4 Linearization and approximation of variance

Anne Massiani

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We want to estimate the variance $\text{var}\left(\widehat{\Theta }\right)$ of an estimator $\widehat{\Theta }$ calculated on the sample of cross-sectional individuals $s_p^B$ with assigned weights $w_j.$ Lavallée (2002, pages 122-123) developed an asymptotic framework for a population surveyed indirectly. This framework lends itself to the use of linearization techniques (cf. Deville 1999) to obtain an approximation of the variance of a complex estimator calculated on a population surveyed indirectly. If $\widehat{\Theta }$ is the estimator of one of the inequality indexes selected by Eurostat, linearization techniques are used to make estimation of the variance of $\widehat{\Theta }$ equivalent to estimation of the variance of a total. The macros of Osier (2009) can be used for this purpose. Osier's linearization formulas are reviewed in Appendix A with respect to the four indicators considered in the numerical application in Section 7. Let ${\ell }_j$ denote the linearized values of $\widehat{\Theta }.$ We then have:

$\text{var}\left(\widehat{\Theta }\right)\simeq \text{var}\left({\displaystyle \sum _{j\in s_p^B}}{w}_j{\ell }_j\right)\mathrm{.}$(4.1)

By using the residuals of the regression of the variable of interest in relation to the calibration variables, we can take account of the calibration effect in the calculations of variance (cf. Deville and Särndal 1992). Since the calibration variables $x_k$ are defined at the household level, we first calculate the following for any household $k$ for the survey year:

${\ell }_k={\displaystyle \sum _{j\in m_k}}{\ell }_j\mathrm{,}$

where $m_k$ designates all the members of household $k,$ then we define $e_k={\ell }_k-x_k^T\beta ,$ with the parameter $\beta$ being calculated here based on all the households present in the population. We then have:

$\text{var}\left(\widehat{\Theta }\right)\simeq \text{var}\left({\displaystyle \sum _{j\in s_p^B}}{w}_j{\ell }_j\right)=\text{var}\left({\displaystyle \sum _{k\in s_m^B}}{w}_k{\ell }_k\right)\simeq \text{var}\left({\displaystyle \sum _{k\in s_m^B}}{w}_k^u{e}_k\right)\mathrm{.}$(4.2)

Note 2: The linearized values ${\ell }_j$ introduce quantities that are calculated for the entire population, as may be seen, for example, in formula (A.6) in Appendix A. In accordance with the usual practice, the linearized values ${\ell }_j$ will ultimately be replaced by estimates ${\widehat{\ell }}_j.$ Similarly, since the quantities $e_k$ are unknown, they will be replaced by estimates

${\widehat{e}}_k={\widehat{\ell }}_k-x_k^T\widehat{\beta }\mathrm{,}$(4.3)

where

${\widehat{\ell }}_k={\displaystyle \sum _{j\in m_k}}{\widehat{\ell }}_j$

and

$\widehat{\beta }={\left({\displaystyle \sum _{k\in s_m^B}}{w}_k{x}_k{x}_k^T\right)}^{-1}\left({\displaystyle \sum _{k\in s_m^B}}{w}_k{x}_k{\widehat{\ell }}_k\right)\mathrm{.}$

Finally, since the four samples $s_m^{B\mathrm{,}\tau },$ for $\tau =\mathrm{1,}\dots ,4,$ which comprise $s_m^B$ are reached through disjoint samples $s_m^{A_1\mathrm{,}{t}_{\tau }},$ they are not strictly independent. However, we make the approximation that these four samples are independent, since the probabilities of selection ${\pi }_k^{A_1}$ are very low. We also assume that the allocation factors ${\varphi }_k^{\tau }$ that appear in formula (3.5) are not random. If we assume, for any household $k:$

${e^{\prime }}_k={\varphi }_k^{\tau }\cdot e_k$(4.4)

and go back to (3.5), we can rewrite the amount that appears in the last member of (4.2) in the following form:

${\displaystyle \sum _{k\in s_m^B}}{w}_k^u{e}_k={\displaystyle \sum _{\tau =1}^4}{\displaystyle \sum _{k\in s_m^{B\mathrm{,}\tau }}}{w}_k^{nr}\left({\varphi }_k^{\tau }{e}_k\right)={\displaystyle \sum _{\tau =1}^4}\left({\displaystyle \sum _{k\in s_m^{B\mathrm{,}\tau }}}{w}_k^{nr}{e^{\prime }}_k\right)\mathrm{.}$(4.5)

This enables us to obtain the following approximation of the variance:

$\text{var}\left(\widehat{\Theta }\right)\simeq {\displaystyle \sum _{\tau =1}^4\text{var(}{\widehat{T}}_{\tau }\text{)}}\mathrm{,}$(4.6)

where

${\widehat{T}}_{\tau }={\displaystyle \sum _{k\in s_m^{B\mathrm{,}\tau }}}{w}_k^{nr}{e^{\prime }}_k\mathrm{.}$(4.7)

The four components of variance that appear in formula (4.6) can be computed and estimated in the same way. In the next section, we give an estimator of the variance of ${\widehat{T}}_{\tau },$ for any $\tau .$

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