Linearization versus bootstrap for variance estimation of the change between Gini indexes
Section 4. Simulation study

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In this section, five artificial populations are first generated as described in Section 4.1. In Section 4.2, the union estimator is compared with the intersection estimator in terms of asymptotic variance. A Monte Carlo experiment is then presented in Section 4.3, and the performances of the linearization and the bootstrap are compared in case of a SI2 sampling design. A similar comparison is made in Section 4.4, in case of the bi-dimensional two-stage sampling design.

4.1  Simulation set-up

We generated 5 finite populations of size $N\mathrm{<mo>=</mo>}$40,000, each containing two study variables $y_1$ and $y_2.$ The $y_{1k}$ values and the $y_{2k}$ values were generated according to the lognormal model

$$y_{dk}\mathrm{<mo>=</mo>}\exp \left({\alpha }_d\mathrm{<mi>\; </mi>}{\epsilon }_k\right)\mathrm{.}(4.1)$$

The ${\epsilon }_k’\text{s}$ were generated according to a standard normal distribution. The values of the Gini coefficients for the five populations are presented in Table 4.1.

In each of the 5 populations, the units were grouped into $M\mathrm{<mo>=</mo>}$500 clusters of equal size $N_0\mathrm{<mo>=</mo>}$80. The clusters were built so that the intra-cluster correlation coefficient with respect to the variable $y_1$ was approximately equal to 0.20 in each population.

4.2  Comparison of the union estimator and of the intersection estimator

In this section, we compare the union estimator with the intersection estimator for the change between Gini indexes in terms of asymptotic variance. We consider two sampling designs: the SI2 design presented in Section 3.1.1 with $\left(n_{1•}\text{}\mathrm{,}{n}_3\text{}\mathrm{,}{n}_{2•}\right)\mathrm{<mo>=</mo>}$ (1,000; 1,000; 1,000), (1,000; 2,000; 1,000) or (1,000; 4,000; 1,000); the MULT2 design presented in Section 3.1.2 with $m\mathrm{<mo>=</mo>}$  300 and $\left(n_{1•}^i\text{}\mathrm{,}{n}_3^i\text{}\mathrm{,}{n}_{2•}^i\right)\mathrm{<mo>=</mo>}$ (10; 10; 10), (10; 20; 10) or (10; 40; 10).

For each population, we compute the asymptotic variance $V_{\text{lin}}\left({\widehat{\Delta G}}^{\text{uni}}\right)$ of the union estimator, and the asymptotic variance $V_{\text{lin}}\left({\widehat{\Delta G}}^{\text{int}}\right)$ of the intersection estimator. So as to compare them, we compute the relative efficiency defined as

$$\text{RE}\left\{{\widehat{\Delta G}}^{\cdot }\right\}\mathrm{<mo>=</mo>}\frac{V_{\text{lin}}\left\{{\widehat{\Delta G}}^{(\cdot )}\right\}}{V_{\text{lin}}\left\{{\widehat{\Delta G}}^{\text{opt}}\right\}}\mathrm{,}(4.2)$$

with ${\widehat{\Delta G}}^{\text{opt}}$ the optimal estimator.

The results are presented in Table 4.2. The union estimator is highly inefficient. Its asymptotic variance is 15 to 244 times higher than that of the intersection estimator for SI2, and 2 to 44 times higher than that of the intersection estimator for MULT2. The difference between both estimators tends to decrease when the sample size of the common sample increases and/or when $\Delta G$ increases. On the other hand, the intersection estimator is slightly less efficient than the optimal estimator for SI2, with RE ranging from 1.33 to 2.46, and approximately as efficient as the optimal estimator for MULT2, with RE ranging from 1.02 to 1.12. This supports the heuristic reasoning in Section 3.1.1. In view of the poor performance of the union estimator, and of the good performance of the intersection estimator, we confine our attention to the latter in the remainder of the simulation study.

4.3  Comparison of linearization and bootstrap for the SI2 design

In this section, we compare the linearization and bootstrap for variance estimation and for producing confidence intervals, in case of the intersection estimator for the change between Gini indexes under the SI2 sampling design. From each population, we selected $B\mathrm{<mo>=</mo>}$  10,000 two-dimensional samples by means of the SI2 design indexed by $\left(n_{1•}\text{}\mathrm{,}{n}_3\text{}\mathrm{,}{n}_{2•}\right)\mathrm{<mo>=</mo>}$ (1,000; 1,000; 1,000), $\left(n_{1•}\text{}\mathrm{,}{n}_3\text{}\mathrm{,}{n}_{2•}\right)\mathrm{<mo>=</mo>}$ (1,000; 2,000; 1,000) or $\left(n_{1•}\text{}\mathrm{,}{n}_3\text{}\mathrm{,}{n}_{2•}\right)\mathrm{<mo>=</mo>}$ (1,000; 4,000; 1,000). In each sample, we computed the intersection estimator ${\widehat{\Delta G}}^{\text{int}}$ of the change between Gini indexes. For this estimator, we computed (i) the linearization variance estimator $v_{\text{int}}\left({\widehat{\Delta G}}^{\text{int}}\right)$ given in (3.24), and (ii) the Bootstrap variance estimator $v_{\text{BWO}}\left(\widehat{\Delta G}\right),$ following the Bootstrap procedure described in Section 3.4.1.

To measure the bias of a variance estimator $v\left(\widehat{\Delta G}\right),$ we used the Monte Carlo Percent Relative Bias

$$\text{RB}\left\{v\left(\widehat{\Delta G}\right)\right\}\mathrm{<mo>=</mo>}100\times \frac{B^{-1}{\displaystyle {\sum }_{b\mathrm{<mo>=</mo>\; 1}}^{B}v\left({\widehat{\Delta G}}_b\right)}-\text{MSE}\left(\widehat{\Delta G}\right)}{\text{MSE}\left(\widehat{\Delta G}\right)}\mathrm{,}(4.3)$$

where $v\left({\widehat{\Delta G}}_b\right)$ denotes the estimator $v\left(\widehat{\Delta G}\right)$ in the $b^{\text{th}}$ sample, and $\text{MSE}\left(\widehat{\Delta G}\right)$ is a simulation-based approximation of the true mean square error of $\widehat{\Delta G},$ obtained from an independent run of 100,000 simulations. As a measure of stability of $v\left(\widehat{\Delta G}\right),$ we used the Relative Stability

$$\text{RS}\left\{v\left(\widehat{\Delta G}\right)\right\}\mathrm{<mo>=</mo>}\frac{{\left[B^{-1}{{\displaystyle {\sum }_{b\mathrm{<mo>=</mo>\; 1}}^B\left\{v\left(\widehat{\Delta G}\right)-\text{MSE}\left(\widehat{\Delta G}\right)\right\}}}^2\right]}^{1/2}}{\text{MSE}\left(\widehat{\Delta G}\right)}\mathrm{.}(4.4)$$

Finally, we compared the coverage rates of (i) the normality-based confidence interval with use of the linearization variance estimator and (ii) the confidence interval associated to the percentile Bootstrap. The bootstrap variance estimators and the bootstrap confidence intervals are based on $C\mathrm{<mo>=</mo>}$1,000 bootstrap replications. Error rates of the confidence intervals (with nominal one-tailed error rate of 2.5% in each tail) are compared. The comparison with nominal error rate of 5% gave no qualitative difference and is thus omitted.

The results are presented in Table 4.3. Both variance estimators are negatively biased. This bias is moderate (less than 5% ) in most cases, except for the smaller sample size $n\mathrm{<mo>=</mo>}$1,000, and for the population $U_5$ with the highest value of $\Delta G.$ The bootstrap variance estimator is systematically slightly more biased than the linearization variance estimator, but the difference decreases as the sample size increases. For both variance estimators, the instability increases with $\Delta G.$ The Bootstrap variance estimator is slightly more stable for the smaller sample size $n\mathrm{<mo>=</mo>}$1,000, but the situation is reversed when the sample size increases. Turning to the coverage of the confidence intervals, both methods lead to under-coverage which is consistent with the negative bias of both variance estimators. The normality-based confidence intervals show a slightly better coverage than the bootstrap percentile confidence intervals. For both confidence intervals, the under-coverage is more acute when $\Delta G$ increases, and reduces when the sample size increases.

4.4  Comparison of linearization and bootstrap for the MULT2 design

In this section, we compare the linearization and bootstrap for variance estimation and for producing confidence intervals, in case of the intersection estimator for the change between Gini indexes under the MULT2 sampling design presented in Section 3.1.2. From each population, we selected $B\mathrm{<mo>=</mo>}$10,000 two-dimensional two-stage samples by means of the MULT2 design indexed by $m\mathrm{<mo>=</mo>}$  300 and $\left(n_{1•}^i\text{}\mathrm{,}{n}_3^i\text{}\mathrm{,}{n}_{2•}^i\right)\mathrm{<mo>=</mo>}$ (10; 10; 10), (10; 20; 10) or (10; 40; 10). In each sample, we computed the intersection estimator ${\widehat{\Delta G}}^{\text{int}}$ of the change between Gini indexes. For this estimator, we computed (i) the linearization variance estimator $v^{\text{HH}}\left\{{\widehat{\Delta G}}^{\text{co}}\left(a\mathrm{,}b\right)\right\}$ given in (3.26), and (ii) the Bootstrap variance estimator $v_{\text{BWR}}\left({\widehat{\Delta G}}^{\text{int}}\right),$ following the Bootstrap procedure described in Section 3.4.2.

To measure the bias of a variance estimator $v\left(\widehat{\Delta G}\right),$ we used the Monte Carlo Percent Relative Bias defined in equation (4.3), and the Relative Stability defined in equation (4.4). The true mean square error of $\widehat{\Delta G}$ was obtained from an independent run of 100,000 simulations. Also, we compared the coverage rates of (i) the normality-based confidence interval with use of the linearization variance estimator and (ii) the confidence interval associated to the percentile Bootstrap. The bootstrap variance estimators and the bootstrap confidence intervals are based on $C\mathrm{<mo>=</mo>}$1,000 bootstrap replications. Error rates of the confidence intervals (with nominal one-tailed error rate of 2.5% in each tail) are compared. The comparison with nominal error rate of 5% gave no qualitative difference and is thus omitted.

The results are presented in Table 4.4. Both variance estimators are approximately unbiased for small values of $\Delta G,$ but show a moderate negative bias which increases with $\Delta G.$ The bootstrap variance estimator is more biased than the linearization variance estimator. For both variance estimators, the instability increases with $\Delta G.$ The Bootstrap variance estimator is slightly more stable than the linearization variance estimator. Both methods lead to an under-coverage which is consistent with the negative bias of both variance estimators. The normality-based confidence intervals perform slightly better. For both confidence intervals, the under-coverage is more acute when $\Delta G$ increases, and reduces when the sample size increases.

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