Sample empirical likelihood approach under complex survey design with scrambled responses
Section 2. Preliminaries

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Suppose the finite population $F_N\mathrm{<mo>=</mo>}\left(X_i\mathrm{,}{Y}_i\mathrm{,}i\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}N\right)$ is generated from some unknown super-population model, where $Y_i$ is a study variable and $X_i$ is a covariate. For ease of presentation, given $F_N,$ a random sample $A$ is assumed to be selected from a single stage unstratified sampling design. Let $I_i$ be the sampling indicator for unit $i$ such that $I_i\mathrm{<mo>=</mo>}1$ if unit $i$ is selected and 0 otherwise. Denote the first-order and second-order inclusion probabilities as ${\pi }_i\mathrm{<mo>=</mo>}E\left(I_i\right)$ and ${\pi }_{ij}\mathrm{<mo>=</mo>}E\left(I_i{I}_j\right)$ for $i\mathrm{,}j\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}N.$ Then, the sampling weight can be written as $d_i\mathrm{<mo>=</mo>}{\pi }_i^{-1}$ and sample size is $n\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N{I}_i}.$ Suppose the parameter of interest is ${\theta }_N\mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N{Y}_i}.$ Due to confidentiality, we plan to use scrambled responses $Z_i$ of $Y_i$ such that $Z_i\mathrm{<mo>=</mo>}{Y}_i{S}_i$ with probability $1-p$ and $Z_i\mathrm{<mo>=</mo>}{Y}_i$ with probability $p,$ where $E\left(S_i\right)\mathrm{<mo>=</mo>}a$ and $V\left(S_i\right)\mathrm{<mo>=</mo>}{b}^2$ with $p,$ $a,$ and $b^2$ known. Bar-lev et al. (2004) and Singh and Kim (2011) considered similar models. Instead of observing $Y_i$ directly, we only observe the scrambled responses $Z_i$ in the data file. Hájek estimator discussed in Hájek (1971) and Fuller (2009) has been used frequently in survey data analysis. Under certain regularity conditions, one can show that the following Hájek (HJ) type estimator is consistent:

$${\widehat{\theta }}_{\text{HJ}}\mathrm{<mo>=</mo>}\frac{1}{\widehat{N}}{\displaystyle \sum _{i\in A}{d}_i{Y}_i^{*}}\mathrm{,}(2.1)$$

where $Y_i^{*}\mathrm{<mo>=</mo>}{Z}_i{\left\{\left(1-p\right)a+p\right\}}^{-1}$ and $\widehat{N}\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in A}{d}_i}$ since $E\left(\text{}\widehat{N}\text{}\right)\mathrm{<mo>=</mo>}N$ and

$$E\left({\displaystyle \sum _{i\in A}{d}_i{Y}_i^{*}}\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N\left\{E\left(Y_i^{*}\right)\right\}}\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N\left[\left\{E\left(Y_i{S}_i\right)\left(1-p\right)+Y_{i}p\right\}{\left\{\left(1-p\right)a+p\right\}}^{-1}\right]}\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N{Y}_i}.$$

The asymptotic properties of ${\widehat{\theta }}_{\text{HJ}}$ are described in the following Theorem 1, and the sketched proof is contained in Appendix B.

Theorem 1. Under the regularity conditions in Appendix A, ${\widehat{\theta }}_{\text{HJ}}$  has the following asymptotic expansion

$${\widehat{\theta }}_{\text{HJ}}\mathrm{<mo>=</mo>}{\theta }_N+\frac{1}{N}{\displaystyle \sum _{i\in A}{d}_i\left(Y_i^{*}-{\theta }_N\right)}+o_p\left(\text{}{n}^{-1/2}\text{}\right)\mathrm{,}(2.2)$$

and

$$V_{\text{HJ}}^{-1/2}\left({\widehat{\theta }}_{\text{HJ}}-{\theta }_N\right){\to }^{d}N\left(\mathrm{0,}1\right)\mathrm{,}(2.3)$$

as $n\mathrm{,}N\to \infty$  with

$$V_{\text{HJ}}\mathrm{<mo>=</mo>}{V}_1+V_2\mathrm{,}(2.4)$$

where

$$V_1\mathrm{<mo>=</mo>}\frac{1}{N^2}{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{j=1}^N\frac{{\pi }_{ij}-{\pi }_i{\pi }_j}{{\pi }_i{\pi }_j}\left(Y_i-{\theta }_N\right)\left(Y_j-{\theta }_N\right),}}$$

and

$$V_2\mathrm{<mo>=</mo>}\frac{\left(1-p\right)\left\{p{\left(a-1\right)}^2+b^2\right\}}{{\left\{\left(1-p\right)a+p\right\}}^2}\frac{1}{N^2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N{Y}_i^2}\mathrm{.}$$

Note that $V_1$ is the design variability of Hájek estimator for population mean ${\theta }_N$ by using the true values and $V_2$ is the additional variability generated by using scrambled responses. According to Theorem 1, the consistent estimator of $V_{\text{HJ}}$ can be written as

$${\widehat{V}}_{\text{HJ}}\mathrm{<mo>=</mo>}\frac{1}{{\widehat{N}}^2}{\displaystyle \sum _{i\in A}{\displaystyle \sum _{j\in A}\frac{{\pi }_{ij}-{\pi }_i{\pi }_j}{{\pi }_{ij}}\frac{Y_i^{*}-{\widehat{\theta }}_{\text{HJ}}}{{\pi }_i}\frac{Y_j^{*}-{\widehat{\theta }}_{\text{HJ}}}{{\pi }_j}}}+\frac{\left(1-p\right)\left\{b^2+p{\left(a-1\right)}^2\right\}}{\left(b^2+a^2\right)\left(1-p\right)+p}\frac{1}{{\widehat{N}}^2}{\displaystyle \sum _{i\in A}{d}_i{Y}_i^{*2}}\mathrm{.}$$

When $n/N\mathrm{<mo>=</mo>}o\left(\text{}1\text{}\right),$ the second term above can be safely ignored. Therefore, we can use a traditional design consistent estimator with transformed variable $Y_i^{*}.$ In the next section, we will propose using the pseudo empirical likelihood method to construct both point estimator and confidence interval when we have aggregated auxiliary information.

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