Variance estimation in multi-phase calibration
Section 5. Concluding remarks

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In this paper we have constructed a novel presentation of multi-phase calibrated weights that enables the presentation of a multi-phase calibrated estimator in the form of a one-phase multi-variate calibrated estimator. This presentation enables the derivation of a closed form approximation for the variance of multi-phase calibrated estimators for any number of phases. A comparison with another approximation known in literature for the two-phase case shows that although the two approximations are consistent yet they differ in their estimates, form and interpretation. We have discussed some advantages of the new approximation in the case of two phases and also demonstrated its consistency in a simulation study for three-phase calibration where it performed very well for all designs investigated. The efficiency of the proposed estimator as a function of the sampling rates and other design parameters is left for future research.

Appendix A

To shorten the notation we will conduct our analysis in matrix form. We shall use a convention that for $j\mathrm{<mo>></mo>}i$ the summation in the scalar products $X_i^{\prime }{w}_j$ and $X_i^{\prime }{D}_j$ (or with $w_j^{*}$ or ${\tilde{w}}_j)$ are over units $k\in s_j$ (and not $s_i),$ i.e., over the sample indicated by the latest set of weights in the scalar product. Hence ${\widehat{Z}}_{ij}={\left(X_i^{\prime }{D}_i^{*}{X}_i\right)}^{-1}{X}_i^{\prime }\left(D_j^{*}-D_{j-1}^{*}\right)X_j$ under this notation.

Proof of Lemma 3.1. The weights that satisfy the calibration equation in the $j^{\text{th}}$ phase with initial weights ${\tilde{w}}_{j-1}$ are given by equation (3.4). Under our matrix notation

$${\tilde{w}}_j\mathrm{<mo>=</mo>}{D}_j^{*}\left[g_1+\dots +g_j-\left(j-1\right)\right]$$

where $g_j\mathrm{<mo>=</mo>\; 1}+X_j{T}_j^{-1}\left(X_j^{\prime }{\tilde{w}}_{j-1}-X_j^{\prime }{D}_j{\tilde{w}}_{j-1}\right)$ (see equation (3.5)). So

$$\begin{array}{ll}{\tilde{w}}_j\hfill & \mathrm{<mo>=</mo>}{D}_{j-1}^{*}{D}_j\left[g_1+\dots +g_{j-1}-\left(j-2\right)+g_j-1\right]\hfill \\ \hfill & \mathrm{<mo>=</mo>}{D}_j\left[{\tilde{w}}_{j-1}+D_{j-1}^{*}\left(g_j-1\right)\right]\mathrm{.}(\text{A}\text{.1})\hfill \end{array}$$

Plugging $g_j$ gives ${\tilde{w}}_j\mathrm{<mo>=</mo>}{D}_j\left[{\tilde{w}}_{j-1}+D_{j-1}^{*}{X}_j{T}_j^{-1}\left(X_j^{\prime }{\tilde{w}}_{j-1}-X_j^{\prime }{D}_j{\tilde{w}}_{j-1}\right)\right]$ which involves the weight ${\tilde{w}}_{j-1}$ from the previous phase of calibration and its scalar product with $X_j^{\prime }$ and $X_j^{\prime }{D}_j\text{},$ while the rest of the multipliers are design parameters. The square brackets contain three summands and thus after $j$ phases of calibration we would have $3^j$ summands that would involve design parameters only. Substituting ${\tilde{w}}_{j-1}$ of (A.1) into $X_j^{\prime }{D}_j{\tilde{w}}_{j-1}$ yields

$$\begin{array}{ll}{X}_j^{\prime }{D}_j{\tilde{w}}_{j-1}\hfill & \mathrm{<mo>=</mo>}{X}_j^{\prime }{D}_j\left\{D_{j-1}{\tilde{w}}_{j-2}+D_{j-1}^{*}\left(g_{j-1}-1\right)\right\}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{X}_j^{\prime }{D}_j{D}_{j-1}{\tilde{w}}_{j-2}+X_j^{\prime }{D}_j{D}_{j-1}^{*}{X}_{j-1}{T}_{j-1}^{-1}\left(X_{j-1}^{\prime }{\tilde{w}}_{j-2}-X_{j-1}^{\prime }{D}_{j-1}{\tilde{w}}_{j-2}\right)(\text{A}\text{.2})\hfill \end{array}$$

and therefore also

$$X_j^{{}^{\prime }}{\tilde{w}}_{j-1}\mathrm{<mo>=</mo>}{X}_j^{\prime }{D}_{j-1}{\tilde{w}}_{j-2}+X_j^{\prime }{D}_{j-1}^{*}{X}_{j-1}{T}_{j-1}^{-1}\left(X_{j-1}^{\prime }{\tilde{w}}_{j-2}-X_{j-1}^{\prime }{D}_{j-1}{\tilde{w}}_{j-2}\right)\mathrm{.}(\text{A}\text{.3})$$

Combining the terms results in an expression for ${\tilde{w}}_j$ that involves calibrated weights from phase $j-2$ only

$$\begin{array}{ll}{\tilde{w}}_j\hfill & \mathrm{<mo>=</mo>}{D}_j{D}_{j-1}{\tilde{w}}_{j-2}\hfill \\ \hfill & +D_j^{*}{X}_{j-1}{T}_{j-1}^{-1}\left(X_{j-1}^{\prime }{\tilde{w}}_{j-2}-X_{j-1}^{\prime }{D}_{j-1}{\tilde{w}}_{j-2}\right)\hfill \\ \hfill & +D_j^{*}{X}_j{T}_j^{-1}\left(X_j^{\prime }{D}_{j-1}{\tilde{w}}_{j-2}-X_j^{\prime }{D}_j{D}_{j-1}{\tilde{w}}_{j-2}\right)\hfill \\ \hfill & -D_j^{*}{X}_j{T}_j^{-1}{\widehat{Z}}_{j-1,j}^{\prime }\left(X_{j-1}^{\prime }{\tilde{w}}_{j-2}-X_{j-1}^{\prime }{D}_{j-1}{\tilde{w}}_{j-2}\right)\mathrm{.}(\text{A}\text{.4})\hfill \end{array}$$

Plugging (A.2) and (A.3) with $j\mathrm{<mo>=</mo>}p$ into (A.1) and recursing $p-1$ times over the respective calibration groups, produces the desired result.

Appendix B

A consistent estimator for the population total in three-phase calibration can be presented by ${\widehat{w}}_3^{ \prime }y\mathrm{<mo>=</mo>}{\widehat{Y}}_{{\text{HT}}_3}+{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^3}{\left({\widehat{t}}_1^{-}-{\widehat{t}}_1^{+}\right)}^{\prime }{\widehat{\gamma }}_i\text{},$ where

$$\begin{array}{ll}{\widehat{\gamma }}_1\hfill & \mathrm{<mo>=</mo>}{\widehat{\beta }}_1-{\widehat{Z}}_{12}{\widehat{\beta }}_2-{\widehat{Z}}_{13}{\widehat{\beta }}_3+{\widehat{Z}}_{12}{\widehat{Z}}_{23}{\widehat{\beta }}_3\hfill \\ {\widehat{\gamma }}_2\hfill & \mathrm{<mo>=</mo>}{\widehat{\beta }}_2-{\widehat{Z}}_{23}{\widehat{\beta }}_3\hfill \\ {\widehat{\gamma }}_3\hfill & \mathrm{<mo>=</mo>}{\widehat{\beta }}_3\mathrm{.}\hfill \end{array}$$

A consistent estimator for the variance is

$$\begin{array}{ll}{\widehat{V}}_P\left({\tilde{w}}_3^{ \prime }y\right)\hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{k\mathrm{,}l\in s_1}}\left(w_{1k}^{*}{w}_{1l}^{*}-w_{1kl}^{*}\right){\widehat{e}}_{1k}{\widehat{e}}_{1l}+\dots +{\displaystyle \sum _{k\text{}\mathrm{,}l\in s_3}}\left(w_{3k}^{*}{w}_{3l}^{*}-w_{3kl}^{*}\right){\widehat{e}}_{3k}{\widehat{e}}_{3l}\hfill \\ \hfill & +2{\displaystyle \sum _{k\in s_1\text{}\mathrm{,}l\in s_2}}{w}_{2l}\left(w_{1k}{w}_{1l}-w_{1kl}\right){\widehat{e}}_{1k}{\widehat{e}}_{2l}+2{\displaystyle \sum _{k\in s_2\text{}\mathrm{,}l\in s_3}}{w}_{3l}\left(w_{2k}^{*}{w}_{2l}^{*}-w_{2kl}^{*}\right){\widehat{e}}_{2k}{\widehat{e}}_{3l}\hfill \\ \hfill & +2{\displaystyle \sum _{k\in s_1\text{}\mathrm{,}l\in s_3}}{w}_{2l}{w}_{3l}\left(w_{3k}^{*}{w}_{3l}^{*}-w_{3kl}^{*}\right){\widehat{e}}_{1k}{\widehat{e}}_{3l}\mathrm{.}\hfill \end{array}$$

where ${\widehat{e}}_{1k}\mathrm{<mo>=</mo>}{x}_{1k}^{\prime }{\widehat{\gamma }}_1-x_{2k}^{\prime }{\widehat{\gamma }}_2\text{},$ ${\widehat{e}}_{2k}\mathrm{<mo>=</mo>}{x}_{2k}^{\prime }{\widehat{\gamma }}_2-x_{3k}^{\prime }{\widehat{\gamma }}_3$ and ${\widehat{e}}_{3k}\mathrm{<mo>=</mo>}{x}_{3k}^{\prime }{\widehat{\gamma }}_3-y_k$ as defined in Theorem 3.1.

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