Sample empirical likelihood approach under complex survey design with scrambled responses
Section 3. Proposed method

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Population-level aggregated information is often available through census or large surveys, such as the American Community Survey (ACS). For instance, we may know the national-level population counts by gender, race, educational level, or income level. Incorporating such information into estimation will often reduce the coverage error and improve the efficiency of the estimators. In this section, we propose using the sample empirical likelihood (SEL) approach proposed by Chen and Kim (2014) to conduct point and interval estimation simultaneously. Suppose a population mean ${\overline{X}}_N\mathrm{<mo>=</mo>}{N}^{-1}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N{X}_i}$ is known through some external resources. Then, the SEL estimator can be obtained by maximizing the following sample empirical log-likelihood function

$$l_s\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in A}\text{log}\left(w_i\right),}(3.1)$$

subject to constraints

$${\displaystyle \sum _{i\in A}{w}_i\mathrm{<mo>=</mo>}1,}{\displaystyle \sum _{i\in A}{w}_i{\pi }_i^{-1}\left(X_i-{\overline{X}}_N\right)\mathrm{<mo>=</mo>}\mathrm{0,}}{w}_i\ge \mathrm{0,}(3.2)$$

and

$${\displaystyle \sum _{i\in A}{w}_i{\pi }_i^{-1}\left(Y_i^{*}-\theta \right)\mathrm{<mo>=</mo>}0.}(3.3)$$

By maximizing objective function (3.1) subject to constraints in (3.2), the SEL weight can be written as

$${\widehat{w}}_i\mathrm{<mo>=</mo>}\frac{1}{n}\frac{1}{1+\widehat{\lambda }{\pi }_i^{-1}\left(X_i-{\overline{X}}_N\right)}\mathrm{,}$$

with $\widehat{\lambda }$ as the Lagrange multiplier, and it can be obtained by solving the second constraint in (3.2). Then, according to (3.3), the SEL estimator of ${\theta }_N$ can be written as

$${\widehat{\theta }}_{\text{SEL}}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{i\in A}{\widehat{w}}_i{\pi }_i^{-1}{Y}_i^{*}}}{{\displaystyle {\sum }_{i\in A}{\widehat{w}}_i{\pi }_i^{-1}}}\mathrm{.}$$

The following Theorem 2 contains asymptotic properties of the proposed SEL estimator ${\widehat{\theta }}_{\text{SEL}}.$ The sketched proof is contained in Appendix C.

Theorem 2. Under the regularity conditions in Appendix A, ${\widehat{\theta }}_{\text{SEL}}$  has the following asymptotic expansion

$${\widehat{\theta }}_{\text{SEL}}={\theta }_N+\frac{1}{N}{\displaystyle \sum _{i\in A}{d}_i\left(Y_i^{*}-{\theta }_N\right)}-B\frac{1}{N}{\displaystyle \sum _{i\in A}{d}_i\left(X_i-{\overline{X}}_N\right)}+o_p\left(\text{}{n}^{-1/2}\right)\mathrm{,}(3.4)$$

where

$$B=\left\{\frac{1}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N{\pi }_i^{-1}\left(Y_i-{\theta }_N\right)\left(X_i-{\overline{X}}_N\right)}\right\}{\left\{\frac{1}{N}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N{\pi }_i^{-1}\left(X_i-{\overline{X}}_N\right){\left(X_i-{\overline{X}}_N\right)}^T}\right\}}^{-1}$$

and

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

as $n\mathrm{,}N\to \infty$  with

$$V_{\text{SEL}}\mathrm{<mo>=</mo>}{V}_1^{*}+V_2\mathrm{,}$$

where $V_2$  is defined in Theorem 1 and

$$V_1^{*}\mathrm{<mo>=</mo>}\frac{1}{N^2}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^N{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^N\frac{{\pi }_{ij}-{\pi }_i{\pi }_j}{{\pi }_i{\pi }_j}{\eta }_i{\eta }_j\mathrm{,}}}$$

with ${\eta }_i\mathrm{<mo>=</mo>}{Y}_i-{\theta }_N-B\left(X_i-{\overline{X}}_N\right).$

Note that $V_1^{*}$ is the design variability of optimal regression estimator which is less than $V_1$ defined in Theorem 1. The optimal regression estimator has been discussed by Fuller and Isaki (1981), Montanari (1987), and Rao (1994). According to Theorem 2, the consistent estimator of $V_{\text{SEL}}$ can be written as

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

where ${\widehat{\eta }}_i\mathrm{<mo>=</mo>}{Y}_i^{*}-{\widehat{\theta }}_{\text{SEL}}-\widehat{B}\left(X_i-{\overline{X}}_N\right)$ with

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

When $n/N\mathrm{<mo>=</mo>}o\left(\text{}1\text{}\right),$ the second term of ${\widehat{V}}_{\text{SEL}}$ can be ignored. Under the simple random sampling (SRS) design, it can be shown that ${\widehat{\theta }}_{\text{SEL}}$ is asymptotically equivalent to the following well-known regression estimator

$${\widehat{\theta }}_{\text{REG}}\mathrm{<mo>=</mo>}\frac{1}{\widehat{N}}{\displaystyle \sum _{i\in A}{d}_i{Y}_i^{*}}-{\widehat{B}}_R\left(\frac{1}{\widehat{N}}{\displaystyle \sum _{i\in A}{d}_i{X}_i-{\overline{X}}_N}\right)\mathrm{,}(3.5)$$

where

$${\widehat{B}}_R\mathrm{<mo>=</mo>}\left\{{\displaystyle \sum _{i\in A}{d}_i\left(Y_i^{*}-{\widehat{\theta }}_{\text{HJ}}\right)\left(X_i-{\overline{X}}_N\right)}\right\}{\left\{{\displaystyle \sum _{i\in A}{d}_i\left(X_i-{\overline{X}}_N\right){\left(X_i-{\overline{X}}_N\right)}^T}\right\}}^{-1}\mathrm{.}$$

However, for general design, ${\widehat{\theta }}_{\text{SEL}}$ is different from ${\widehat{\theta }}_{\text{REG}}.$ Under Poisson sampling design, it can be shown that ${\widehat{\theta }}_{\text{SEL}}$ is more efficient than ${\widehat{\theta }}_{\text{REG}}.$ Theorem 1 and Theorem 2 can be used to construct a Wald-type confidence interval for ${\theta }_N.$ The following Theorem 3 can be used to construct a Wilk-type confidence interval. The sketched proof of Theorem 3 is contained in Appendix D.

Theorem 3. Define $R_n\left({\theta }_N\right)\mathrm{<mo>=</mo>}2\left\{l_s\left({\widehat{\theta }}_{\text{SEL}}\right)-l_s\left({\theta }_N\right)\right\},$  where $l_s\left(\text{}\theta \text{}\right)$  is defined in (3.1) with $w_i$  satisfying (3.2) and (3.3). Then under the regularity conditions listed in Appendix A, as $n\mathrm{,}N\to \infty ,$   $c_1{c}_2^{-1}{R}_n\left({\theta }_N\right){\to }^d{\chi }_1^2,$  where $c_1\mathrm{<mo>=</mo>}{N}^{-2}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N{\pi }_i^{-1}{\eta }_i^{*2}}$  with ${\eta }_i^{*}\mathrm{<mo>=</mo>}{Y}_i^{*}-{\theta }_N-B\left(X_i-{\overline{X}}_N\right)$  and $c_2\mathrm{<mo>=</mo>}{V}_{\text{SEL}}.$

The estimator of $c_1$ and $c_2$ can be written as

$${\widehat{c}}_1\mathrm{<mo>=</mo>}{\widehat{N}}^{-2}{\displaystyle \sum _{i\in A}{\pi }_i^{-2}{\left\{Y_i^{*}-{\widehat{\theta }}_{\text{SEL}}-\widehat{B}\left(X_i-{\overline{X}}_N\right)\right\}}^2\mathrm{,}}$$

and ${\widehat{c}}_2\mathrm{<mo>=</mo>}{\widehat{V}}_{\text{SEL}}.$ Theorem 3 can be used to construct a Wilk-type confidence interval for ${\theta }_N.$

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