estimators (BLUE), denoted by ${\hat{t}}_{y}^{B}$ and ${\hat{t}}_{x}^{B},$ which are minimum-variance linear unbiased combinations of the four estimators ${\tilde{t}}_{y},$ ${\hat{t}}_{x},$ ${\tilde{t}}_{x},$ $t_{x_{1}} - {\hat{t}}_{x_{1}}$ and given in matrix form by

${({\hat{t}}_{y}^{B^{'}}, {\hat{t}}_{x}^{B^{'}})}^{'} = P {({\tilde{t}}_{y}^{'}, {\hat{t}}_{x}^{'}, {\tilde{t}}_{x}^{'}, t_{x_{1}}^{'} - {\hat{t}}_{x_{1}}^{'}_{x})}^{'}, (3.1)$

where $P = {(W^{'} V^{- 1} W)}^{- 1} W^{'} V^{- 1},$ the matrix $W$ has entries 1’s and 0’s and satisfies $E [{({\tilde{t}}_{y}^{'}, {\hat{t}}_{x}^{'}, {\tilde{t}}_{x}^{'}, t_{x_{1}}^{'} - {\hat{t}}_{x_{1}}^{'})}^{'}] = W {(t_{y}^{'}, t_{x}^{'})}^{'},$ and $V$ is the covariance matrix of ${({\tilde{t}}_{y}^{'}, {\hat{t}}_{x}^{'}, {\tilde{t}}_{x}^{'}, t_{x_{1}}^{'} - {\hat{t}}_{x_{1}}^{'})}^{'} .$ It follows that $Var [{({\hat{t}}_{y}^{B^{'}}, {\hat{t}}_{x}^{B^{'}})}^{'}] = {(W^{'} V^{- 1} W)}^{- 1} .$ This typical formulation of best linear unbiased estimation has been explored in two other areas of survey sampling; see Wolter (1979), Jones (1980), Fuller (1990), and Chipperfield and Steel (2009). In the present context, a more practical formulation, which leads also to the representation of the BLUE as a calibration estimator, is as follows.

Writing the two linear combinations in (3.1) in expanded form and using the condition of unbiasedness $E ({\hat{t}}_{y}^{B}) = t_{y}$ and $E ({\hat{t}}_{x}^{B}) = t_{x},$ it is easy to show that the matrix $P$ of the coefficients in these linear combinations satisfies

$P = (\begin{matrix} B_{1 y} & B_{2 y} & B_{3 y} & B_{4 y} \\ B_{1 x} & B_{2 x} & B_{3 x} & B_{4 x} \end{matrix}) = (\begin{matrix} I & B_{2 y} & - B_{2 y} & B_{4 y} \\ 0 & B_{2 x} & I - B_{2 x} & B_{4 x} \end{matrix}),$

and then the two components of the BLUE in (3.1) are written in the regression form

$\begin{array}{l} {\hat{t}}_{y}^{B} & = {\tilde{t}}_{y} + B_{2 y} ({\hat{t}}_{x} - {\tilde{t}}_{x}) + B_{4 y} (t_{x_{1}} - {\hat{t}}_{x_{1}}) \\ {\hat{t}}_{x}^{B} & = {\tilde{t}}_{x} + B_{2 x} ({\hat{t}}_{x} - {\tilde{t}}_{x}) + B_{4 x} (t_{x_{1}} - {\hat{t}}_{x_{1}}). \end{array} (3.2)$

Now we can write (3.1) as

$(\begin{matrix} {\hat{t}}_{y}^{B} \\ {\hat{t}}_{x}^{B} \end{matrix}) = (\begin{matrix} {\tilde{t}}_{y} \\ {\tilde{t}}_{x} \end{matrix}) + B (\begin{matrix} {\hat{t}}_{x} - {\tilde{t}}_{x} \\ t_{x_{1}} - {\hat{t}}_{x_{1}} \end{matrix}), (3.3)$

where the matrix $B$ consists of the second and fourth columns of $P,$ and has the easily derived variance-minimizing value

$B = - Cov [(\begin{matrix} {\tilde{t}}_{y} \\ {\tilde{t}}_{x} \end{matrix}), (\begin{matrix} {\hat{t}}_{x} - {\tilde{t}}_{x} \\ t_{x_{1}} - {\hat{t}}_{x_{1}} \end{matrix})] {[Var (\begin{matrix} {\hat{t}}_{x} - {\tilde{t}}_{x} \\ t_{x_{1}} - {\hat{t}}_{x_{1}} \end{matrix})]}^{- 1} . (3.4)$

Next write

$w^{*} = (\begin{matrix} w_{1} \\ w \end{matrix}), X = (\begin{matrix} - X_{1} & X_{11} \\ X_{2} & 0 \end{matrix}), Ψ = (\begin{matrix} 0 & 0 \\ Y_{2} & X_{2} \end{matrix}), (3.5)$

so that

$X^{'} w^{*} = (\begin{matrix} {\tilde{t}}_{x} - {\hat{t}}_{x} \\ {\hat{t}}_{x_{1}} \end{matrix}), Ψ^{'} w^{*} = (\begin{matrix} {\tilde{t}}_{y} \\ {\tilde{t}}_{x} \end{matrix}), (3.6)$

and $B$ may then be expressed as $B = Cov (Ψ^{'} w^{*}, X^{'} w^{*}) {[Var (X^{'} w^{*})]}^{- 1} .$ For the calculation of variances and covariances we define $w^{*}$ at the population level as $w_{U}^{*} = {(w_{1 U}^{'}, w_{U}^{'})}^{'},$ where the $k^{th}$ element of $w_{1 U}$ is $w_{1 U_{k}} = (1 / π_{1 k}) I_{1 k},$ the indicator variable $I_{1}$ denoting inclusion of a population unit in $s_{1},$ and the $k^{th}$ element of $w_{U}$ is $w_{U_{k}} = [1 / (π_{1 k} π_{2 k})] I_{1 k} I_{2 k},$ the indicator variable $I_{2}$ denoting inclusion of a population unit in $s_{2}$ conditional on the selection of sample $s_{1} .$ We may now write $X^{'} w^{*} = X_{U}^{'} w_{U}^{*}$ and $Ψ^{'} w^{*} = Ψ_{U}^{'} w_{U}^{*},$ where $X_{U}$ and $Ψ_{U}$ are the population counterparts of $X$ and $Ψ,$ respectively; all submatrices in $X$ and $Ψ$ are expanded to population level, having $N$ rows. Then, denoting ${\hat{t}}_{Ψ} = Ψ^{'} w^{*}$ and ${\hat{t}}_{X} = X^{'} w^{*},$ we get

$B = Cov ({\hat{t}}_{Ψ}, {\hat{t}}_{X}) {[Var ({\hat{t}}_{X})]}^{- 1} = Ψ_{U}^{'} Var (w_{U}^{*}) X_{U} {[X_{U}^{'} Var (w_{U}^{*}) X_{U}]}^{- 1} . (3.7)$

A useful more analytic expression of $B$ is then obtained using the following Lemma; the proof is in the Appendix.

Lemma 1

$Var (w_{U}^{*}) = (\begin{matrix} Var (w_{1 U}) & Var (w_{1 U}) \\ Var (w_{1 U}) & Var (w_{U}) \end{matrix}), (3.8)$

where $Var (w_{1 U}) = {(π_{1 k l} - π_{1 k} π_{1 l}) / π_{1 k} π_{1 l}},$ $Var (w_{U}) = {(π_{1 k l} π_{2 k l} - π_{1 k} π_{2 k} π_{1 l} π_{2 l}) / π_{1 k} π_{2 k} π_{1 l} π_{2 l}} .$

Using (3.7) and (3.8), it is easy to show that (3.4) is expressed as

$B = [\begin{matrix} - Cov ({\tilde{t}}_{y}, {\hat{t}}_{x} - {\tilde{t}}_{x}) {[Var ({\hat{t}}_{x} - {\tilde{t}}_{x})]}^{- 1} & Cov ({\tilde{t}}_{y}, {\hat{t}}_{x_{1}}) {[Var ({\hat{t}}_{x_{1}})]}^{- 1} \\ I & Cov ({\tilde{t}}_{x}, {\hat{t}}_{x_{1}}) {[Var ({\hat{t}}_{x_{1}})]}^{- 1} \end{matrix}] . (3.9)$

Implicit in this representation of $B$ is the property $Cov ({\tilde{t}}_{x}, {\hat{t}}_{x}) = Var ({\hat{t}}_{x}),$ following from (3.8), implying that $Var ({\hat{t}}_{x} - {\tilde{t}}_{x}) = Var ({\tilde{t}}_{x}) - Var ({\hat{t}}_{x}),$ and the property $Cov ({\tilde{t}}_{x}, {\hat{t}}_{x_{1}}) = Cov ({\hat{t}}_{x}, {\hat{t}}_{x_{1}}),$ implying $Cov ({\hat{t}}_{x_{1}}, {\hat{t}}_{x} - {\tilde{t}}_{x}) = 0$ (this covariance being the off-diagonal block of $X_{U}^{'} Var (w_{U}^{*}) X_{U}) .$ Then (3.2) can be written explicitly as

$\begin{array}{l} {\hat{t}}_{y}^{B} & = {\tilde{t}}_{y} - Cov ({\tilde{t}}_{y}, {\hat{t}}_{x} - {\tilde{t}}_{x}) {[Var ({\hat{t}}_{x} - {\tilde{t}}_{x})]}^{- 1} ({\hat{t}}_{x} - {\tilde{t}}_{x}) \\ + Cov ({\tilde{t}}_{y}, {\hat{t}}_{x_{1}}) {[Var ({\hat{t}}_{x_{1}})]}^{- 1} (t_{x_{1}} - {\hat{t}}_{x_{1}}) \\ {\hat{t}}_{x}^{B} & = {\hat{t}}_{x} + Cov ({\hat{t}}_{x}, {\hat{t}}_{x_{1}}) {[Var ({\hat{t}}_{x_{1}})]}^{- 1} (t_{x_{1}} - {\hat{t}}_{x_{1}}). \end{array} (3.10)$

In view of the property $Cov ({\hat{t}}_{x_{1}}, {\hat{t}}_{x} - {\tilde{t}}_{x}) = 0,$ it follows immediately that

$\begin{array}{l} Var ({\hat{t}}_{y}^{B}) & = Var ({\tilde{t}}_{y}) - Cov ({\tilde{t}}_{y}, {\hat{t}}_{x} - {\tilde{t}}_{x}) {[Var ({\hat{t}}_{x} - {\tilde{t}}_{x})]}^{- 1} {Cov}^{'} ({\tilde{t}}_{y}, {\hat{t}}_{x} - {\tilde{t}}_{x}) \\ - Cov ({\tilde{t}}_{y}, {\hat{t}}_{x_{1}}) {[Var ({\hat{t}}_{x_{1}})]}^{- 1} {Cov}^{'} ({\tilde{t}}_{y}, {\hat{t}}_{x_{1}}) \\ Var ({\hat{t}}_{x}^{B}) & = Var ({\hat{t}}_{x}) - Cov ({\hat{t}}_{x}, {\hat{t}}_{x_{1}}) {[Var ({\hat{t}}_{x_{1}})]}^{- 1} {Cov}^{'} ({\hat{t}}_{x}, {\hat{t}}_{x_{1}}). \end{array} (3.11)$

Remark 3.1. Every component or linear combination of components of ${\hat{t}}_{y}^{B}$ is BLUE for the corresponding total. Also, as evident from (3.11), the efficiency of ${\hat{t}}_{y}^{B},$ relative to ${\tilde{t}}_{y},$ depends on the strength of correlation of $y$ with $x = (x_{1}, x_{2}),$ as well as on the difference in sample size (and possibly in sampling design) for the samples $s_{1}$ and $s_{2} .$

Remark 3.2. Because of the orthogonality property $Cov ({\hat{t}}_{x_{1}}, {\hat{t}}_{x} - {\tilde{t}}_{x}) = 0,$ the coefficient of any of the terms ${\hat{t}}_{x} - {\tilde{t}}_{x}$ and $t_{x_{1}} - {\hat{t}}_{x_{1}}$ in (3.10) would not change if the other one would be set equal to $0$ in (3.2). For instance, the BLUE for $t_{y}$ based on $({\tilde{t}}_{y}, {\hat{t}}_{x}, {\tilde{t}}_{x})$ would be ${\hat{t}}_{y}^{B}$ as in (3.10) but without the last term. This is easily worked out as a special case of the full setup $({\tilde{t}}_{y}, {\hat{t}}_{x}, {\tilde{t}}_{x}, t_{x_{1}} - {\hat{t}}_{x_{1}}).$ This orthogonality property explains the additive reduction of variance noticed in the first equation of (3.11).

Remark 3.3. The BLUE ${\hat{t}}_{x}^{B}$ in (3.10) can also be produced using the reduced setup $({\hat{t}}_{x}, t_{x_{1}} - {\hat{t}}_{x_{1}})$ in (3.1). The same best linear estimator, for a single target variable, has been derived differently in the context of general single-phase sampling by Fuller and Isaki (1981) and Montanari (1987). In particular, for the auxiliary variable $x_{1}$ we have ${\hat{t}}_{x_{1}}^{B} = t_{x_{1}} .$ Next, it can be easily verified that the BLUE in (3.1) can be alternatively derived in two steps of best linear unbiased estimation using the setup $({\tilde{t}}_{y}^{B}, {\hat{t}}_{x}^{B}, {\tilde{t}}_{x}^{B}),$ where ${\tilde{t}}_{y}^{B} = {\tilde{t}}_{y} + Cov ({\tilde{t}}_{y}, {\hat{t}}_{x_{1}}) {[Var ({\hat{t}}_{x_{1}})]}^{- 1} (t_{x_{1}} - {\hat{t}}_{x_{1}})$ and ${\tilde{t}}_{x}^{B} = {\tilde{t}}_{x} + Cov ({\tilde{t}}_{x}, {\hat{t}}_{x_{1}}) {[Var ({\hat{t}}_{x_{1}})]}^{- 1} (t_{x_{1}} - {\hat{t}}_{x_{1}})$ are the BLUEs generated by the one-phase setups $({\tilde{t}}_{y}, t_{x_{1}} - {\hat{t}}_{x_{1}})$ and $({\tilde{t}}_{x}, t_{x_{1}} - {\hat{t}}_{x_{1}}),$ respectively. It can be shown through tedious algebra, that another, more explicit, BLUE setup that is equivalent to that in (3.1) is $({\tilde{t}}_{y}, {\hat{t}}_{x_{2}}, {\tilde{t}}_{x_{2}}, t_{x_{1}} - {\hat{t}}_{x_{1}}, t_{x_{1}} - {\tilde{t}}_{x_{1}}).$ This attests that the compact setup in (3.1) provides the most efficient linear estimation of $t_{y}$ and $t_{x}$ using all available relevant estimates.

3.2 The two-phase BLUE as calibration estimator

Using the notation leading to (3.7), and setting ${\hat{t}}_{Ψ}^{B} = ({\hat{t}}_{y}^{B^{'}}, {\hat{t}}_{x}^{B^{'}})^{'}$ and $Δ = Var (w_{U}^{*}),$ we may express the BLUE in (3.3) as ${\hat{t}}_{Ψ}^{B} = {\hat{t}}_{Ψ} + B (t_{X} - {\hat{t}}_{X}),$ where $B = Ψ_{U}^{'} Δ X_{U} {(X_{U}^{'} Δ X_{U})}^{- 1}$ and $t_{X} = {(0^{'}, t_{x_{1}}^{'})}^{'},$ or in the more suggestive form

${\hat{t}}_{Ψ}^{B} = Ψ_{U}^{'} [w_{U}^{*} + Δ X_{U} {(X_{U}^{'} Δ X_{U})}^{- 1} (t_{X} - X_{U}^{'} w_{U}^{*})] . (3.12)$

It appears from (3.12) that ${\hat{t}}_{Ψ}^{B}$ has the form of a calibration estimator, with population vector of calibrated weights $c_{U}^{*} = w_{U}^{*} + Δ X_{U} {(X_{U}^{'} Δ X_{U})}^{- 1} (t_{X} - X_{U}^{'} w_{U}^{*})$ and vector of calibration totals $t_{X} .$ This is formalized in the following theorem; the proof is in the Appendix.

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

Theorem 1 states that best linear unbiased estimation using the setup $({\tilde{t}}_{y}, {\hat{t}}_{x}, {\tilde{t}}_{x}, t_{x_{1}} - {\hat{t}}_{x_{1}})$ is essentially a calibration procedure whereby the two estimates ${\hat{t}}_{x}$ and ${\tilde{t}}_{x}$ of $t_{x}$ are calibrated to each other, i.e., they are aligned, and the estimate ${\hat{t}}_{x_{1}}$ is calibrated to the total $t_{x_{1}} .$ We may now write formally the BLUE ${\hat{t}}_{Ψ}^{B}$ as a calibration estimator ${\hat{t}}_{Ψ}^{B} = Ψ_{U}^{'} c_{U}^{*},$ with its two components given in the simple linear forms ${\hat{t}}_{y}^{B} = Y_{U}^{'} c_{U}$ and ${\hat{t}}_{x}^{B} = X_{U}^{'} c_{U} .$

The alternative two-step construction of the BLUE noted in Remark 3.3 above can also be carried out through a two-step calibration procedure involving $w_{U}^{*}$ in both steps. Indeed, partitioning $X_{U}$ by its two column submatrices as $X_{U} = (X_{12 U}, X_{1 U}),$ and noting that $X_{12 U}^{'} Δ X_{1 U} = Cov ({\hat{t}}_{x_{1}}, {\hat{t}}_{x} - {\tilde{t}}_{x}) = 0,$ it is easy to decompose the vector $c_{U}^{*}$ as

$\begin{array}{l} c_{U}^{*} & = w_{U}^{*} + Δ X_{12 U} {(X_{12 U}^{'} Δ X_{12 U})}^{- 1} (0 - X_{12 U}^{'} w_{U}^{*}) \\ + Δ X_{1 U} {(X_{1 U}^{'} Δ X_{1 U})}^{- 1} (t_{x_{1}} - X_{1 U}^{'} w_{U}^{*}) . \end{array} (3.13)$

In the right hand side of (3.13), the sum of the first and second terms results from calibration with constraint $X_{12 U}^{'} c_{U}^{*} = X_{U}^{'} c_{1 U} - X_{U}^{'} c_{U} = 0$ only, while the sum of the first and third terms results from calibration with constraint $X_{1 U}^{'} c_{U}^{*} = t_{x_{1}}$ only.

Now setting $Δ_{1} = Var (w_{1 U})$ and $Δ_{2} = Var (w_{U}),$ these variances being specified by (3.8), it follows easily from (3.13) that the optimal calibration estimators ${\hat{t}}_{y}^{B}$ and ${\hat{t}}_{x}^{B}$ in (3.10) can be written in the explicit form, which will be recalled later,

$\begin{array}{l} {\hat{t}}_{y}^{B} & = {\tilde{t}}_{y} + [Y_{U}^{'} Δ_{2} X_{U} - Y_{U}^{'} Δ_{1} X_{U}] {[X_{U}^{'} Δ_{2} X_{U} - X_{U}^{'} Δ_{1} X_{U}]}^{- 1} ({\hat{t}}_{x} - {\tilde{t}}_{x}) \\ + Y_{U}^{'} Δ_{1} X_{1 U} {(X_{1 U}^{'} Δ_{1} X_{1 U})}^{- 1} (t_{x_{1}} - {\hat{t}}_{x_{1}}) \\ {\hat{t}}_{x}^{B} & = {\hat{t}}_{x} + X_{U}^{'} Δ_{1} X_{1 U} {(X_{1 U}^{'} Δ_{1} X_{1 U})}^{- 1} (t_{x_{1}} - {\hat{t}}_{x_{1}}). \end{array} (3.14)$

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Optimal linear estimation in two-phase sampling
Section 3. Best linear unbiased estimation in two-phase
sampling

3.1 An analytic form of the best linear unbiased estimator

3.2 The two-phase BLUE as calibration estimator

Optimal linear estimation in two-phase sampling Section 3. Best linear unbiased estimation in two-phase sampling

3.1 An analytic form of the best linear unbiased estimator

3.2 The two-phase BLUE 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 3. Best linear unbiased estimation in two-phase
sampling