Sample empirical likelihood approach under complex survey design with scrambled responses
Section 4. Simulation study

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In the simulation study, we consider finite population $\left(X_i\mathrm{,}{Y}_i\right)\mathrm{,}i\mathrm{<mo>=</mo>}\mathrm{1,}\mathrm{2,}\dots \mathrm{,}N$ for $N\mathrm{<mo>=</mo>}$ 10,000. $X_i$ is uniformly distributed over [0, 1] and $Y_i\mathrm{<mo>=</mo>}m\left(X_i\right)+{\epsilon }_i$ with ${\epsilon }_i~N\left(\mathrm{0,}\text{0}\text{.01}\right).$ Four functions $m\left(\text{}x\text{}\right)$ are listed below:

We generated $B\mathrm{<mo>=</mo>}$ 5,000 Monte Carlo samples from Poisson sampling with inclusion probabilities ${\pi }_i\mathrm{<mo>=</mo>}n{k}_i/{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^N{k}_j},$ where the size variable $k_j\mathrm{<mo>=</mo>}\text{max}\left(\text{0}\text{.5}{Y}_j+\mathrm{2,}1\right)+u_j$ with $u_j~{\chi }^2\left(\text{}1\text{}\right).$ We considered sample sizes $n\mathrm{<mo>=</mo>}$ 40, 50, 100 and 200. For each Monte Carlo sample, the scrambled responses $Z_i$ were generated with $p\mathrm{<mo>=</mo>}$ 0.6, and $S_i~N\left(\text{1}\text{.5}\mathrm{,}\text{0}\text{.2}/\text{1}\text{.5}\right).$ Suppose we only observe $X_i$ and $Z_i$ in the sample. The performance of the HJ estimator and the proposed SEL estimator were compared with the estimate population mean of $Y,$ which is ${\theta }_0\mathrm{<mo>=</mo>}E\left(\text{}Y\text{}\right).$ The results are shown in Table 4.1.

We computed Monte Carlo bias $\text{MCB}\mathrm{<mo>=</mo>}{B}^{-1}{\displaystyle {\sum }_{b\mathrm{<mo>=</mo>\; 1}}^B{\widehat{\theta }}_b-{\theta }_0},$ Monte Carlo standard error $\text{MCSE}\mathrm{<mo>=</mo>}{\left\{B^{-1}{\displaystyle {\sum }_{b\mathrm{<mo>=</mo>\; 1}}^B{\left({\widehat{\theta }}_b-\overline{\theta }\right)}^2}\right\}}^{1/2}$ with $\overline{\theta }\mathrm{<mo>=</mo>}{B}^{-1}{\displaystyle {\sum }_{b\mathrm{<mo>=</mo>\; 1}}^B{\widehat{\theta }}_b}$ and Monte Carlo mean squared error $\text{MCMSE}\mathrm{<mo>=</mo>}{\left\{B^{-1}{\displaystyle {\sum }_{b\mathrm{<mo>=</mo>\; 1}}^B{\left({\widehat{\theta }}_b-{\theta }_0\right)}^2}\right\}}^{1/2}.$ For variance estimation, we calculated coverage rate, average length of interval estimates, and percentage of relative bias of variance estimators $\text{RB}\mathrm{<mo>=</mo>}100\times \left[\left(B^{-1}{\displaystyle {\sum }_{b\mathrm{<mo>=</mo>\; 1}}^B{\widehat{V}}_b}\right){\left\{B^{-1}{\displaystyle {\sum }_{b\mathrm{<mo>=</mo>\; 1}}^B{\left({\widehat{\theta }}_b-\overline{\theta }\right)}^2}\right\}}^{-1}-1\right].$ Results obtained from the simulation are given in Table 4.1.

For model $m_1,$ $m_3,$ and $m_4,$ SEL has a smaller Monte Carlo bias, Monte Carlo standard error, and Monte Carlo mean squared error, especially for small sample sizes $(n\mathrm{<mo>=</mo>}$ 40 or 50). For model $m_2,$ the two methods have comparable performance. For all four models, we found that, for most of the cases (14 of 16) the SEL estimators had a coverage rate higher than or equal to that of the HJ estimator, while the average length of confidence interval was shorter compared with the average length obtained with the HJ estimator. Both methods provided small relative biases of variance estimators. Overall, the proposed SEL outperformed HJ for most cases.

To test the sensitivity of the proposed approach, under current simulation study setups, we added noise, $W_i,$ to the simulation. Then, $Y_i\mathrm{<mo>=</mo>}m\left(\alpha X_i+\left(1-\alpha \right)W_i\right)+{\epsilon }_i$ with $\alpha \mathrm{<mo>=</mo>}$ 0, 0.1, 0.3, 0.5, 0.7, 0.9, 1, $X_i~\text{Uniform}\left(\mathrm{0,}1\right),$ $W_i~N\left(\mathrm{0,}1\right),$ and ${\epsilon }_i~N\left(\mathrm{0,}\text{0}\text{.01}\right).$ Suppose we only observe $X_i$ and $Z_i$ (the scrambled response of $Y_i)$ in the sample, the HJ estimator and SEL estimator were again compared. The results are shown in Tables 4.2 and 4.3. We found that as $\alpha$ decreases, the coverage rates of the SEL estimator are smaller than those of the HJ estimator, and the average length of CI for SEL estimator is not shorter than that of the HJ estimator. Therefore, the SEL estimator has better performance than the HJ estimator, provided that most of the information is contained in the current covariate.

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