Linearization versus bootstrap for variance estimation of the change between Gini indexes
Section 5. Conclusion

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In this paper, we considered the estimation of the change between Gini indexes. We presented the class of composite estimators introduced by Goga, Deville and Ruiz-Gazen (2009), and studied more particularly the intersection estimator which makes use of the common sample only, and the union estimator which makes use of the whole available samples. We justified both heuristically and through the simulation study in Section 4.2 that the intersection estimator can be close to the optimal estimator, while the union estimator exhibits poor performances in all the scenarios considered. The intersection estimator is also easy to compute, while the optimal estimator involves unknown quantities which need to be estimated in practice. We therefore advocate for the use of the intersection estimator for estimating the change between Gini indexes.

We also compared linearization and bootstrap for variance estimation and for producing confidence intervals. In the scenarios that we considered in the simulation study, the linearization performed better with usually smaller relative biases for the variance estimator, and better coverage rates with normality-based confidence intervals than with percentile confidence intervals. Bootstrap $-t$ confidence intervals (not considered in the simulation study) would be a competitor of interest, but due to the intensive computational work involved, they are less attractive for a data user. Linearization has also the advantage to offer a unified approach suitable for any sampling design, while a specific sampling design usually requires a specific bootstrap procedure, as illustrated with the BWO for SI sampling and the BWR for multistage sampling.

From the simulation study, we note that the coverage rates may not be well respected neither with linearization nor bootstrap, particularly in the multistage context and even with large sample sizes. There is a need for confidence intervals with better coverage rates under a reasonable computational burden. This is a matter for further research.

Acknowledgements

We thank Anne Ruiz-Gazen for helpful discussion. We also thank two referees and an Associate Editor for useful comments and suggestions which led to a significant improvement of the paper.

Appendix

Proof of equation (3.6)

From (3.3), we have ${\widehat{\Delta t}}^{\text{co}}=N\left(A^{\top }X\right),$ where $X$$=$$\left({\overline{y}}_{\mathrm{2,}{s}_{2•}}-{\overline{y}}_{\mathrm{2,}{s}_3}\text{}\mathrm{,}{\overline{y}}_{\mathrm{1,}{s}_{1•}}-{\overline{y}}_{\mathrm{1,}{s}_3}\text{}\mathrm{,}{\overline{y}}_{\mathrm{2,}{s}_3}-{\overline{y}}_{\mathrm{1,}{s}_3}\right)$${}_{}^{\top }$ and $A\mathrm{<mo>=</mo>}{\left(b\mathrm{,}-a\mathrm{,}1\right)}^{\top }\text{}.$ This leads to

$$V\left\{{\widehat{\Delta t}}^{\text{co}}\right\}\mathrm{<mo>=</mo>}{N}^2\left\{A^{\top }V\left(X\right)A\right\}\mathrm{.}(\text{A}.1)$$

We compute the elements in $V\left(X\right)$ separately. We have

$$\begin{array}{ll}V\left({\overline{y}}_{\mathrm{2,}{s}_3}-{\overline{y}}_{\mathrm{1,}{s}_3}\right)\hfill & \mathrm{<mo>=</mo>}\left(\frac{1}{n_3}-\frac{1}{N}\right)S_{y_2-y_1\text{}\mathrm{,}U}^2\hfill \\ \hfill & \mathrm{<mo>=</mo>}\left(\frac{1}{n_3}-\frac{1}{N}\right)\left(S_{y_2\text{}\mathrm{,}U}^2+S_{y_1\text{}\mathrm{,}U}^2-2S_{y_1{y}_2\text{}\mathrm{,}U}\right)\mathrm{.}\hfill \end{array}$$

Also, since $E\left({\overline{y}}_{\mathrm{2,}{s}_{2•}}-{\overline{y}}_{\mathrm{2,}{s}_3}|s_2\right)\mathrm{<mo>=</mo>\; 0},$ we have

$$\begin{array}{ll}V\left({\overline{y}}_{\mathrm{2,}{s}_{2•}}-{\overline{y}}_{\mathrm{2,}{s}_3}\right)\hfill & \mathrm{<mo>=</mo>}\text{EV}\left({\overline{y}}_{\mathrm{2,}{s}_{2•}}-{\overline{y}}_{\mathrm{2,}{s}_3}|s_2\right)\mathrm{,}\hfill \\ \hfill & \mathrm{<mo>=</mo>}\text{EV}\left(\frac{n_2}{n_{2•}}{\overline{y}}_{\mathrm{2,}{s}_2}-\frac{n_3}{n_{2•}}{\overline{y}}_{\mathrm{2,}{s}_3}-{\overline{y}}_{\mathrm{2,}{s}_3}|s_2\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\left(1+\frac{n_3}{n_{2•}}\right)}^2\text{EV}\left({\overline{y}}_{\mathrm{2,}{s}_3}|s_2\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\left(1+\frac{n_3}{n_{2•}}\right)}^2\left(\frac{1}{n_3}-\frac{1}{n_2}\right)S_{y_2\text{}\mathrm{,}U}^2\hfill \\ \hfill & \mathrm{<mo>=</mo>}\frac{n_2}{n_3\left(n_2-n_3\right)}{S}_{y_2\text{}\mathrm{,}U}^2\hfill \end{array}$$

and

$$\begin{array}{ll}\text{Cov}\left({\overline{y}}_{\mathrm{2,}{s}_{2•}}-{\overline{y}}_{\mathrm{2,}{s}_3}\text{},{\overline{y}}_{\mathrm{2,}{s}_3}-{\overline{y}}_{\mathrm{1,}{s}_3}\right)\hfill & \mathrm{<mo>=</mo>}\text{ECov}\left({\overline{y}}_{\mathrm{2,}{s}_{2•}}-{\overline{y}}_{\mathrm{2,}{s}_3}\text{},{\overline{y}}_{\mathrm{2,}{s}_3}-{\overline{y}}_{\mathrm{1,}{s}_3}|s_2\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}\text{ECov}\left(\frac{n_2}{n_{2•}}{\overline{y}}_{\mathrm{2,}{s}_2}-\frac{n_3}{n_{2•}}{\overline{y}}_{\mathrm{2,}{s}_3}-{\overline{y}}_{\mathrm{2,}{s}_3}\text{},{\overline{y}}_{\mathrm{2,}{s}_3}-{\overline{y}}_{\mathrm{1,}{s}_3}|s_2\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}-\left(1+\frac{n_3}{n_{2•}}\right)\text{ECov}\left({\overline{y}}_{\mathrm{2,}{s}_3}\text{},{\overline{y}}_{\mathrm{2,}{s}_3}-{\overline{y}}_{\mathrm{1,}{s}_3}\text{}|s_2\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}\left(1+\frac{n_3}{n_{2•}}\right)\left(\frac{1}{n_3}-\frac{1}{n_2}\right)\left(S_{y_2\text{}\mathrm{,}U}^2-S_{y_1{y}_2\text{}\mathrm{,}U}\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}-\frac{1}{n_3}\left(S_{y_2\text{}\mathrm{,}U}^2-S_{y_1{y}_2\text{}\mathrm{,}U}\right).\hfill \end{array}$$

Similar arguments lead to

$$\begin{array}{ccc}V\left({\overline{y}}_{1,s_{1•}}-{\overline{y}}_{1,s_3}\right)& =& \frac{n_1}{n_3\left(n_1-n_3\right)}{S}_{y_1\text{}\mathrm{,}U\text{}\text{}}^2,\text{}\text{}\text{}\\ \text{Cov}\left({\overline{y}}_{1,s_{1•}}-{\overline{y}}_{1,s_3}\text{},{\overline{y}}_{2,s_3}-{\overline{y}}_{1,s_3}\right)& =& \frac{1}{n_3}\left(S_{y_1\text{}\mathrm{,}U}^2-S_{y_1{y}_2\text{}\mathrm{,}U}\right).\\ & & \end{array}$$

Finally, we consider $\text{Cov}\left({\overline{y}}_{2,s_{2•}}-{\overline{y}}_{2,s_3}\text{},{\overline{y}}_{1,s_{1•}}-{\overline{y}}_{1,s_3}\right).$ We first compute $\text{Cov}\left({\overline{y}}_{2,s_{2•}}\text{},{\overline{y}}_{1,s_{1•}}\right),$ which may be written as

$$\begin{array}{ll}\text{Cov}\left({\overline{y}}_{\mathrm{2,}{s}_{2•}}\text{}\mathrm{,}{\overline{y}}_{\mathrm{1,}{s}_{1•}}\right)\hfill & \mathrm{<mo>=</mo>}\text{Cov}\left(E\left({\overline{y}}_{\mathrm{2,}{s}_{2•}}|s_{1•}\right)\mathrm{,}E\left({\overline{y}}_{\mathrm{1,}{s}_{1•}}|s_{1•}\right)\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}\text{Cov}\left({\overline{y}}_{\mathrm{2,}U\backslash s_{1•}}\text{}\mathrm{,}{\overline{y}}_{\mathrm{1,}{s}_{1•}}\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}\text{Cov}\left(\frac{N}{N-n_{1•}}{\overline{y}}_{\mathrm{2,}U}-\frac{n_{1•}}{N-n_{1•}}{\overline{y}}_{\mathrm{2,}{s}_{1•}}\text{}\mathrm{,}{\overline{y}}_{\mathrm{1,}{s}_{1•}}\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}-\frac{n_{1•}}{N-n_{1•}}\text{Cov}\left({\overline{y}}_{\mathrm{2,}{s}_{1•}}\mathrm{,}{\overline{y}}_{\mathrm{1,}{s}_{1•}}\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}-\frac{n_{1•}}{N-n_{1•}}\left(\frac{1}{n_{1•}}-\frac{1}{N}\right)S_{y_1{y}_2\text{}\mathrm{,}U}\hfill \\ \hfill & \mathrm{<mo>=</mo>}-\frac{1}{N}{S}_{y_1{y}_2\text{}\mathrm{,}U}\text{}\mathrm{.}\hfill \end{array}$$

Similar arguments lead to

$$\text{Cov}\left({\overline{y}}_{\mathrm{2,}{s}_{2•}}\text{}\mathrm{,}{\overline{y}}_{\mathrm{1,}{s}_3}\right)\mathrm{<mo>=</mo>}\text{Cov}\left({\overline{y}}_{\mathrm{2,}{s}_3}\text{}\mathrm{,}{\overline{y}}_{\mathrm{1,}{s}_{1•}}\right)\mathrm{<mo>=</mo>}-\frac{1}{N}{S}_{y_1{y}_2\text{}\mathrm{,}U}\mathrm{.}$$

We obtain

$$\begin{array}{ll}\text{Cov}\left({\overline{y}}_{\mathrm{2,}{s}_{2•}}-{\overline{y}}_{\mathrm{2,}{s}_3}\text{}\mathrm{,}{\overline{y}}_{\mathrm{1,}{s}_{1•}}-{\overline{y}}_{\mathrm{1,}{s}_3}\right)\hfill & \mathrm{<mo>=</mo>}\frac{1}{N}{S}_{y_1{y}_2\text{}\mathrm{,}U}+\text{Cov}\left({\overline{y}}_{\mathrm{2,}{s}_3}\text{}\mathrm{,}{\overline{y}}_{\mathrm{1,}{s}_3}\right)\hfill \\ \hfill & \mathrm{<mo>=</mo>}\frac{1}{N}{S}_{y_1{y}_2\text{}\mathrm{,}U}+\left(\frac{1}{n_3}-\frac{1}{N}\right)S_{y_1{y}_2\text{}\mathrm{,}U}\hfill \\ \hfill & \mathrm{<mo>=</mo>}\frac{1}{n_3}{S}_{y_1{y}_2\text{}\mathrm{,}U}\text{}\mathrm{.}\hfill \end{array}$$

In summary, we obtain

$$V\left(X\right)\mathrm{<mo>=</mo>}\left(\begin{array}{ccc}\frac{n_2}{n_3\left(n_2-n_3\right)}{S}_{y_2\text{}\mathrm{,}U}^2& \frac{1}{n_3}{S}_{y_1{y}_2\text{}\mathrm{,}U}& -\frac{1}{n_3}\left(S_{y_2\text{}\mathrm{,}U}^2-S_{y_1{y}_2\text{}\mathrm{,}U}\right)\\ & \frac{n_1}{n_3\left(n_1-n_3\right)}{S}_{y_1\text{}\mathrm{,}U}^2& \frac{1}{n_3}\left(S_{y_1\text{}\mathrm{,}U}^2-S_{y_1{y}_2\text{}\mathrm{,}U}\right)\\ & & \left(\frac{1}{n_3}-\frac{1}{N}\right)\left(S_{y_2\text{}\mathrm{,}U}^2+S_{y_1\text{}\mathrm{,}U}^2-2S_{y_1{y}_2\text{}\mathrm{,}U}\right)\end{array}\right)$$

which, along with (A.1), leads to (3.6).

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