Small area estimation methods under cut-off sampling
Section 4. Calibration estimators

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Calibration is traditionally applied when the true totals of certain auxiliary variables, which are potentially correlated with the study variable, are known. The idea of calibration is to adjust the design weights $w_{j|i},$ so that the corresponding expansion estimators of the available true totals have zero error. If the adjusted weights provide estimators of the available totals of the auxiliary variables that are absent of error, then one expects that they will also decrease the error in the estimation of the total of the study variable, provided that it is linearly related with the auxiliary variables. Even if there is an underlying linear model, in the absence of cut-off sampling, calibration estimators are design-consistent as the area sample size $n_i$ increases even if the model does not hold. In this sense, they are model-assisted and their properties are typically evaluated under the design-based setup. However, if $n_i$ is small, the estimates may suffer from small sample bias.

As we shall see below, calibration estimators reduce the bias due to cut-off sampling if the underlying linear model holds for the whole population (included and excluded units). However, for small domains, they might have unacceptably large sampling errors, apart from non-negligible small sample bias.

Let us denote by $x_{ij}$ the vector of auxiliary variables for unit $j$ within domain $i.$ Depending on whether the domain totals or only the population totals of these auxiliary variables are available, we can apply different calibration approaches. First, consider the case whereby the vector of domain totals $X_i={\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^{N_i}}{x}_{ij}$ is available. Note that $X_i$ is the total in the whole domain $U_i\mathrm{<mo>=</mo>}{U}_{iI}\cup U_{iE}.$ Then, one approach to calibration is to determine calibration weights $h_{j|i},$ $j\in s_i,$ that minimize

$$\begin{array}{l}{\displaystyle \sum _{j\in s_i}}{\left(h_{j|i}-w_{j|i}\right)}^2/w_{j|i}(4.1)\hfill \\ \text{s}\text{.t}\text{.}{\displaystyle \sum _{j\in s_i}}{h}_{j|i}{x}_{ij}\mathrm{<mo>=</mo>}{X}_i\mathrm{.}\hfill \end{array}$$

The resulting calibration weights $h_{j|i}$ are given by

$$h_{j|i}=w_{j|i}\left\{1+{\left(X_i-{\widehat{X}}_i\right)}^{\prime }{\left({\displaystyle \sum _{j\in s_i}}{w}_{j|i}{x}_{ij}{x}_{ij}^{\prime }\right)}^{-1}{x}_{ij}\right\}\mathrm{,}j\in s_i\mathrm{,}(4.2)$$

provided that ${\displaystyle {\sum }_{j\in s_i}}{w}_{j|i}{x}_{ij}{x}_{ij}^{\prime }$ is non-singular. The calibration estimator of the domain total $Y_i$ is then given by

$${\widehat{Y}}_i^{\text{LCAL}}\mathrm{<mo>=</mo>}{\displaystyle \sum _{j\in s_i}}{h}_{j|i}{y}_{ij}\mathrm{<mo>=</mo>}{\widehat{Y}}_i+{\left(X_i-{\widehat{X}}_i\right)}^{\prime }{\widehat{B}}_i\mathrm{,}(4.3)$$

which is the well-known generalized regression (GREG) estimator of $Y_i,$ where

$${\widehat{B}}_i\mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{j\in s_i}}{w}_{j|i}{x}_{ij}{x}_{ij}^{\prime }\right)}^{-1}{\displaystyle \sum _{j\in s_i}}{w}_{j|i}{x}_{ij}{y}_{ij}\mathrm{.}$$

The Hájek estimator ${\widehat{Y}}_i^{\text{HA}}$ is a special case of (4.3), with $x_{ij}\mathrm{<mo>=</mo>}1,$ $j\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}{N}_i.$ In the absence of cut-off sampling, the above GREG estimator is design-consistent as the domain sample size $n_i$ increases, although it may suffer from small sample bias. It reduces the variance if the calibration variables are linearly correlated with the outcome and the correlation is strong. Under cut-off sampling, the second term on the right-hand side of (4.3) corrects for the bias of the basic expansion estimator ${\widehat{Y}}_i$ as estimator of $Y_i$ with the help of the known domain totals in $X_i.$ However, for small domain sample size $n_i,$ this reduction in cut-off sampling bias might be transferred to an increase in variance.

In the above procedure, we have a different calibration problem for each domain. In the case that only the overall population total $X\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^m}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^{N_i}}{x}_{ij}$ is available, we may seek calibration weights for all the domains at once, $g_{j|i},$ $j\in s_i,$ $i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}m,$ by solving only one calibration problem:

$$\begin{array}{l}\underset{\left\{g_{j|i}\mathrm{<mo>:</mo>}j\in s_i\text{}\mathrm{,}i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}m\right\}}{\text{min}}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}{\displaystyle \sum _{j\in s_i}}{\left(g_{j|i}-w_{j|i}\right)}^2/w_{j|i}(4.4)\hfill \\ \text{s}\text{.t}\text{.}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}{\displaystyle \sum _{j\in s_i}}{g}_{j|i}{x}_{ij}\mathrm{<mo>=</mo>}X\mathrm{.}\hfill \end{array}$$

In this case, the calibration weights $g_{j|i}$ are given by

$$g_{j|i}\mathrm{<mo>=</mo>}{w}_{j|i}\left\{1+{\left(X-\widehat{X}\right)}^{\prime }{\left({\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^m}{\displaystyle \sum _{j\in s_i}}{w}_{j|i}{x}_{ij}{x}_{ij}^{\prime }\right)}^{-1}{x}_{ij}\right\}\mathrm{,}j\in s_i\mathrm{,}i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}m\mathrm{,}(4.5)$$

provided that ${\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^m}{\displaystyle {\sum }_{j\in s_i}}{w}_{j|i}{x}_{ij}{x}_{ij}^{\prime }$ is non-singular. The resulting calibration estimator of the domain total $Y_i$ is then obtained as

$${\widehat{Y}}_i^{\text{LCALN}}\mathrm{<mo>=</mo>}{\displaystyle \sum _{j\in s_i}}{g}_{j|i}{y}_{ij}\mathrm{<mo>=</mo>}{\widehat{Y}}_i+{\left(X-\widehat{X}\right)}^{\prime }{\widehat{B}}_i^N\text{}\mathrm{,}(4.6)$$

where

$${\widehat{B}}_i^N\mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{l=1}^m}{\displaystyle \sum _{j\in s_l}}{w}_{j|l}{x}_{lj}{x}_{lj}^{\prime }\right)}^{-1}{\displaystyle \sum _{j\in s_i}}{w}_{j|i}{x}_{ij}{y}_{ij}\mathrm{.}$$

In contrast with the GREG estimator, the correction of ${\widehat{Y}}_i$ in ${\widehat{Y}}_i^{\text{LCALN}}$ uses the overall population total $X$ and its corresponding expansion estimator.

The LCAL (or GREG) estimator (4.3) is expected to have smaller cut-off sampling bias than (4.6) because it uses auxiliary information from each particular domain $i.$ On the other hand, for domains with small sample sizes $n_i,$ its variance (and small sample bias) may be large since it uses only domain-specific data. The alternative calibration estimator given in (4.6) is expected to have slightly larger cut-off sampling bias because it uses only aggregated auxiliary information at the national level, but its design-variance is expected to be smaller. We now study the properties of (4.3). To this end, consider the theoretical version of LCAL estimator (4.3), given by

$${\tilde{Y}}_i^{\text{LCAL}}\mathrm{<mo>=</mo>}{\widehat{Y}}_i+{\left(X_i-{\widehat{X}}_i\right)}^{\prime }{B}_{iI}\mathrm{.}(4.7)$$

Here, $B_{iI}\mathrm{<mo>=</mo>}{\left({\displaystyle {\sum }_{j\in U_{iI}}}{x}_{ij}{x}_{ij}^{\prime }\right)}^{-1}$ ${\displaystyle {\sum }_{j\in U_{iI}}}{x}_{ij}{y}_{ij}$ is the census version of ${\widehat{B}}_i$ based on the set of included units from domain $i.$ Note that the sample $s_i$ is drawn only from $U_{iI}$ and thus ${\widehat{B}}_i$ estimates $B_{iI}.$ We decompose the bias of ${\widehat{Y}}_i^{\text{LCAL}}$ as

$$\begin{array}{ll}{B}_{\pi }\left({\widehat{Y}}_i^{\text{LCAL}}\right)\hfill & =E_{\pi }\left({\widehat{Y}}_i^{\text{LCAL}}-{\tilde{Y}}_i^{\text{LCAL}}\right)+B_{\pi }\left({\tilde{Y}}_i^{\text{LCAL}}\right)\mathrm{,}\hfill \\ \hfill & =E_{\pi }\left\{{\left(X_i-{\widehat{X}}_i\right)}^{\prime }\left({\widehat{B}}_i-B_{iI}\right)\right\}+B_{\pi }\left({\tilde{Y}}_i^{\text{LCAL}}\right)\mathrm{.}(4.8)\hfill \end{array}$$

The term $E_{\pi }\left\{\left(X_i-{\widehat{X}}_i\right)\left({\widehat{B}}_i-B_{iI}\right)\right\}/N_i$ tends to zero as $n_i\to \infty$ regardless of whether cut-off sampling is applied or not, since ${\widehat{B}}_i$ tends to $B_{iI}.$ However, for small $n_i$ this term may not be negligible; that is, the LCAL estimator has small sample bias even if $U_{iE}\mathrm{<mo>=</mo>}\varnothing .$ In the absence of cut-off sampling, the bias term $B_{\pi }\left({\tilde{Y}}_i^{\text{LCAL}}\right)$ in (4.8) is exactly equal to zero. Under cut-off sampling, we know that $E_{\pi }\left({\widehat{Y}}_i\right)\mathrm{<mo>=</mo>}{Y}_{iI}$ and $E_{\pi }\left({\widehat{X}}_i\right)\mathrm{<mo>=</mo>}{X}_{iI},$ where $X_{iI}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{j\in U_{iI}}}{x}_{ij}.$ Noting that $X_i-X_{iI}\mathrm{<mo>=</mo>}{X}_{iE},$ for $X_{iE}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{j\in U_{iE}}}{x}_{ij},$ we obtain the design-bias of this LCAL theoretical estimator, given in absolute and relative terms by

$$B_{\pi }\left({\tilde{Y}}_i^{\text{LCAL}}\right)\mathrm{<mo>=</mo>}-N_{iE}\left({\overline{Y}}_{iE}-{\overline{X}}_{iE}^{\prime }{B}_{iI}\right)\mathrm{,}{\text{RB}}_{\pi }\left({\tilde{Y}}_i^{\text{LCAL}}\right)\mathrm{<mo>=</mo>}-\frac{N_{iE}}{N_i}\frac{{\overline{Y}}_{iE}-{\overline{X}}_{iE}^{\prime }{B}_{iI}}{{\overline{Y}}_i}\mathrm{.}(4.9)$$

This bias is small when the same model holds for the included and excluded individuals.

Since the calibration estimator ${\widehat{Y}}_i^{\text{LCAL}}$ is intended to estimate $Y_i$ (and not $Y_{iI}),$ for the domain mean ${\overline{Y}}_i\mathrm{<mo>=</mo>}{Y}_i/N_i$ we consider the estimator obtained simply dividing ${\widehat{Y}}_i^{\text{CAL}}$ by $N_i$ (instead of $N_{iI}),$ ${\widehat{\overline{Y}}}_i^{\text{LCAL}}\mathrm{<mo>=</mo>}{\widehat{Y}}_i^{\text{LCAL}}/N_i.$ The asymptotic bias of ${\widehat{\overline{Y}}}_i^{\text{LCAL}}$ is given by (4.9) divided by $N_i.$

We now analyze properties under the model and the sampling replication mechanism. Note that ${\widehat{B}}_i$ in the GREG estimator is the weighted least squares (WLS) estimator of the vector of regression coefficients ${\beta }_i$ in the following linear regression model for the units in domain $i:$

$$y_{ij}\mathrm{<mo>=</mo>}{x}_{ij}^{\prime }{\beta }_i+{\epsilon }_{ij}\mathrm{,}{E}_m\left({\epsilon }_{ij}\right)\mathrm{<mo>=</mo>}\mathrm{0,}{E}_m\left({\epsilon }_{ij}^2\right)\mathrm{<mo>=</mo>}{\sigma }_{\epsilon }^2\mathrm{,}j\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}{N}_i\mathrm{,}(4.10)$$

where model errors ${\epsilon }_{ij}$ are all mutually independent. We wish to see the value added by the model to the design properties of the estimators; that is, how much would be gained if data were actually generated (at least approximately) by the assumed model. Let $E_m$ denote expectation under model (4.10). If the linear regression model (4.10) actually holds for all the units in the domain (included and excluded), then $E_m\left(B_{iI}\right)\mathrm{<mo>=</mo>}{\beta }_i$ and taking expectation of the bias term in (4.9) under model (4.10), we obtain the model-design bias,

$$B_{m\mathrm{,}\pi }\left({\tilde{Y}}_i^{\text{LCAL}}\right)\mathrm{<mo>=</mo>}-N_{iE}\left\{E_m\left({\overline{Y}}_{iE}\right)-{\overline{X}}_{iE}^{\prime }{E}_m\left(B_{iI}\right)\right\}\mathrm{<mo>=</mo>}-N_{iE}\left({\overline{X}}_{iE}^{\prime }{\beta }_i-{\overline{X}}_{iE}^{\prime }{\beta }_i\right)\mathrm{<mo>=</mo>\; 0.}(4.11)$$

In contrast, assuming exactly the same regression model, the bias of the basic direct estimator ${\widehat{\overline{Y}}}_i^{\text{HA}}$ under cut-off sampling is not zero unless the means of the auxiliary variables for the excluded and included units are equal. Indeed,

$$B_{m\mathrm{,}\pi }\left({\widehat{Y}}_i^{\text{HA}}\right)\mathrm{<mo>=</mo>}{N}_{iE}{E}_m\left({\overline{Y}}_{iI}-{\overline{Y}}_{iE}\right)\mathrm{<mo>=</mo>}{N}_{iE}{\left({\overline{X}}_{iI}-{\overline{X}}_{iE}\right)}^{\prime }{\beta }_i\mathrm{.}(4.12)$$

Thus, the condition under which the LCAL estimator is design-unbiased, namely that the linear model (4.10) holds without error for all the units in the domain, is much weaker than the requirements for the basic direct estimator to be design-unbiased. This means that calibration estimators will tend to be less biased than the basic direct estimator and can reduce substantially the cut-off sampling bias if the outcome is generated by the above domain-specific linear regression model.

Turning now to LCALN estimator (4.6), we define the corresponding theoretical version

$${\tilde{Y}}_i^{\text{LCALN}}\mathrm{<mo>=</mo>}{\widehat{Y}}_i+{\left(X-\widehat{X}\right)}^{\prime }{B}_{iI}^N\mathrm{,}(4.13)$$

where $B_i^N$ is the census version for the included units,

$$B_i^N\mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{l\mathrm{<mo>=</mo>\; 1}}^m}{\displaystyle \sum _{j\in U_{lI}}}{x}_{lj}{x}_{lj}^{\prime }\right)}^{-1}{\displaystyle \sum _{j\in U_{iI}}}{x}_{ij}{y}_{ij}\mathrm{.}$$

Decomposing the bias similarly as in (4.8), we obtain

$$B_{\pi }\left({\widehat{Y}}_i^{\text{LCALN}}\right)\mathrm{<mo>=</mo>}{E}_{\pi }\left\{{\left(X-\widehat{X}\right)}^{\prime }\left({\widehat{B}}_i^N-B_{iI}^N\right)\right\}+B_{\pi }\left({\tilde{Y}}_i^{\text{LCALN}}\right)\mathrm{.}(4.14)$$

Again, $E_{\pi }\left\{\left(X-\widehat{X}\right)\left({\widehat{B}}_i^N-B_{iI}^N\right)\right\}/N_i$ is not zero for small $n_i$ but it tends to zero as $n_i\to \infty$ even under cut-off sampling, whereas $B_{\pi }\left({\tilde{Y}}_i^{\text{LCALN}}\right)\mathrm{<mo>=</mo>}0$ only in the absence of cut-off sampling bias. In general, using the decomposition $X\mathrm{<mo>=</mo>}{X}_I+X_E,$ where $X_I$ and $X_E$ are the national totals for the included and excluded units respectively, the design bias of ${\tilde{Y}}_i^{\text{LCALN}}$ is given by

$$B_{\pi }\left({\tilde{Y}}_i^{\text{LCALN}}\right)\mathrm{<mo>=</mo>}-\left(Y_{iE}-X_E^{\prime }{B}_{iI}^N\right)\mathrm{.}(4.15)$$

Consider now the linear model with constant regression coefficients for all the population units, called model $m_2\text{:}$

$$y_{ij}\mathrm{<mo>=</mo>}{x}_{ij}^{\prime }\beta +{\epsilon }_{ij}\mathrm{,}{E}_{m_2}\text{}\left({\epsilon }_{ij}\right)\mathrm{<mo>=</mo>}\mathrm{0,}{E}_{m_2}\text{}\left({\epsilon }_{ij}^2\right)\mathrm{<mo>=</mo>}{\sigma }_{\epsilon }^2\mathrm{,}j\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}{N}_i\mathrm{,}i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}m\mathrm{,}(4.16)$$

where again the model errors ${\epsilon }_{ij}$ are mutually independent. Note that, under this model, $E_{m_2}\left(B_{iI}^N\right)\ne \beta$ in general, but if we consider the sum $B_I\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^m}{B}_{iI}^N$ instead, we have $E_{m_2}\text{}\left(B_I\right)\mathrm{<mo>=</mo>}\beta .$ This means that the theoretical LCALN estimator for a particular domain, ${\tilde{Y}}_i^{\text{LCALN}},$ is not model-design unbiased, because

$$B_{m_2\text{}\mathrm{,}\pi }\left({\tilde{Y}}_i^{\text{LCALN}}\right)\mathrm{<mo>=</mo>}-\left\{X_{iE}^{\prime }\beta -X_E^{\prime }{E}_{m_2}\text{}\left(B_{iI}^N\right)\right\}\mathrm{,}$$

is not necessarily equal to zero. However, the national estimator obtained adding those of the domains, ${\tilde{Y}}^{\text{LCALN}}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^m}{\tilde{Y}}_i^{\text{LCALN}}\mathrm{<mo>=</mo>}\widehat{Y}+{\left(X-\widehat{X}\right)}^{\prime }\text{}{B}_I,$ is actually model-design unbiased, because

$$B_{m_2\text{}\mathrm{,}\pi }\left(\text{}{\tilde{Y}}^{\text{LCALN}}\right)\mathrm{<mo>=</mo>}-\left\{X_E^{\prime }\beta -X_E^{\prime }{E}_{m_2}\left(B_I\right)\right\}\mathrm{<mo>=</mo>}0.$$

Hence, under model (4.16) with constant regression coefficients for all the population units, the LCALN estimator is not model-design unbiased for a particular domain, but it is unbiased when aggregating for all the domains, provided that the same model holds for the included and excluded units in all domains. For the mean ${\overline{Y}}_i,$ the bias of the theoretical estimator ${\tilde{\overline{Y}}}_i^{\text{LCALN}}\mathrm{<mo>=</mo>}{\tilde{Y}}_i^{\text{LCALN}}/N_i$ is given by (4.15) divided by $N_i.$

We now study the variances. For the theoretical LCAL estimator (4.7), the design-variance is given by

$$V_{\pi }\left({\tilde{Y}}_i^{\text{LCAL}}\right)\mathrm{<mo>=</mo>}{V}_{\pi }\left({\widehat{Y}}_i-{\widehat{X}}_i^{\prime }{B}_{iI}\right)\mathrm{<mo>=</mo>}{V}_{\pi }\left({\displaystyle \sum _{j\in s_i}}{w}_{j|i}{E}_{ij}\right)\mathrm{,}(4.17)$$

where $E_{ij}\mathrm{<mo>=</mo>}{y}_{ij}-x_{ij}^{\prime }{B}_{iI},$ $j\in U_{iI}.$ We can then apply the usual variance estimators for expansion type estimators. In the case of LCALN given in (4.13), the variance is given by

$$V_{\pi }\left({\tilde{Y}}_i^{\text{LCALN}}\right)\mathrm{<mo>=</mo>}{V}_{\pi }\left({\widehat{Y}}_i-{\widehat{X}}^{\prime }{B}_{iI}^N\right)\mathrm{.}$$

Note that $\widehat{X}$ is based on the $n$ sample units, whereas ${\widehat{X}}_i$ uses only the $n_i$ units in domain $i.$ As a consequence, the contribution of $\widehat{X}$ to the variance of LCALN should be much smaller than the contribution of ${\widehat{X}}_i$ in (4.17). This means that, provided that the domain and national regression lines are similar, the variance of LCALN estimator, obtained from the calibration at the national level, should be smaller than that of the domain-specific calibration estimator LCAL.

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