Assessing the coverage of confidence intervals under nonresponse. A case study on income mean and quantiles in some municipalities from the 2015 Mexican Intercensal Survey
Section 4. Simulated populations

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In order to control the asymmetry of the distribution of $y,$ the correlation $\text{corr}\mathrm{<mo\; stretchy="false">(</mo>}y\mathrm{,}\varphi \mathrm{<mo\; stretchy="false">)</mo>}$ and the percentage of nonresponse, we simulated two symmetric and two asymmetric populations of size $N\mathrm{<mo>=</mo>}$50,000 with a variable of interest $y$ and six auxiliary variables $x_1\mathrm{,}\dots \mathrm{,}{x}_6,$ as follows.

1. 50,000 simple random samples for each of the variables $x_1\mathrm{,}\dots ,x_6$ were generated independently from a $N\mathrm{<mo\; stretchy="false">(</mo>0,}\mathrm{1<mo\; stretchy="false">)</mo>}$ distribution.
2. The response probabilities ${\varphi }_k$ were obtained using a logistic regression with ${\beta }_1\mathrm{<mo>=</mo>}\cdots \mathrm{<mo>=</mo>}{\beta }_6\mathrm{<mo>=</mo>}-\text{0}\text{.3}$ and ${\beta }_0$ chosen so that the average nonresponse was equal to 24.8%.
3. Two settings were considered for the distribution of $y:$
   • i) Symmetric. $y_k$ was generated from a $N\mathrm{<mo\; stretchy="false">(</mo>}\mu \mathrm{,}{\sigma }^2\mathrm{<mo\; stretchy="false">)</mo>},$ with $\mu \mathrm{<mo>=</mo>}1+$ 2.16 $\text{corr}\mathrm{<mo\; stretchy="false">(</mo>}y\mathrm{,}x\mathrm{<mo\; stretchy="false">)</mo>}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>1}}^6{x}_{kj}}$ and ${\sigma }^2\mathrm{<mo>=</mo>}$ 4.67 $*\mathrm{<mo\; stretchy="false">(</mo>1}-6{\text{corr}}^2\mathrm{<mo\; stretchy="false">(</mo>}y\mathrm{,}x\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>},$ where $\text{corr}\mathrm{<mo\; stretchy="false">(</mo>}y\mathrm{,}x\mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}\mathrm{corr}\mathrm{<mo\; stretchy="false">(</mo>}y\mathrm{,}{x}_j\mathrm{<mo\; stretchy="false">)</mo>,}j\in \mathrm{<mo>\{</mo>1,}\dots \mathrm{,}\mathrm{6<mo>\}</mo>.}$
   • ii) Asymmetric. $y_k\mathrm{<mo>=</mo>}\exp \mathrm{<mo\; stretchy="false">(</mo>}{z}_k\mathrm{<mo\; stretchy="false">)</mo>},$ where $z_k$ was generated from a $N\mathrm{<mo\; stretchy="false">(</mo>}\mu \mathrm{,}{\sigma }^2\mathrm{<mo\; stretchy="false">)</mo>},$ with $\mu \mathrm{<mo>=</mo>}\text{corr}\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{,}x\mathrm{<mo\; stretchy="false">)</mo>}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>1}}^6{x}_{kj}}$ and ${\sigma }^2\mathrm{<mo>=</mo>}1-6{\text{corr}}^2\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{,}x\mathrm{<mo\; stretchy="false">)</mo>},$ where $\text{corr}\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{,}x\mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}\text{corr}\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{,}{x}_j\mathrm{<mo\; stretchy="false">)</mo>,}j\in \mathrm{<mo>\{</mo>1,}\dots \mathrm{,}\mathrm{6<mo>\}</mo>.}$
   Both $\text{corr}\mathrm{<mo\; stretchy="false">(</mo>}y\mathrm{,}x\mathrm{<mo\; stretchy="false">)</mo>}$ and $\text{corr}\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{,}x\mathrm{<mo\; stretchy="false">)</mo>}$ were chosen so that $\text{corr}\mathrm{<mo\; stretchy="false">(</mo>}y\mathrm{,}\varphi \mathrm{<mo\; stretchy="false">)</mo>}$ was approximately equal to -0.2 or -0.8.

4.1 Numerical results

The coverage rate of the CIs was computed as in Section 3.1, with $n\mathrm{<mo>=</mo>}500$ and using a logistic regression without interactions to obtain ${\widehat{\varphi }}_k\mathrm{,}k\in s.$

Table 4.1 reports the results for the populations with $\text{corr}\mathrm{<mo\; stretchy="false">(</mo>}y\mathrm{,}\varphi \mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}-\mathrm{0.8.}$ Table 4.2 shows the $|$ RB $|$ and $|$ RRMSE $|$ of the estimators. Numerical results for populations with $\text{corr}\mathrm{<mo\; stretchy="false">(</mo>}y\mathrm{,}\varphi \mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}-\text{0}\text{.2}$ are omitted since they were similar to those obtained for populations with $\mathrm{corr}\mathrm{<mo\; stretchy="false">(</mo>}y\mathrm{,}\varphi \mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}-\text{0}\text{.8}\text{.}$ We make the following observations:

1. EL and Lin methods have a similar reasonable performance for $\overline{Y},$ though the upper tail error rates are larger than 5% in the asymmetric population.
2. For quantiles, Lin method has the lowest and the highest coverage of 85.7 and 97.9% respectively. Unlike the distribution of ${\widehat{Y}}_q$ shown in Figure 3.1, the distribution of ${\widehat{Y}}_q$ for the simulated populations is symmetric and unimodal in all cases. EL and W methods perform well, reaching the nominal level in all cases with comparable tail error rates and CI average length.
3. Weighting adjustment in the nonresponse scenario helps to get a RB similar to that of the full response.
4. The coverage rate of the CI for each method is larger in the nonresponse scenario than in the full response, and in some cases it is also larger than the nominal level. This might be related to the impact of having treated the ${\widehat{\varphi }}_k$ as fixed in the nonresponse case.

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