Criteria for choosing between calibration weighting and survey weighting
Section 2. Estimator of a variable of interest total

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$U=\left\{1,\dots ,N\right\}$ for a population size $N$ from which sample $s$ of size $n$ is selected based on survey design $p\left(s\right).$ $S$ designates a random variable such as $p\left(s\right)=P\left(S=s\right),$ and ${\pi }_k$ and ${\pi }_{kl}$ respectively designate the first and second probabilities of inclusion in survey design $p\left(s\right).$ We are interested in a variable of interest $Y={\left(y_1,\dots ,y_k,\dots ,y_N\right)}^{\prime },$ with the objective of estimating its total $t_y={\displaystyle {\sum }_{k\in U}{y}_k}.$ To do that, we consider the category of linear estimators ${\widehat{t}}_{yw}={\displaystyle {\sum }_{k\in S}{w}_{kS}{y}_k}$ where $w_{kS}$ are the weights that can depend on sample $S$ and the auxiliary variables available. The basic weights used are the sampling weights generated by $d_k=1/{\pi }_k.$ They correspond to the Horvitz-Thompson estimator ${\widehat{t}}_{y\pi }$ (1952).

It is assumed that we have $p$ auxiliary variables $X_1,\dots ,X_p,$ for which the values may be represented by vectors $x_k={\left(x_{k1},\dots ,x_{kp}\right)}^{\prime }$ and for which the vector of their totals $t_x={\displaystyle {\sum }_{k\in U}{x}_k}$ is known. The category of calibration estimators is defined by ${\widehat{t}}_{yC}={\displaystyle {\sum }_{k\in S}{w}_{kS\text{},C}{y}_k}$ where $w_{kS\text{},C},$ referred to as calibration weights, verify the calibration equation given by

$${\displaystyle {\sum }_{k\in S}{w}_{kS\text{},C}{x}_k}={\displaystyle {\sum }_{k\in U}{x}_k}.(2.1)$$

Calibration helps to reduce the variance of a total estimator, particularly for variables of interest that are linked to the auxiliary variables used in calibration. However, calibration results in an estimator with a bias other than zero. That is why the calibration weights are determined so that they are as close as possible to the sampling weights in order to manage bias.

2.1  Precision of a linear total estimator

In order to measure the precision of a linear total estimator, we will consider the design and model-based approach. In addition to the design distribution, this approach consists of assuming that values $y_1,\dots ,y_k\text{},\dots ,y_N$ for the variable of interest $Y$ are the product of a random vector ${\left(Y_1,\dots ,Y_k,\dots ,Y_N\right)}^{\prime }$ whose joint probability distribution is given by the Superpopulation model $\xi$ defined by:

$$Y_k=x_k^{\prime }\beta +{\epsilon }_k(2.2)$$

with

$$E_{\xi }\left({\epsilon }_k\right)=0,\text{V}\text{a}{\text{r}}_{\xi }\left({\epsilon }_k\right)={\sigma }_k^2{\text{ and Cov}}_{\xi }\left({\epsilon }_k,{\epsilon }_l\right)=0$$

where $\beta ={\left({\beta }_1,\dots ,{\beta }_p\right)}^{\prime }\text{},$ ${\sigma }_k^2$ $\left(k\in U\right)$ are unknown parameters. $E_{\xi },$ ${\text{Var}}_{\xi }$ and ${\text{Cov}}_{\xi }$ represent respectively the expectation, variance and covariance for the model. Vector estimator $\beta$ for the regression coefficients is produced by

$$\widehat{\beta }={\left(X_S^{\prime }{\Pi }_S^{-1}{V}_S^{-1}{X}_S\right)}^{-1}{X}_S^{\prime }{\Pi }_S^{-1}{V}_S^{-1}{Y}_S$$

where $X_S^{\prime }$ is the matrix of $x_k^{\prime }$ values for $k\in S,$ ${\Pi }_S=\text{diag}{\left({\pi }_k\right)}_{k\in S}$ and $V_S=\text{diag}{\left({\sigma }_k^2\right)}_{k\in S}\text{}.$ Under the the design and model-based approach, the criterion used to measure the precision of a linear total estimator is

$${\text{MSE}}_{p\xi }\left({\widehat{t}}_{yw}\right)=E_p{E}_{\xi }{\left({\widehat{t}}_{yw}-t_y\right)}^2(2.3)$$

which corresponds to the mean square error (MSE) for the design and model, also referred to as the anticipated mean square error (AMSE). This is based on the assumption that the design is not informative. We can then show that the AMSE for linear estimator ${\widehat{t}}_{yw}$ is (Nedyalkova and Tillé, 2008):

$${\text{MSE}}_{p\xi }\left({\widehat{t}}_{yw}\right)=E_p{\left({\displaystyle \sum _{k\in s}{w}_{kS}{x}_k^{\prime }\beta }-{\displaystyle \sum _{k\in U}{x}_k^{\prime }\beta }\right)}^2+{\displaystyle \sum _{k\in U}{\sigma }_k^2\left[{\mathrm{var}}_p\left(w_{kS}{I}_k\right)+{\left(R_{kS}-1\right)}^2\right]}(2.4)$$

where

$$R_{kS}=\frac{E\left(w_{kS}|I_k=1\right)}{d_k}$$

with $d_k=1/{\pi }_k$ (sampling weight) and $I_k=1$ for $k\in S$ and $I_k=0$ otherwise. Ratio $R_{kS}$ equals 1 when linear estimator ${\widehat{t}}_{yw}$ is unbiased according to the design.

2.2  AMSE for the calibration estimator

For the calibration estimator, verifying the calibration equation renders it unbiased under the model:

$$E_{\xi }\left({\widehat{t}}_{yC}-t_y\right)={\displaystyle \sum _{k\in S}{w}_{kS\text{},C}{x}_k^{\prime }\beta }-{\displaystyle \sum _{k\in U}{x}_k^{\prime }\beta }=0.$$

Consequently, the AMSE is expressed as:

$$\begin{array}{ll}{\text{MSE}}_{p\xi }\left({\widehat{t}}_{yC}\right)\hfill & ={\displaystyle \sum _{k\in U}{\sigma }_k^2\left[{\mathrm{var}}_p\left(w_{kS\text{},C}{I}_k\right)+{\left(R_k-1\right)}^2\right]}\hfill \\ \hfill & \text{=}{\displaystyle \sum _{k\in U}{\sigma }_k^2\left[\frac{V_k}{d_k}+R_k^2\left(d_k-1\right)+{\left(R_k-1\right)}^2\right](2.5)}\hfill \end{array}$$

where $V_k={\mathrm{var}}_p\left(w_{kS\text{},C}|I_k=1\right)$ and $R_k=E_p\left(w_{kS\text{},C}|I_k=1\right)/d_k\text{}.$

Giving

$$\begin{array}{ll}{\mathrm{var}}_p\left(w_{kS\text{},C}{I}_k\right)\hfill & =E_p\left[{\mathrm{var}}_p\left(w_{kS\text{},C}{I}_k|I_k\right)\right]+{\mathrm{var}}_p\left[E_p\left(w_{kS\text{},C}{I}_k|I_k\right)\right]\hfill \\ \hfill & ={\pi }_k{\mathrm{var}}_p\left(w_{kS\text{},C}|I_k=1\right)+{\pi }_k{\left[E_p\left(w_{kS\text{},C}|I_k=1\right)\right]}^2-{\left[E_p\left(w_{kS\text{},C}{I}_k\right)\right]}^2\hfill \\ \hfill & =\frac{V_k}{d_k}+R_k^2\left(d_k-1\right).(2.6)\hfill \end{array}$$

Note that the expression (2.5) of ${\text{MSE}}_{p\xi }\left({\widehat{t}}_{yC}\right)$ makes it possible to underscore the two criteria that determine the accuracy of calibration estimator ${\widehat{t}}_{yC}.$ The first corresponds to Superpopulation model $\xi$ through its residual variance ${\sigma }_k^2,$ which decreases when the variable of interest and the calibration variables are correlated (variance reduction ${\widehat{t}}_{yC}).$ The second criterion is represented by weight ratios $R_k,$ which become important when the calibration weights are very different from the sampling weights (bias increase ${\widehat{t}}_{yC}).$

2.3  AMSE for the HT estimator

In order to develop our criterion for choosing between calibration weighting and sample weighting, we need to determine the expression of the AMSE for the HT estimator. Since the latter is unbiased under the design $\left(R_{kS}=1\right),$ its AMSE is given by:

$$\begin{array}{ll}{\text{MSE}}_{p\xi }\left({\widehat{t}}_{y\pi }\right)\hfill & ={\mathrm{var}}_p\left({\displaystyle \sum _{k\in s}{d}_k{x}_k^{\prime }\beta }\right)+{\displaystyle \sum _{k\in U}{\sigma }_k^2{d}_k\left(1-{\pi }_k\right)}\hfill \\ \hfill & ={\displaystyle \sum _{k\in U}{\displaystyle \sum _{l\in U}\left({\pi }_{kl}-{\pi }_k{\pi }_l\right)d_k{x}_k^{\prime }\beta d_l{x}_l^{\prime }\beta }}+{\displaystyle \sum _{k\in U}{\sigma }_k^2{d}_k\left(1-{\pi }_k\right).(2.7)}\hfill \end{array}$$

It should be noted that the expression of the AMSE for ${\widehat{t}}_{y\pi }$ depends on probabilities ${\pi }_{kl},$ which are generally unknown and difficult to calculate for unequal probability sampling designs. Several approximations for these probabilities have been proposed in literature, enabling us to obtain several possible estimators for the variance of the HT estimator. However, Matei and Tillé (2005) showed, through a series of simulations, that these estimators are almost equivalent and allow us to effectively estimate the exact expression of the variance under design ${\widehat{t}}_{y\pi }.$

An approximation of ${\mathrm{var}}_p\left({\displaystyle {\sum }_{k\in s}{d}_k{x}_k^{\prime }\beta }\right)$ can be obtained by considering the one proposed by Hájek (1981) for the variance of the HT estimator, produced by:

$$V_{\text{Approx}}={\displaystyle \sum _{k\in U}{c}_k{\left(d_k{x}_k^{\prime }\beta \right)}^2}-\frac{1}{h}{\left({\displaystyle \sum _{k\in U}{c}_k{d}_k{x}_k^{\prime }\beta }\right)}^2(2.8)$$

where $h={\displaystyle {\sum }_{k\in U}{c}_k}$ and $c_k=N{\pi }_k\left(1-{\pi }_k\right)/\left(N-1\right).$ The latter is obtained from the following approximation of probabilities ${\pi }_{kl}$ (see Deville and Tillé, 2005; Tirari, 2003):

$${\pi }_{kl}-{\pi }_k{\pi }_l\approx \{\begin{array}{ll}{c}_k-\frac{c_k^2}{h}\hfill & \text{if}k=l\hfill \\ -\frac{c_k{c}_l}{h}\hfill & \text{if}k\ne l.\hfill \end{array}(2.9)$$

Consequently, the AMSE for ${\widehat{t}}_{y\pi }$ can be approximated by:

$${\widetilde{\text{MSE}}}_{p\xi }\left({\widehat{t}}_{y\pi }\right)=V_{\text{Approx}}+{\displaystyle \sum _{k\in U}{\sigma }_k^2{d}_k\left(1-{\pi }_k\right).}(2.10)$$

It should be noted that for simple designs, such as Poisson design or simple stratified random design, joint probability can be calculated precisely without the need for an approximation. In the next section, we will be basing calibration and HT estimators on the AMSE to develop a new measurement of the impact of using calibration weights.

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