# Variance estimation under monotone non-response for a panel survey Section 3. Calibration and complex parameters

In most surveys, a calibration step is used to obtain adjusted weights which enable to improve the accuracy of total estimates. Such calibrated estimators are considered in Section 3.1. Also, more complex parameters than totals are frequently of interest, and a linearization step can be used for variance estimation. This is the purpose of Section 3.2. The estimation of complex parameters with calibrated weights is treated in Section 3.3. In each case, explicit formulas for variance estimation and simplified variance estimation are derived, and the bias of the simplified variance estimator is discussed.

### 3.1  Variance estimation for calibrated total estimators

Assume that a vector ${x}_{i}$ of auxiliary variables is available for any unit $i\in {s}_{t},$ and that the vector of totals $X$ on the population $U$ is known. Then an additional calibration step (Deville and Särndal, 1992) is usually applied to ${\stackrel{^}{Y}}_{t}.$ It consists in modifying the weights ${d}_{ti}={\pi }_{i}^{-1}{\left({\stackrel{^}{p}}_{i}^{1\to t}\right)}^{-1}$ to obtain calibrated weights ${w}_{ti}$ which enable to match the real total $X,$ in the sense that

$\sum _{i\in {s}_{t}}\text{\hspace{0.17em}}{w}_{ti}{x}_{i}=X.\text{ }\text{ }\text{ }\text{ }\text{ }\left(3.1\right)$

The new calibrated weights are chosen to minimize a distance function with the original weights, while satisfying (3.1). This leads to the calibrated estimator

${\stackrel{^}{Y}}_{wt}=\sum _{i\in {s}_{t}}\text{\hspace{0.17em}}{w}_{ti}{y}_{it}.\text{ }\text{ }\text{ }\text{ }\text{ }\left(3.2\right)$

The estimated residual for the weighted regression of ${y}_{it}$ on ${x}_{i}$ is denoted by

${e}_{it}={y}_{it}-{\stackrel{^}{b}}_{t}{x}_{i}\text{ }\text{ }\text{ }\text{ }\text{ }\left(3.3\right)$

with

${\stackrel{^}{b}}_{t}={\left(\sum _{i\in {s}_{t}}\frac{1}{{\pi }_{i}{\stackrel{^}{p}}_{i}^{1\text{\hspace{0.17em}}\to \text{\hspace{0.17em}}t}}{x}_{i}{x}_{i}^{\top }\right)}^{-1}\sum _{i\in {s}_{t}}\frac{1}{{\pi }_{i}{\stackrel{^}{p}}_{i}^{1\text{\hspace{0.17em}}\to \text{\hspace{0.17em}}t}}{x}_{i}{y}_{it}.\text{ }\text{ }\text{ }\text{ }\text{ }\left(3.4\right)$

Replacing in (2.11) the variable ${y}_{it}$ with ${e}_{it}$ yields the estimator of the variance due to the sampling design

${\stackrel{^}{V}}_{t}^{p}\left({\stackrel{^}{Y}}_{wt}\right)=\sum _{i,\text{\hspace{0.17em}}j\in {s}_{t}}\frac{{\Delta }_{ij}}{{\pi }_{ij}}\frac{1}{{\stackrel{^}{p}}_{ij}^{1\text{\hspace{0.17em}}\to \text{\hspace{0.17em}}t}}\frac{{e}_{it}}{{\pi }_{i}}\frac{{e}_{jt}}{{\pi }_{j}}.\text{ }\text{ }\text{ }\text{ }\text{ }\left(3.5\right)$

Similarly, replacing in (2.12) the variable ${y}_{it}$ with ${e}_{it}$ yields the estimator of the variance due to the non-response

${\stackrel{^}{V}}_{t}^{\text{nr}}\left({\stackrel{^}{Y}}_{wt}\right)=\sum _{\delta =1}^{t}\sum _{i\in {s}_{t}}\frac{{\stackrel{^}{p}}_{i}^{\delta }\left(1-{\stackrel{^}{p}}_{i}^{\delta }\right)}{{\stackrel{^}{p}}_{i}^{\delta \text{\hspace{0.17em}}\to \text{\hspace{0.17em}}t}}{\left(\frac{{e}_{it}}{{\pi }_{i}{\stackrel{^}{p}}_{i}^{1\text{\hspace{0.17em}}\to \text{\hspace{0.17em}}\delta }}-{k}_{i}^{\delta }{\left({\stackrel{^}{h}}_{i}^{\delta }\right)}^{\top }{\stackrel{^}{\gamma }}_{te}^{\delta }\right)}^{2}\text{ }\text{ }\text{ }\text{ }\left(3.6\right)$

$γ ^ te δ = { ∑ i∈ s t k i δ p ^ i δ ( 1− p ^ i δ ) p ^ i δ → t h ^ i δ ( h ^ i δ ) ⊤ } −1 ∑ i∈ s t 1− p ^ i δ p ^ i 1 → t h ^ i δ e it π i . (3.7) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaHZoWzgaqcamaaDaaaleaacaWG0b Gaamyzaaqaaiabes7aKbaakiaai2dadaGadaqaamaaqafabeWcbaGa amyAaiabgIGiolaadohadaWgaaadbaGaamiDaaqabaaaleqaniabgg HiLdGccaaMe8Uaam4AamaaDaaaleaacaWGPbaabaGaeqiTdqgaaOWa aSaaaeaaceWGWbGbaKaadaqhaaWcbaGaamyAaaqaaiabes7aKbaakm aabmaabaGaaGymaiabgkHiTiqadchagaqcamaaDaaaleaacaWGPbaa baGaeqiTdqgaaaGccaGLOaGaayzkaaaabaGabmiCayaajaWaa0baaS qaaiaadMgaaeaacqaH0oazcaaMc8UaeyOKH4QaaGPaVlaadshaaaaa aOGabmiAayaajaWaa0baaSqaaiaadMgaaeaacqaH0oazaaGcdaqada qaaiqadIgagaqcamaaDaaaleaacaWGPbaabaGaeqiTdqgaaaGccaGL OaGaayzkaaWaaWbaaSqabeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0 uy0Hgip5wzaGabbiab=rQivcaaaOGaay5Eaiaaw2haamaaCaaaleqa baGaeyOeI0IaaGymaaaakmaaqafabeWcbaGaamyAaiabgIGiolaado hadaWgaaadbaGaamiDaaqabaaaleqaniabggHiLdGcdaWcaaqaaiaa igdacqGHsislceWGWbGbaKaadaqhaaWcbaGaamyAaaqaaiabes7aKb aaaOqaaiqadchagaqcamaaDaaaleaacaWGPbaabaGaaGymaiaaykW7 cqGHsgIRcaaMc8UaamiDaaaaaaGccaaMe8UabmiAayaajaWaa0baaS qaaiaadMgaaeaacqaH0oazaaGccaaMe8+aaSaaaeaacaWGLbWaaSba aSqaaiaadMgacaWG0baabeaaaOqaaiabec8aWnaaBaaaleaacaWGPb aabeaaaaGccaaIUaGaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaI ZaGaaiOlaiaaiEdacaGGPaaaaa@9E0D@$

The global variance estimator for ${\stackrel{^}{Y}}_{wt}$ is

${\stackrel{^}{V}}_{t}\left({\stackrel{^}{Y}}_{wt}\right)={\stackrel{^}{V}}_{t}^{p}\left({\stackrel{^}{Y}}_{wt}\right)+{\stackrel{^}{V}}_{t}^{\text{nr}}\left({\stackrel{^}{Y}}_{wt}\right).\text{ }\text{ }\text{ }\text{ }\text{ }\left(3.8\right)$

The simplified estimator of the variance due to non-response is

${\stackrel{^}{V}}_{t,\text{\hspace{0.17em}}\text{simp}}^{\text{nr}}\left({\stackrel{^}{Y}}_{wt}\right)=\sum _{i\in {s}_{t}}\frac{1-{\stackrel{^}{p}}_{i}^{1\text{\hspace{0.17em}}\to \text{\hspace{0.17em}}t}}{{\left({\stackrel{^}{p}}_{i}^{1\text{\hspace{0.17em}}\to \text{\hspace{0.17em}}t}\right)}^{2}}{\left(\frac{{e}_{it}}{{\pi }_{i}}\right)}^{2}\text{​}.\text{ }\text{ }\text{ }\text{ }\text{ }\left(3.9\right)$

Here again, this simplified variance estimator ignores the prediction terms ${k}_{i}^{\delta }{\left({\stackrel{^}{h}}_{i}^{\delta }\right)}^{\top }{\stackrel{^}{\gamma }}_{te}^{\delta }.$ If the underlying calibration model is appropriate, then the explanatory power of ${\stackrel{^}{h}}_{i}^{\delta }$ for ${e}_{it}$ is expected to be small, as well as the bias of the simplified variance estimator. On the other hand, if there remains in ${e}_{it}$ some significant part of ${y}_{it}$ that may not been explained by ${x}_{i},$ the bias of the simplified variance estimator may be non-negligible. This may occur in case of domain estimation, when the calibration variables do not include any auxiliary information specific of the domain.

### 3.2  Variance estimation for complex parameters

We may be interested in estimating more complex parameters than totals. Suppose that the variable of interest ${y}_{it}$ is $q\text{\hspace{0.17em}}-$ multivariate, and that the parameter of interest is $\theta \left(t\right)=f\left\{Y\left(t\right)\right\}$ with $f\left(\cdot \right)$ a known function. At time $t,$ substituting ${\stackrel{^}{Y}}_{t}$ into $\theta \left(t\right)$ yields the plug-in estimator ${\stackrel{^}{\theta }}_{t}=f\left({\stackrel{^}{Y}}_{t}\right).$

The estimated linearized variable of $\theta \left(t\right)$ is

${u}_{it}={\left\{{f}^{\prime }\left({\stackrel{^}{Y}}_{t}\right)\right\}}^{\top }\text{​}{y}_{it},\text{ }\text{ }\text{ }\text{ }\text{ }\left(3.10\right)$

with ${f}^{\prime }\left({\stackrel{^}{Y}}_{t}\right)$ the $q\text{\hspace{0.17em}}-$ vector of first derivatives of $f$ at point ${\stackrel{^}{Y}}_{t}.$ Replacing in (2.11) the variable ${y}_{it}$ with ${u}_{it}$ yields the estimator of the variance due to the sampling design

${\stackrel{^}{V}}_{t}^{p}\left({\stackrel{^}{\theta }}_{t}\right)=\sum _{i,\text{\hspace{0.17em}}j\in {s}_{t}}\frac{{\Delta }_{ij}}{{\pi }_{ij}}\frac{1}{{\stackrel{^}{p}}_{ij}^{1\text{\hspace{0.17em}}\to \text{\hspace{0.17em}}t}}\frac{{u}_{it}}{{\pi }_{i}}\frac{{u}_{jt}}{{\pi }_{j}}.\text{ }\text{ }\text{ }\text{ }\text{ }\left(3.11\right)$

Similarly, replacing in (2.12) the variable ${y}_{it}$ with ${u}_{it}$ yields the estimator of the variance due to the non-response

${\stackrel{^}{V}}_{t}^{\text{nr}}\left({\stackrel{^}{\theta }}_{t}\right)=\sum _{\delta =1}^{t}\text{\hspace{0.17em}}\sum _{i\in {s}_{t}}\text{\hspace{0.17em}}\frac{{\stackrel{^}{p}}_{i}^{\delta }\left(1-{\stackrel{^}{p}}_{i}^{\delta }\right)}{{\stackrel{^}{p}}_{i}^{\delta \text{\hspace{0.17em}}\to \text{\hspace{0.17em}}t}}{\left(\frac{{u}_{it}}{{\pi }_{i}{\stackrel{^}{p}}_{i}^{1\text{\hspace{0.17em}}\to \text{\hspace{0.17em}}\delta }}-{k}_{i}^{\delta }{\left({\stackrel{^}{h}}_{i}^{\delta }\right)}^{\top }\text{​}{\stackrel{^}{\gamma }}_{t\theta }^{\delta }\right)}^{2}\text{ }\text{ }\text{ }\text{ }\left(3.12\right)$

$γ ^ tθ δ = { ∑ i∈ s t k i δ p ^ i δ ( 1− p ^ i δ ) p ^ i δ → t h ^ i δ ( h ^ i δ ) ⊤ } −1 ∑ i∈ s t 1− p ^ i δ p ^ i 1 → t h ^ i δ u it π i . (3.13) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaHZoWzgaqcamaaDaaaleaacaWG0b GaeqiUdehabaGaeqiTdqgaaOGaaGypamaacmaabaWaaabuaeqaleaa caWGPbGaeyicI4Saam4CamaaBaaameaacaWG0baabeaaaSqab0Gaey yeIuoakiaaysW7caWGRbWaa0baaSqaaiaadMgaaeaacqaH0oazaaGc daWcaaqaaiqadchagaqcamaaDaaaleaacaWGPbaabaGaeqiTdqgaaO WaaeWaaeaacaaIXaGaeyOeI0IabmiCayaajaWaa0baaSqaaiaadMga aeaacqaH0oazaaaakiaawIcacaGLPaaaaeaaceWGWbGbaKaadaqhaa WcbaGaamyAaaqaaiabes7aKjaaykW7cqGHsgIRcaaMc8UaamiDaaaa aaGcceWGObGbaKaadaqhaaWcbaGaamyAaaqaaiabes7aKbaakmaabm aabaGabmiAayaajaWaa0baaSqaaiaadMgaaeaacqaH0oazaaaakiaa wIcacaGLPaaadaahaaWcbeqaamrr1ngBPrwtHrhAXaqeguuDJXwAKb stHrhAG8KBLbaceeGae8hPIujaaaGccaGL7bGaayzFaaWaaWbaaSqa beaacqGHsislcaaIXaaaaOWaaabuaeqaleaacaWGPbGaeyicI4Saam 4CamaaBaaameaacaWG0baabeaaaSqab0GaeyyeIuoakiaaysW7daWc aaqaaiaaigdacqGHsislceWGWbGbaKaadaqhaaWcbaGaamyAaaqaai abes7aKbaaaOqaaiqadchagaqcamaaDaaaleaacaWGPbaabaGaaGym aiaaykW7cqGHsgIRcaaMc8UaamiDaaaaaaGccaaMe8UaaGjbVlaays W7ceWGObGbaKaadaqhaaWcbaGaamyAaaqaaiabes7aKbaakiaaysW7 caaMe8UaaGjbVpaalaaabaGaamyDamaaBaaaleaacaWGPbGaamiDaa qabaaakeaacqaHapaCdaWgaaWcbaGaamyAaaqabaaaaOGaaGOlaiaa ywW7caaMf8UaaGzbVlaaywW7caGGOaGaaG4maiaac6cacaaIXaGaaG 4maiaacMcaaaa@A761@$

The global variance estimator for ${\stackrel{^}{\theta }}_{t}$ is

${\stackrel{^}{V}}_{t}\left({\stackrel{^}{\theta }}_{t}\right)={\stackrel{^}{V}}_{t}^{p}\left({\stackrel{^}{\theta }}_{t}\right)+{\stackrel{^}{V}}_{t}^{\text{nr}}\left({\stackrel{^}{\theta }}_{t}\right).\text{ }\text{ }\text{ }\text{ }\text{ }\left(3.14\right)$

The simplified estimator of the variance due to non-response is

${\stackrel{^}{V}}_{t,\text{\hspace{0.17em}}\text{simp}}^{\text{nr}}\left({\stackrel{^}{\theta }}_{t}\right)=\sum _{i\in {s}_{t}}\text{\hspace{0.17em}}\frac{1-{\stackrel{^}{p}}_{i}^{1\text{\hspace{0.17em}}\to \text{\hspace{0.17em}}t}}{{\left({\stackrel{^}{p}}_{i}^{1\text{\hspace{0.17em}}\to \text{\hspace{0.17em}}t}\right)}^{2}}\text{\hspace{0.17em}}{\left(\frac{{u}_{it}}{{\pi }_{i}}\right)}^{2}\text{​}.\text{ }\text{ }\text{ }\text{ }\text{ }\left(3.15\right)$

The bias of this simplified variance estimator will depend on the explanatory power for ${\stackrel{^}{h}}_{i}^{\delta }$ on the linearized variable ${u}_{it}.$

### 3.3  Variance estimation for complex parameters under calibration

The calibrated weights ${w}_{ti}$ may be used to obtain an estimator of the parameter $\theta \left(t\right).$ Substituting ${\stackrel{^}{Y}}_{wt}$ into $\theta \left(t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}f\text{ }\left\{Y\left(t\right)\right\}$ yields the calibrated plug-in estimator ${\stackrel{^}{\theta }}_{wt}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}f\text{ }\left({\stackrel{^}{Y}}_{wt}\right).$ To obtain a variance estimator for ${\stackrel{^}{\theta }}_{wt},$ we first compute the estimated linearized variable ${u}_{it}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left\{{f}^{\prime }\text{ }\left({\stackrel{^}{Y}}_{t}\right)\right\}}^{\top }\text{​}{y}_{it}$ and take

${e}_{\theta it}={u}_{it}-{\stackrel{^}{b}}_{\theta t}{x}_{i}\text{ }\text{ }\text{ }\text{ }\text{ }\left(3.16\right)$

with

${\stackrel{^}{b}}_{\theta t}={\left(\sum _{i\in {s}_{t}}\frac{1}{{\pi }_{i}{\stackrel{^}{p}}_{i}^{1\text{\hspace{0.17em}}\to \text{\hspace{0.17em}}t}}{x}_{i}{x}_{i}^{\top }\right)}^{-1}\sum _{i\in {s}_{t}}\text{\hspace{0.17em}}\frac{1}{{\pi }_{i}{\stackrel{^}{p}}_{i}^{1\text{\hspace{0.17em}}\to \text{\hspace{0.17em}}t}}\text{\hspace{0.17em}}{x}_{i}{u}_{it}\text{​}.\text{ }\text{ }\text{ }\text{ }\text{ }\left(3.17\right)$

Replacing in (2.11) the variable ${y}_{it}$ with ${e}_{\theta it}$ yields the estimator of the variance due to the sampling design

${\stackrel{^}{V}}_{t}^{p}\left({\stackrel{^}{\theta }}_{wt}\right)=\sum _{i,\text{\hspace{0.17em}}j\in {s}_{t}}\text{\hspace{0.17em}}\frac{{\Delta }_{ij}}{{\pi }_{ij}}\text{\hspace{0.17em}}\frac{1}{{\stackrel{^}{p}}_{ij}^{1\text{\hspace{0.17em}}\to \text{\hspace{0.17em}}t}}\text{\hspace{0.17em}}\frac{{e}_{\theta it}}{{\pi }_{i}}\text{\hspace{0.17em}}\frac{{e}_{\theta jt}}{{\pi }_{j}}.\text{ }\text{ }\text{ }\text{ }\text{ }\left(3.18\right)$

Similarly, replacing in (2.12) the variable ${y}_{it}$ with ${e}_{\theta it}$ yields the estimator of the variance due to the non-response

${\stackrel{^}{V}}_{t}^{\text{nr}}\left({\stackrel{^}{\theta }}_{wt}\right)=\sum _{\delta =1}^{t}\text{\hspace{0.17em}}\sum _{i\in {s}_{t}}\text{\hspace{0.17em}}\frac{{\stackrel{^}{p}}_{i}^{\delta }\left(1-{\stackrel{^}{p}}_{i}^{\delta }\right)}{{\stackrel{^}{p}}_{i}^{\delta \text{\hspace{0.17em}}\to \text{\hspace{0.17em}}t}}{\left(\frac{{e}_{\theta it}}{{\pi }_{i}{\stackrel{^}{p}}_{i}^{1\text{\hspace{0.17em}}\to \text{\hspace{0.17em}}\delta }}-{k}_{i}^{\delta }{\left({\stackrel{^}{h}}_{i}^{\delta }\right)}^{\top }\text{​}{\stackrel{^}{\gamma }}_{te\theta }^{\delta }\right)}^{2}\text{ }\text{ }\text{ }\text{ }\left(3.19\right)$

$γ ^ teθ δ = { ∑ i∈ s t k i δ p ^ i δ ( 1− p ^ i δ ) p ^ i δ→t h ^ i δ ( h ^ i δ ) ⊤ } −1 ∑ i∈ s t 1− p ^ i δ p ^ i 1 → t h ^ i δ e θit π i . (3.20) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaHZoWzgaqcamaaDaaaleaacaWG0b GaamyzaiabeI7aXbqaaiabes7aKbaakiaai2dadaGadaqaamaaqafa beWcbaGaamyAaiabgIGiolaadohadaWgaaadbaGaamiDaaqabaaale qaniabggHiLdGccaaMe8Uaam4AamaaDaaaleaacaWGPbaabaGaeqiT dqgaaOWaaSaaaeaaceWGWbGbaKaadaqhaaWcbaGaamyAaaqaaiabes 7aKbaakmaabmaabaGaaGymaiabgkHiTiqadchagaqcamaaDaaaleaa caWGPbaabaGaeqiTdqgaaaGccaGLOaGaayzkaaaabaGabmiCayaaja Waa0baaSqaaiaadMgaaeaacqaH0oazcqGHsgIRcaWG0baaaaaakiqa dIgagaqcamaaDaaaleaacaWGPbaabaGaeqiTdqgaaOWaaeWaaeaace WGObGbaKaadaqhaaWcbaGaamyAaaqaaiabes7aKbaaaOGaayjkaiaa wMcaamaaCaaaleqabaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDOb YtUvgaiqqacqWFKksLaaaakiaawUhacaGL9baadaahaaWcbeqaaiab gkHiTiaaigdaaaGcdaaeqbqabSqaaiaadMgacqGHiiIZcaWGZbWaaS baaWqaaiaadshaaeqaaaWcbeqdcqGHris5aOGaaGjbVpaalaaabaGa aGymaiabgkHiTiqadchagaqcamaaDaaaleaacaWGPbaabaGaeqiTdq gaaaGcbaGabmiCayaajaWaa0baaSqaaiaadMgaaeaacaaIXaGaaGPa VlabgkziUkaaykW7caWG0baaaaaakiaaysW7caaMe8UabmiAayaaja Waa0baaSqaaiaadMgaaeaacqaH0oazaaGccaaMe8+aaSaaaeaacaWG LbWaaSbaaSqaaiabeI7aXjaadMgacaWG0baabeaaaOqaaiabec8aWn aaBaaaleaacaWGPbaabeaaaaGccaaIUaGaaGzbVlaaywW7caaMf8Ua aGzbVlaacIcacaaIZaGaaiOlaiaaikdacaaIWaGaaiykaaaa@A232@$

The global variance estimator for ${\stackrel{^}{\theta }}_{wt}$ is

${\stackrel{^}{V}}_{t}\left({\stackrel{^}{\theta }}_{wt}\right)={\stackrel{^}{V}}_{t}^{p}\left({\stackrel{^}{\theta }}_{wt}\right)+{\stackrel{^}{V}}_{t}^{\text{nr}}\left({\stackrel{^}{\theta }}_{wt}\right).\text{ }\text{ }\text{ }\text{ }\text{ }\left(3.21\right)$

The simplified estimator of the variance due to non-response is

${\stackrel{^}{V}}_{t,\text{\hspace{0.17em}}\text{simp}}^{\text{nr}}\left({\stackrel{^}{\theta }}_{wt}\right)=\sum _{i\in {s}_{t}}\text{\hspace{0.17em}}\frac{1-{\stackrel{^}{p}}_{i}^{1\text{\hspace{0.17em}}\to \text{\hspace{0.17em}}t}}{{\left({\stackrel{^}{p}}_{i}^{1\text{\hspace{0.17em}}\to \text{\hspace{0.17em}}t}\right)}^{2}}{\left(\frac{{e}_{\theta it}}{{\pi }_{i}}\right)}^{2}\text{​}.\text{ }\text{ }\text{ }\text{ }\text{ }\left(3.22\right)$

Since the variable ${e}_{\theta it}$ is obtained as the residual in the regression of the linearized variable ${u}_{it}$ on the calibration variables ${x}_{i},$ the explanatory power for ${\stackrel{^}{h}}_{i}^{\delta }$ on ${e}_{\theta it}$ is expected to be small in practice, and the bias of the simplified variance estimator is expected to be small as well.

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