Conclusion

1. At-risk-of-poverty rate

The at-risk-of-poverty threshold, ARPT, is calculated on the basis of the median income $\text{MED}$ of the population $U:$

$$\text{ARPT}=0.6\times \text{MED}\mathrm{.}\text{ (A.4)}$$

The at-risk-of-poverty rate, $\text{ARPR,}$ is defined as follows:

$$\text{ARPR}=\frac{1}{J}{\displaystyle \sum _{j\in U}{1}_{\{R_j\le \text{ARPT}\}}}\mathrm{.}\text{ (A.5)}$$

The linearized ${\ell }_j$ that appears in the approximation of the variance of the at-risk-of-poverty rate estimator, given in formula (4.1), is written as follows for each individual $j:$

$${\ell }_j=\frac{1}{J}\left[1_{\{R_j\le \text{ARPT}\}}-\text{ARPR}\right]-\frac{0.6}{J}\cdot \frac{{F^{\prime }}_K\left(\text{ARPT}\right)}{{F^{\prime }}_K\left(\text{MED}\right)}\cdot [1_{\{R_j\le \text{MED}\}}-0.5]\mathrm{.}\text{ (A.6)}$$

In Section 7, the poverty rate was also estimated within different sub-populations. Formula (A.6) is easily generalized in the case of sub-populations and is not reviewed here.

2. Quintile share ratio

Let ${\gamma }_{+}$ be the quantile of order 0.8 and let ${\gamma }_{-}$ be the quantile of order 0.2, and then let:

$$R={\displaystyle \sum _{j\in U}{R}_j}\text{ and }S\left(x\right)={\displaystyle \sum _{j\in U}{R}_j{1}_{\{R_j\le x\}}}\mathrm{.}\text{ (A.7)}$$

The quintile share ratio, denoted $\text{QSR},$ is defined as follows:

$$\text{QSR=}\frac{R-S\left({\gamma }_{+}\right)}{S\left({\gamma }_{-}\right)}\mathrm{.}\text{ (A.8)}$$

Similar to what was done for the function $F,$ we approach the function $S$ by the derivable function $S_K$ defined as:

$$S_K\left(x\right)={\displaystyle \int }S\left(z\right)K\left(x\mathrm{,}z\right)dz\mathrm{.}\text{ (A.9)}$$

The linearized ${\ell }_j$ that appears in the approximation of the variance of the quintile share ratio estimator, which was given in formula (4.1), is written as follows for each individual $j:$

$${\ell }_j=\frac{S\left({\gamma }_{-}\right)\left[R_j-Q_j\left({\gamma }_{+}\right)\right]-\left[R-S\left({\gamma }_{+}\right)\right]Q_j\left({\gamma }_{-}\right)}{{\left[S\left({\gamma }_{-}\right)\right]}^2}\mathrm{,}\text{ (A.10)}$$

where the quantity $Q_j\left(\gamma \right)$ is defined, for any quantile $\gamma$ of order $\alpha ,$ as follows:

$$Q_j\left(\gamma \right)=R_j{1}_{\{R_j\le \gamma \}}-\frac{{S^{\prime }}_K\left(\gamma \right)}{J{F}_K\left(\gamma \right)}[1_{\{R_j\le \gamma \}}-\alpha ]\mathrm{.}\text{ (A.11)}$$

3. Gini coefficient

The Gini coefficient, denoted $\text{G,}$ is defined as:

$$\text{G=}\frac{2M-R}{JR}-\mathrm{1,}\text{ (A.12)}$$

where

$$M={\displaystyle \sum _{j\in U}{R}_j}\left({\displaystyle \sum _{j^{\prime }\in U}{1}_{\{R_{j^{\prime }\le R_j}\}}}\right)\text{ (A.13)}$$

and $R$ is defined by (A.7). The linearized ${\ell }_j$ appearing in the approximation of the variance of the Gini coefficient estimator, given in formula (4.1), is written as follows for each individual $j:$

$${\ell }_j=\frac{JR\left(2U_j-R_j\right)-\left(2M-R\right)\left(R+J{R}_j\right)}{{\left(JR\right)}^2}\mathrm{,}\text{ (A.14)}$$

where the quantity $U_j$ is defined by

$$U_j={\displaystyle \sum _{j^{\prime }\in U}{R}_{j^{\prime }}{1}_{\{R_j\le R_{j^{\prime }}\}}}+R_j{\displaystyle \sum _{j^{\prime }\in U}{1}_{\{R_{j^{\prime }}\le R_j\}}}\mathrm{.}\text{ (A.15)}$$

4. Relative median at-risk-of-poverty gap

Relative median at-risk-of-poverty gap, denoted by $\text{RMPG},$ is defined by:

$$\text{RMPG=}\frac{\text{ARPT}-{\text{MED}}^p}{\text{ARPT}}\mathrm{,}\text{ (A.16)}$$

where as previously noted, ARPT designates the poverty threshold and where ${\text{MED}}^p$ is the median income calculated for individuals with an income below the poverty threshold. The linearized ${\ell }_j$ appearing in the approximation of the variance of the RMPG estimator, given in formula (4.1), is written as follows for each individual $j:$

$${\ell }_j=-\frac{\text{ARPT}\cdot Y_j-{\text{MED}}^p\cdot W_j}{{\left(\text{ARPT}\right)}^2}\mathrm{,}\text{ (A.17)}$$

where $W_j$ is defined by

$$W_j=-\frac{0.6}{{F^{\prime }}_K\left(\text{MED}\right)}\frac{1}{J}[1_{\{R_j\le \text{MED}\}}-0.5]\text{ (A.18)}$$

and $Y_j$ verifies the equation

$$\begin{array}{c}{F^{\prime }}_K\left({\text{MED}}^p\right)Y_j=\frac{1}{2}\left\{\frac{1}{J}\left[1_{\{R_j\le \text{ARPT}\}}-F\left(\text{ARPT}\right)\right]+{F^{\prime }}_K\left(\text{ARPT}\right)\cdot W_j\right\}\text{ (A.19)}\\ -\frac{1}{J}\left[1_{\{R_j\le {\text{MED}}^p\}}-F\left({\text{MED}}^p\right)\right]\mathrm{.}\end{array}$$

1. Definition of ${\widehat{\text{cov}}}_{j{j}^{\prime }}$

For all longitudinals $j$ and $j^{\prime }$ within $s_p^{A_2\mathrm{,}{t}_{\tau }}$ and belonging to households $k$ and $k^{\prime }$ respectively during the survey year, let

$${\widehat{\text{cov}}}_{j{j}^{\prime }}=\{\begin{array}{ll}0\hfill & \text{if }k\ne k^{\prime }\hfill \\ \frac{1}{{\left(L_k+P_k\right)}^2}{\left({e^{\prime }}_k\right)}^2\frac{1-q_k^b{q}_k^c}{{\left(q_k^b{q}_k^c\right)}^2}{1}_{\{k\in s_m^{B\mathrm{,}\tau }\}}\hfill & \text{if }k=k^{\prime }\mathrm{,}\hfill \end{array}\text{ (B.1)}$$

with the result that

$$\text{E}\left[{\widehat{\text{cov}}}_{j{j}^{\prime }}|s_p^{A_2\mathrm{,}{t}_{\tau }}\right]{\text{=cov}}_{j{j}^{\prime }}\mathrm{.}\text{ (B.2)}$$

2. Estimation of $V_{A_2}^{\tau }$

If the $Z_j$ values were known for $s_p^{A_2\mathrm{,}{t}_{\tau }},$ we could estimate $V_{A_2}^{\tau }$ without bias by ${\tilde{V}}_{A_2}^{\tau }\left(Z_1\mathrm{,}{Z}_2\mathrm{,}\dots \right).$ Since this is not the case, we estimate $V_{A_2}^{\tau }$ without bias by:

$${\widehat{V}}_{A_2}^{\tau }={\displaystyle \sum _{j\in s_p^{A_2\mathrm{,}{t}_{\tau }}}}{\displaystyle \sum _{j^{\prime }\in s_p^{A_2\mathrm{,}{t}_{\tau }}}}\frac{{\pi }_{j{j}^{\prime }}^{A_2}-{\pi }_j^{A_2}{\pi }_{j^{\prime }}^{A_2}}{{\pi }_j^{A_2}{\pi }_{j^{\prime }}^{A_2}}\frac{1}{{\pi }_{j{j}^{\prime }}^{A_2}}\left[{\widehat{Z}}_j{\widehat{Z}}_{j^{\prime }}-{\widehat{\text{cov}}}_{j{j}^{\prime }}\right]\mathrm{.}$$

Indeed, for all longitudinals $j$ and $j^{\prime }$ belonging to $s_p^{A_2\mathrm{,}{t}_{\tau }},$ we have:

$$\text{E}\left[{\widehat{Z}}_j{\widehat{Z}}_{j^{\prime }}-{\widehat{\text{cov}}}_{j{j}^{\prime }}|s_p^{A_2\mathrm{,}{t}_{\tau }}\right]=Z_j{Z}_{j^{\prime }}\mathrm{,}$$

which implies that

$$\text{E}\left({\widehat{V}}_{A_2}^{\tau }\right)={\text{E}}_{s_p^{A_2\mathrm{,}{t}_{\tau }}}\left[\text{E}\left({\widehat{V}}_{A_2}^{\tau }|s_p^{A_2\mathrm{,}{t}_{\tau }}\right)\right]={\text{E}}_{s_p^{A_2\mathrm{,}{t}_{\tau }}}\left[{\tilde{V}}_{A_2}^{\tau }\left(Z_1\mathrm{,}{Z}_2\mathrm{,}\dots \right)\right]=V_{A_2}^{\tau }\mathrm{.}$$

3. Estimation of $V_{\text{CNR}}^{\tau }$

We can estimate $V_{\text{CNR}}^{\tau }$ by:

$${\widehat{V}}_{\text{CNR}}^{\tau }={\displaystyle \sum _{j\in s_p^{A_2\mathrm{,}{t}_{\tau }}}}{\displaystyle \sum _{j^{\prime }\in s_p^{A_2\mathrm{,}{t}_{\tau }}}}\frac{1}{{\pi }_j^{A_2}}\frac{1}{{\pi }_{j^{\prime }}^{A_2}}{\widehat{\text{cov}}}_{j{j}^{\prime }}\mathrm{.}$$

4. Final formula

We estimate $\text{var}\left({\widehat{T}}_{\tau }\right)$ by:

$$\widehat{\text{var}}\left({\widehat{T}}_{\tau }\right)={\widehat{V}}_{A_2}^{\tau }+{\widehat{V}}_{\text{CNR}}^{\tau }\mathrm{.}$$

After a few simplifications, we have:

$$\widehat{\text{var}}\left({\widehat{T}}_{\tau }\right)={\tilde{V}}_{A_2}^{\tau }\left({\widehat{Z}}_1\mathrm{,}{\widehat{Z}}_2\mathrm{,}\dots \right)+{\displaystyle \sum _{j\in s_p^{A_2\mathrm{,}{t}_{\tau }}}}{\displaystyle \sum _{j^{\prime }\in s_p^{A_2\mathrm{,}{t}_{\tau }}}}\frac{1}{{\pi }_{j{j}^{\prime }}^{A_2}}{\widehat{\text{cov}}}_{j{j}^{\prime }}\mathrm{.}\text{ (B.3)}$$

Using (B.1), we can simplify the second term of the right member, so as to obtain formula (5.19) given in proposition 1.