Sample empirical likelihood approach under complex survey design with scrambled responses
Section 6. Conclusions

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In this paper, we proposed a sample empirical likelihood (SEL)-based approach using scrambled responses to protect the confidentiality of complex survey data. The proposed SEL approach is easy to implement in practice and can be used as a general tool for statistical disclosure control. The idea of our proposed approach is to replace the true values by some scrambled values through random device, then the existing sample empirical likelihood approach can be applied with scrambled values to obtain the point estimation. However, the variance estimation and confidence interval estimation are different from that by treating the scrambled values as true values since we need to incorporate the randomness due to random device in the statistical inference. Such theoretical properties have been investigated and verified through simulation study and real data application. The SEL outperforms traditional approaches, such as HJ, by improving coverage rates and reducing the coverage lengths of confidence intervals. Chen and Kim (2014) has compared Wald-type CI and Wilk-type CI in the simulation studies by using sample empirical likelihood method. In general, the Wilk-type confidence intervals show better coverage properties than the Wald-type confidence intervals in terms of coverage rates. We would expect similar results by using our proposed approaches here. In future research, we will extend the proposed approach to estimate more general parameters, such as population quantiles and distribution functions. The corresponding statistical computational tools, such as R package, will also be developed.

Acknowledgements

Dr. Sixia Chen was partially supported by the Oklahoma Shared Clinical and Translational Resources (U54GM104938) with an Institutional Development Award (IDeA) from National Institute of General Medical Sciences. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. The research of Yichuan Zhao was supported by the National Security Agency (NSA) Grant (H98230-12-1-0209) and the National Science Foundation Grant (DMS-1613176).

Appendix

A. Regularity conditions

We present the regularity conditions needed for proving Theorem 1 to Theorem 3 as following:

B. Sketched proof of Theorem 1

${\widehat{\theta }}_{\text{HJ}}$ can be written as the solution of estimating equation ${\widehat{U}}_{\text{HJ}}\left(\text{}\theta \text{}\right)\mathrm{<mo>=</mo>}\mathrm{0,}$ where

$${\widehat{U}}_{\text{HJ}}\left(\text{}\theta \text{}\right)\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _{i\in A}{d}_i\left(Y_i^{*}-\theta \right).}$$

Under the assumptions that ${\widehat{U}}_{\text{HJ}}\left(\text{}\theta \text{}\right)$ converges to $U_{\text{HJ}}\left(\text{}\theta \text{}\right)\mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N\left(Y_i-\theta \right)}$ uniformly, $E\left(\text{}{Y}^2\right)\mathrm{<mo><</mo>}\infty ,$ and because of $U_{\text{HJ}}\left({\theta }_N\right)\mathrm{<mo>=</mo>}0,$ it can be shown that ${\widehat{\theta }}_{\text{HJ}}{\to }^p{\theta }_N.$ By using a Taylor expansion,

$$0\mathrm{<mo>=</mo>}{\widehat{U}}_{\text{HJ}}\left({\widehat{\theta }}_{\text{HJ}}\right)\mathrm{<mo>=</mo>}{\widehat{U}}_{\text{HJ}}\left({\theta }_N\right)+\frac{\partial {\widehat{U}}_{\text{HJ}}\left({\theta }_N\right)}{\partial \theta }\left({\widehat{\theta }}_{\text{HJ}}-{\theta }_N\right)+o_p\left(\text{}{n}^{-1/2}\right)\mathrm{.}$$

After some algebra, it can be shown that

$${\widehat{\theta }}_{\text{HJ}}\mathrm{<mo>=</mo>}{\theta }_N+\frac{1}{N}{\displaystyle \sum _{i\in A}{d}_i\left(Y_i^{*}-{\theta }_N\right)+o_p\left(\text{}{n}^{-1/2}\right)}.$$

Because

$$E\left(Y_i^{*}\right)\mathrm{<mo>=</mo>}{Y}_i\mathrm{,}V\left(Y_i^{*}\right)\mathrm{<mo>=</mo>}\frac{\left(1-p\right)\left\{b^2+p{\left(a-1\right)}^2\right\}}{{\left\{\left(1-p\right)a+p\right\}}^2}{Y}_i^2\mathrm{,}(\text{B}\text{.1})$$

$$\begin{array}{ll}V\left\{\frac{1}{N}{\displaystyle \sum _{i\in A}{d}_i\left(Y_i^{*}-{\theta }_N\right)}\right\}\hfill & \mathrm{<mo>=</mo>}E\left\{\frac{1}{N^2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^N\frac{{\pi }_{ij}-{\pi }_i{\pi }_j}{{\pi }_i{\pi }_j}\left(Y_i^{*}-{\theta }_N\right)\left(Y_j^{*}-{\theta }_N\right)}}\right\}\hfill \\ \hfill & +V\left\{\frac{1}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N\left(Y_i^{*}-{\theta }_N\right)}\right\}\mathrm{.}(\text{B}\text{.2})\hfill \end{array}$$

According to (B.1), (B.2), and after some algebra, we can show that

$$V\left\{\frac{1}{N}{\displaystyle \sum _{i\in A}{d}_i\left(Y_i^{*}-{\theta }_N\right)}\right\}\mathrm{<mo>=</mo>}{V}_{\text{HJ}}\mathrm{,}$$

where $V_{\text{HJ}}$ is defined in equation (2.4). Under the regularity conditions in Fuller and Isaki (1981), the asymptotic normality can be derived.

C. Sketched proof of Theorem 2

Define

$${\widehat{U}}_1\left(\text{}\lambda \text{}\right)\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _{i\in A}\frac{{\pi }_i^{-1}\left(X_i-{\overline{X}}_N\right)}{1+\lambda {\pi }_i^{-1}\left(X_i-{\overline{X}}_N\right)}}$$

and

$${\widehat{U}}_2\left(\lambda \mathrm{,}\theta \right)\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _{i\in A}\frac{{\pi }_i^{-1}\left(Y_i^{*}-\theta \right)}{1+\lambda {\pi }_i^{-1}\left(X_i-{\overline{X}}_N\right)}\mathrm{.}}$$

Then, ${\widehat{\theta }}_{\text{SEL}}$ and $\widehat{\lambda }$ are the solutions of ${\widehat{U}}_1\left(\text{}\lambda \text{}\right)\mathrm{<mo>=</mo>}{\widehat{U}}_2\left(\lambda \mathrm{,}\theta \right)\mathrm{<mo>=</mo>}0.$ By using techniques similar to those of Chen and Kim (2014), it can be shown that $\widehat{\lambda }\mathrm{<mo>=</mo>}{O}_p\left(\text{}{n}^{-1/2}\right)$ and ${\widehat{\theta }}_{\text{SEL}}{\to }^p{\theta }_N.$ Then, by using Taylor expansion, we have

$$0\mathrm{<mo>=</mo>}{\widehat{U}}_1\left(\text{}\widehat{\lambda }\text{}\right)\mathrm{<mo>=</mo>}{\widehat{U}}_1\left(\text{}0\text{}\right)+\frac{\partial {\widehat{U}}_1\left(\text{}0\text{}\right)}{\partial \lambda }\widehat{\lambda }+o_p\left(\text{}{n}^{-1/2}\right)\mathrm{,}(\text{C}\text{.1})$$

and

$$0\mathrm{<mo>=</mo>}{\widehat{U}}_2\left(\widehat{\lambda }\mathrm{,}{\widehat{\theta }}_{\text{SEL}}\right)\mathrm{<mo>=</mo>}{\widehat{U}}_2\left(\mathrm{0,}{\theta }_N\right)+\frac{\partial {\widehat{U}}_2\left(\mathrm{0,}{\theta }_N\right)}{\partial \theta }\left({\widehat{\theta }}_{\text{SEL}}-{\theta }_N\right)+\frac{\partial {\widehat{U}}_2\left(\mathrm{0,}{\theta }_N\right)}{\partial \lambda }\widehat{\lambda }+o_p\left(\text{}{n}^{-1/2}\right)\mathrm{.}(\text{C}\text{.2})$$

According to (C.1), (C.2), and after some algebra, it can be shown that

$$\widehat{\lambda }\mathrm{<mo>=</mo>}{\left\{\frac{1}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N{\pi }_i^{-1}\left(X_i-{\overline{X}}_N\right){\left(X_i-{\overline{X}}_N\right)}^T}\right\}}^{-1}\frac{1}{N}{\displaystyle \sum _{i\in A}{d}_i\left(X_i-{\overline{X}}_N\right)+o_p\left(\text{}{n}^{-1/2}\right)}(\text{C}\text{.3})$$

and

$${\widehat{\theta }}_{\text{SEL}}-{\theta }_N\mathrm{<mo>=</mo>}\frac{1}{N}{\displaystyle \sum _{i\in A}{d}_i\left(Y_i^{*}-{\theta }_N\right)}-B\frac{1}{N}{\displaystyle \sum _{i\in A}{d}_i\left(X_i-{\overline{X}}_N\right)+o_p\left(\text{}{n}^{-1/2}\right)\mathrm{,}}$$

where $B$ is defined in Theorem 2. Because

$$\begin{array}{ll}V\left({\widehat{\theta }}_{\text{SEL}}\right)\mathrm{<mo>=</mo>}V\left(\frac{1}{N}{\displaystyle \sum _{i\in A}{d}_i{\eta }_i^{*}}\right)+o\left(\text{}{n}^{-1}\right)\hfill & \mathrm{<mo>=</mo>}V\left(\frac{1}{N}{\displaystyle \sum _{i\in A}{d}_i{\eta }_i}\right)+E\left\{V\left(\frac{1}{N}{\displaystyle \sum _{i\in A}{d}_i{\eta }_i^{*}}|A\right)\right\}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{V}_2+\frac{1}{N^2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^N\frac{{\pi }_{ij}-{\pi }_i{\pi }_j}{{\pi }_i{\pi }_j}{\eta }_i{\eta }_j+o\left(\text{}{n}^{-1}\right)\mathrm{,}}}\hfill \end{array}$$

where $V_2$ is defined in Theorem 1, ${\eta }_i$ is defined in Theorem 2 and ${\eta }_i^{*}\mathrm{<mo>=</mo>}{Y}_i^{*}-{\theta }_N-B\left(X_i-{\overline{X}}_N\right).$ After some algebra, we can show that

$${\widehat{V}}_{\text{SEL}}\mathrm{<mo>=</mo>}{V}_{\text{SEL}}+o\left(\text{}{n}^{-1}\right)\mathrm{,}$$

with $V_{\text{SEL}}$ defined in Theorem 2. Furthermore, under the regularity conditions in Fuller and Isaki (1981), we obtain the asymptotic normality.

D. Sketched proof of Theorem 3

Because $\widehat{\lambda }\mathrm{<mo>=</mo>}{O}_p\left(\text{}{n}^{-\mathrm{1\; <mo>/</mo>\; 2}}\right)$ and by using a Taylor expansion of $\text{log}\left(1+x\right)$ at $x\mathrm{<mo>=</mo>}\widehat{\lambda }{\pi }_i^{-1}\left(X_i-{\overline{X}}_N\right)$ and (C.3), we have

$$\begin{array}{ll}{l}_s\left({\widehat{\theta }}_{\text{SEL}}\right)\hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in A}}\log \left\{\frac{1}{n}\frac{1}{1+\widehat{\lambda }{\pi }_i^{-1}\left(X_i-{\overline{X}}_N\right)}\right\}\hfill \\ \hfill & =n\log \left(\frac{1}{n}\right)-{\displaystyle \sum _{i\in A}\log \left\{1+\widehat{\lambda }{\pi }_i^{-1}\left(X_i-{\overline{X}}_N\right)\right\}}\hfill \\ \hfill & =n\log \left(\frac{1}{n}\right)-{\displaystyle \sum _{i\in A}\left\{{\widehat{\lambda }}^T{\pi }_i^{-1}\left(X_i-{\overline{X}}_N\right)-\frac{1}{2}{\widehat{\lambda }}^T{\pi }_i^{-2}{\left(X_i-{\overline{X}}_N\right)}^{\otimes 2}\widehat{\lambda }\right\}+o_p\left(\text{}1\text{}\right)}\hfill \\ \hfill & =n\log \left(\frac{1}{n}\right)-\frac{N}{2}\frac{1}{N}{\displaystyle \sum _{i\in A}{\pi }_i^{-1}{\left(X_i-{\overline{X}}_N\right)}^T{\left\{\frac{1}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N{\pi }_i^{-1}{\left(X_i-{\overline{X}}_N\right)}^{\otimes 2}}\right\}}^{-1}}\hfill \\ \hfill & \times \frac{1}{N}{\displaystyle \sum _{i\in A}{\pi }_i^{-1}\left(X_i-{\overline{X}}_N\right)+o_p\left(\text{}1\text{}\right)\mathrm{,}}(\text{D}\text{.1})\hfill \end{array}$$

with $a^{\otimes 2}\mathrm{<mo>=</mo>}a{a}^T.$ We now consider to maximize $l_s\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in A}\log \left(w_i\right)},$ subject to the following constraints

$${\displaystyle \sum _{i\in A}{w}_i\mathrm{<mo>=</mo>}1,}{\displaystyle \sum _{i\in A}{w}_i{\pi }_i^{-1}\left(X_i-{\overline{X}}_N\right)\mathrm{<mo>=</mo>}\mathrm{0,}}(\text{D}\text{.2})$$

and

$${\displaystyle \sum _{i\in A}{w}_i{\pi }_i^{-1}{\eta }_i^{*}\mathrm{<mo>=</mo>}\mathrm{0,}}(\text{D}\text{.3})$$

where ${\eta }_i^{*}\mathrm{<mo>=</mo>}{Y}_i^{*}-{\theta }_N-B\left(X_i-{\overline{X}}_N\right).$ The above constraints are equivalent with the original constraints (3.2) and (3.3). Define $u_i\mathrm{<mo>=</mo>}{\left(X_i^T-{\overline{X}}_N^T\mathrm{,}{\eta }_i^{*}\right)}^T$ . Therefore, by using a similar argument, we have

$$\begin{array}{ll}{l}_s\left({\theta }_N\right)\hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in A}\log \left\{\frac{1}{n}\frac{1}{1+\widehat{\lambda }\left({\theta }_N\right){\pi }_i^{-1}{u}_i}\right\}}\hfill \\ \hfill & =n\log \left(\frac{1}{n}\right)-{\displaystyle \sum _{i\in A}\log \left\{1+\widehat{\lambda }\left({\theta }_N\right){\pi }_i^{-1}{u}_i\right\}}\hfill \\ \hfill & =n\log \left(\frac{1}{n}\right)-{\displaystyle \sum _{i\in A}\left\{{\widehat{\lambda }}^T\left({\theta }_N\right){\pi }_i^{-1}{u}_i-\frac{1}{2}{\widehat{\lambda }}^T\left({\theta }_N\right){\pi }_i^{-2}{u}_i^{\otimes 2}\widehat{\lambda }\left({\theta }_N\right)\right\}+o_p\left(\text{}1\text{}\right)}\hfill \\ \hfill & =n\log \left(\frac{1}{n}\right)-\frac{N}{2}\frac{1}{N}{{\displaystyle \sum _{i\in A}{\pi }_i^{-1}{u}_i^T\left\{\frac{1}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N{\pi }_i^{-1}{u}_i^{\otimes 2}}\right\}}}^{-1}\times \frac{1}{N}{\displaystyle \sum _{i\in A}{\pi }_i^{-1}{u}_i+o_p\left(\text{}1\text{}\right)}\hfill \\ \hfill & =n\log \left(\frac{1}{n}\right)-\frac{N}{2}\frac{1}{N}{\displaystyle \sum _{i\in A}{\pi }_i^{-1}{\left(X_i-{\overline{X}}_N\right)}^T{\left\{\frac{1}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N{\pi }_i^{-1}{\left(X_i-{\overline{X}}_N\right)}^{\otimes 2}}\right\}}^{-1}}\hfill \\ \hfill & \times \frac{1}{N}{\displaystyle \sum _{i\in A}{\pi }_i^{-1}\left(X_i-{\overline{X}}_N\right)}-\frac{N}{2}\frac{1}{N}{\displaystyle \sum _{i\in A}{\pi }_i^{-1}{\eta }_i^{*T}{\left\{\frac{1}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N{\pi }_i^{-1}{\eta }_i^{*\otimes 2}}\right\}}^{-1}}\times \frac{1}{N}{\displaystyle \sum _{i\in A}{\pi }_i^{-1}{\eta }_i^{*}+o_p\left(\text{}1\text{}\right)}(\text{D}\text{.4})\hfill \end{array}$$

provided ${\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N{\pi }_i^{-1}\left(X_i-{\overline{X}}_N\right){\eta }_i\mathrm{<mo>=</mo>}0}.$ According to (D.1), (D.4), and after some algebra, we have

$$\begin{array}{ll}\frac{c_1}{c_2}{R}_n\left({\theta }_N\right)\hfill & \mathrm{<mo>=</mo>}\frac{2c_1}{c_2}\left\{l_s\left({\widehat{\theta }}_{\text{SEL}}\right)-l_s\left({\theta }_N\right)\right\}\mathrm{<mo>=</mo>}\frac{c_1}{c_2}{\left\{\frac{1}{N}{\displaystyle \sum _{i\in A}{\pi }_i^{-1}{\eta }_i^{*}}\right\}}^2{\left(\frac{1}{N^2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N{\pi }_i^{-1}{\eta }_i^{*2}}\right)}^{-1}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\left\{V{\left(\frac{1}{N}{\displaystyle \sum _{i\in A}{\pi }_i^{-1}{\eta }_i^{*}}\right)}^{-1/2}\frac{1}{N}{\displaystyle \sum _{i\in A}{\pi }_i^{-1}{\eta }_i^{*}}\right\}}^2{\to }^d{\chi }_1^2\mathrm{.}(\text{D}\text{.5})\hfill \end{array}$$

Therefore, Theorem 3 is proven.

References

Bar-Lev, S.K., Bobovitch, E. and Boukai, B. (2004). A note on randomized response models for quantitative data. Metrika, 60, 255-260.

Berger, Y.G. (2018a). Empirical likelihood approaches in survey sampling. The Survey Statistician, 78, 22-31.

Berger, Y.G. (2018b). An empirical likelihood approach under cluster sampling with missing observations. Annals of the Institute of Statistical Mathematics, doi:10.1007/s10463-018-0681-x.

Berger, Y.G., and Torres, O.D.L.R. (2016). An empirical likelihood approach for inference under complex sampling design. Journal of the Royal Statistical Society, Series B, 78(2), 319-341.

Chen, S., and Kim, J.K. (2014). Population empirical likelihood for nonparametric inference in survey sampling. Statistica Sinica, 24, 335-355.

Cochran, W.G. (1977). Sampling Techniques, 3^rd Ed. New York: John Wiley & Sons, Inc.

Eichhorn, B.H., and Hayre, L.S. (1983). Scrambled randomized response methods for obtaining sensitive quantitative data. Journal of Statistical Planning and Inference, 7, 307-316.

Fienberg, S.E., and McIntyre, J. (2005). Data swapping: Variations on a theme by Dalenius and Reiss. Journal of Official Statistics, 21, 309-323.

Fox, J.A., and Tracy, P.E. (1986). Randomized Response: A Method for Sensitive Surveys. Beverly Hills, CA: Sage.

Fuller, W.A. (2009). Sampling Statistics. Hoboken, NJ: John Wiley & Sons, Inc.

Fuller, W.A., and Isaki, C.T. (1981). Survey design under superpopulation models. In Current Topics in Survey Sampling, (Eds., D. Krewski, J.N.K. Rao, and R. Platek). New York: Academic Press, 199-226.

Gouweleeuw, J.M., Kooiman, P., Willenborg, L.C.R. and Wolf, P. (1998). Post randomization for statistical disclosure control: Theory and implementation. Journal of Official Statistics, 14, 463-478.

Hájek, J. (1971). Comment on “An essay on the logical foundations of survey sampling, Part one”. In The Foundations of Survey Sampling, (Eds., V.P. Godambe and D.A. Sprott), Holt, Rinehart, and Winston, 236.

Hartley, H.O., and Rao, J.N.K. (1968). A new estimation theory for sample surveys. Biometrika, 55, 547-557.

Horvitz, D.G., Shah, B.V. and Simmons, W.R. (1967). The unrelated question randomized response model. In Proceedings of the Social Statistics Section, American Statistical Association, 65-72.

Hundepool, A., Domingo-Ferrer, J., Franconi, L., Giessing, S., Nordholt, E., Spicer, K. and Wolf, P. (2012). Statistical Disclosure Control, Wiley Series In Survey Methodology.

Kim, J.K., and Yang, S. (2017). A note on multiple imputation under informative sampling. Biometrika, 104, 221-228.

Kish, L. (1965). Survey Sampling. New York: John Wiley & Sons, Inc.

Krenzke, T., Li, J., Freedman, M., Judkins, D., Hubble, D., Roisman, R. and Larsen, M. (2011). Producing transportation data products from the American Community Survey that comply with disclosure rules. Washington, DC: National Cooperative Highway Research Program, Transportation Research Board, National Academy of Sciences.

Meng, X.L. (1994). Multiple-imputation inferences with uncongenial sources of input (with discussion). Statistical Science, 9, 538-573.

Montanari, G.E. (1987). Post-sampling efficient Q-R prediction in large-sample surveys. International Statistical Review, 55, 191-202.

Owen, A.B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika, 75, 237-249.

Owen, A.B. (2001). Empirical Likelihood. New York: Chapman and Hall.

Qin, J., and Lawless, J. (1994). Empirical likelihood and general estimating equations. The Annals of Statistics, 22, 300-325.

Raghunathan, T.E., Reiter, J.P. and Rubin, D.B. (2003). Multiple imputation for statistical disclosure limitation. Journal of Official Statistics, 19, 1-16.

Raghunathan, T.E., Lepkowski, J.M., van Hoewyk, J., and Solenberger, P. (2001). A multivariate technique for multiply imputing missing values using a sequence of regression models. Survey Methodology, 27, 1, 85-95. Paper available at https://www150.statcan.gc.ca/n1/en/pub/12-001-x/2001001/article/5857-eng.pdf.

Rao, J.N.K. (1994). Estimating totals and distribution functions using auxiliary information at the estimation stage. Journal of Official Statistics, 10, 153-165.

Saha, A. (2011). An optional scrambled randomized response technique for practical surveys. Metrika, 73, 139-149.

Singh, S., and Kim, J.M. (2011). A pseudo-empirical log-likelihood estimator using scrambled responses. Statistics and Probability Letters, 81, 345-351.

Tracy, D.S., and Mangat, N.S. (1996). Some developments in randomized response sampling during the last decade-a follow up of review by Chaudhuri and Mukerjee. Journal of Applied Statistical Science, 4, 147-158.

Warner, S.L. (1965). Randomized response: A survey technique for eliminating evasive answer bias. Journal of the American Statistical Association, 60, 63-69.

Wu, C., and Rao, J.N.K. (2006). Pseudo-empirical likelihood ratio confidence intervals for complex surveys. Canadian Journal of Statistics, 34, 359-375.

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