Empirical likelihood inference for missing survey data under unequal probability sampling
Section 3. EL inference under imputation

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Based on the imputed function ${\tilde{h}}_i\left(\mu \right)$ for all $i\in s,$ we now propose an EL method for inference about the population mean $\mu$ of $h\left(y_i\right).$ Define the EL under imputation (2.1) as $L_n\left(q\right)\mathrm{<mo>=</mo>}{\displaystyle {\prod }_{i\in s}{q}_i},$ where $q_i$ satisfy $q_i\ge 0$ for all $i,$ ${\displaystyle {\sum }_{i\in s}{q}_i\mathrm{<mo>=</mo>\; 1}},$ and ${\displaystyle {\sum }_{i\in s}{q}_i{\tilde{h}}_i\left(\mu \right)\mathrm{<mo>=</mo>\; 0}}.$ The corresponding profile log-EL at a given value of $\mu$ is defined as

$$l_n\left(\mu \right)\mathrm{<mo>=</mo>}\underset{\left\{q_i\right\}}{{\displaystyle \sup }}\left\{{\displaystyle \sum _{i\in s}\log q_i\text{}\mathrm{<mo>:</mo>}{q}_i}\ge \mathrm{0,}{\displaystyle \sum _{i\in s}{q}_i\mathrm{<mtext>\hspace{0.17em}</mtext>\; <mo>=</mo><mtext>\hspace{0.17em}</mtext>\; 1}}\mathrm{,}{\displaystyle \sum _{i\in s}{q}_i{\tilde{h}}_i\left(\mu \right)\mathrm{<mtext>\hspace{0.17em}</mtext><mo>=</mo><mtext>\hspace{0.17em}</mtext>\; 0}}\right\}\mathrm{.}(3.1)$$

Note that although the above $l_n\left(\mu \right)$ takes the same form as the EL of the mean for IID data (Owen, 2001), the design weights $d_i$ are in fact subsumed in the definition (2.1) of ${\tilde{h}}_i\left(\mu \right),$ which makes $l_n\left(\mu \right)$ suitable for survey data. Solving the above maximization problem using the method of Lagrange multipliers, we obtain

$$q_i\mathrm{<mo>=</mo>}\frac{1}{n\left\{1+\lambda {\tilde{h}}_i\left(\mu \right)\right\}}\mathrm{,}$$

where $\lambda$ is the solution to the equation $n^{-1}{\displaystyle {\sum }_{i\in s}{\tilde{h}}_i\left(\mu \right)}/\left\{1+\lambda {\tilde{h}}_i\left(\mu \right)\right\}\mathrm{<mo>=</mo>\; 0}.$ Consequently, we get

$$l_n\left(\mu \right)\mathrm{<mo>=</mo>}-{\displaystyle \sum _{i\in s}}\log \left\{1+\lambda {\tilde{h}}_i\left(\mu \right)\right\}\mathrm{.}(3.2)$$

We then define the MELE of $\mu$ as

$$\widehat{\mu }\mathrm{<mo>=</mo>}\underset{\mu }{\text{argmax}}{l}_n\left(\mu \right)\mathrm{.}$$

It can be shown that the maximum of $l_n\left(\mu \right)$ is attained when $q_i\mathrm{<mo>=</mo>}1/n$ for all $i\in s.$ Hence, by the third constraint in (3.1), the MELE $\widehat{\mu }$ is the solution to the equation $n^{-1}{\displaystyle {\sum }_{i\in s}{\tilde{h}}_i\left(\mu \right)\mathrm{<mo>=</mo>\; 0}},$ which is given by

$$\widehat{\mu }\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{i\in s}{\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^{K}1\left(z_i\mathrm{<mo>=</mo>}k\right)\left\{{\delta }_i{d}_{i}h\left(y_i\right)+\left(1-{\delta }_i\right)J^{-1}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^J{d}_{ij}^{*}h\left(y_{ij}^{*}\right)}\right\}}}}{{\displaystyle {\sum }_{i\in s}{\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^{K}1\left(z_i\mathrm{<mo>=</mo>}k\right)}}\left\{{\delta }_i{d}_i+\left(1-{\delta }_i\right)J^{-1}{\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^J{d}_{ij}^{*}}\right\}}\mathrm{.}(3.3)$$

Our Theorem 1 below presents the asymptotic normality of the MELE $\widehat{\mu }.$ For the asymptotic investigation, we consider the case where both the population size $N$ and the sample size $n$ increase to $\infty$ as an index $\nu$ increases to $\infty ,$ as assumed by Chen and Rao (2007).

Theorem 1. Assume that the regularity conditions (R.1)-(R.2) in Appendix A hold. Under PPS sampling with replacement,

$$\sqrt{n}{\sigma }_N^{-1}\left(\widehat{\mu }-{\mu }_N\right)\stackrel{d}{\to }N\left(\mathrm{0,}1\right)$$

as $n\to \infty ,$  where ${\mu }_N$  is the true parameter value and ${\sigma }_N$  is a constant.

An expression of ${\sigma }_N$ is given in Appendix A and the proof of Theorem 1 is given in Appendix B. Theorem 1 also applies to PPS sampling without replacement with negligible sampling fractions, which is asymptotically equivalent to PPS sampling with replacement. Note that increasing $J$ reduces the asymptotic variance ${\sigma }_N,$ leading to a more efficient estimator. However, a larger $J$ does not necessarily imply better coverage probabilities of CIs, as shown in the simulation study in Section 5.

Theorem 1 suggests that we can construct a Wald-type CI for ${\mu }_N$ if given a design-consistent estimator of ${\sigma }_N.$ However, accurately estimating ${\sigma }_N$ under an unequal probability sampling design is not an easy task, especially when ${\sigma }_N$ has a complicated algebraic expression. An alternative approach is to construct a likelihood-ratio-like quantity as in parametric likelihood inference. Toward this end, we define an EL ratio as

$$R\left(\mu \right)\mathrm{<mo>=</mo>}-2l_n\left(\mu \right)\mathrm{.}(3.4)$$

The asymptotic distribution of $R\left(\mu \right)$ is given by Theorem 2.

Theorem 2. Assume that the regularity conditions (R.1)-(R.3) in Appendix A hold. Under PPS sampling with replacement,

$$R\left({\mu }_N\right)\stackrel{d}{\to }{c}_N{\chi }_1^2$$

as $n\to \infty ,$  where ${\chi }_1^2$  is a chi-square random variable with 1 degree of freedom and $c_N$  is a constant depending on ${\mu }_N.$

The proof of Theorem 2 is given in Appendix B. Again, Theorem 2 also applies to PPS sampling without replacement with negligible sampling fractions. The expression of the scaling constant $c_N$ is given in Appendix A. The value of $c_N$ is not 1 in general. Hence, to construct an EL ratio CI for ${\mu }_N,$ we need to estimate an unknown constant as in the case of Wald-type test. A design-consistent estimator of $c_N$ is given in Appendix A, and we use it to construct an EL ratio CI in the simulation study in Section 5. To avoid estimating the scaling constant, we explore proper bootstrap procedures in Section 4.

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