Empirical likelihood inference for missing survey data under unequal probability sampling
Section 4. Bootstrap EL intervals

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We now propose two bootstrap procedures to construct EL CIs on the population mean of $h\left(y\right).$ First, draw a bootstrap sample of size $n$ using simple random sampling with replacement from the sample quadruples $\left\{(y_i\mathrm{,}{d}_i\mathrm{,}{z}_i\mathrm{,}{\delta }_i):i\in s\right\},$ and denote the bootstrap sample as $\left\{({{\displaystyle y}}_{b,i},{{\displaystyle d}}_{b,i},{{\displaystyle z}}_{b,i},{{\displaystyle \delta }}_{b,i}):i=1,\mathrm{...},n\right\}.$ Second, perform the imputation introduced in Section 2 on the bootstrap sample. That is, if ${\delta }_{b\mathrm{,}i}\mathrm{<mo>=</mo>\; 0}$ for some $i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}n$ and $z_{b\text{}\mathrm{,}i}\mathrm{<mo>=</mo>}k\in \left\{\mathrm{1,}\dots \mathrm{,}K\right\},$ select $J$ values at random with replacement from the bootstrap donor set of class $k,$ ${{\displaystyle \mathcal{R}}}_{b,k}=\left\{({{\displaystyle y}}_{b,i},{{\displaystyle d}}_{b,i}):{{\displaystyle \delta }}_{b,i}=1,{{\displaystyle z}}_{b,i}=k,i=1,\mathrm{...},n\right\},$ and denote these values as ${{\displaystyle y}}_{b,ij}^{*},$ $j\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}J.$ Then, similar to (2.1), define a bootstrap version of the imputed estimating function, denoted ${\tilde{h}}_{b\text{}\mathrm{,}i}\left(\mu \right),$ for all $i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}n$ as

$${\tilde{h}}_{b\text{}\mathrm{,}i}\left(\mu \right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^{K}1\left(z_{b\text{}\mathrm{,}i}\mathrm{<mo>=</mo>}k\right)\left[{\delta }_{b\text{}\mathrm{,}i}{d}_{b\text{}\mathrm{,}i}\left\{h\left(y_{b\text{}\mathrm{,}i}\right)-\mu \right\}+\left(1-{\delta }_{b\text{}\mathrm{,}i}\right)J^{-1}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^J{d}_{b\text{}\mathrm{,}ij}^{*}\left\{h\left(y_{b\text{}\mathrm{,}ij}^{*}\right)-\mu \right\}}\right]}.$$

Finally, obtain a bootstrap version of the profile log-EL, denoted $l_{b\text{}\mathrm{,}n}\left(\mu \right),$ by replacing ${\tilde{h}}_i\left(\mu \right)$ in (3.1) with ${\tilde{h}}_{b\text{}\mathrm{,}i}\left(\mu \right),$ and define the bootstrap MELE as ${\widehat{\mu }}_b\mathrm{<mo>=</mo>}{\text{argmin}}_{\mu }{l}_{b\text{}\mathrm{,}n}\left(\mu \right)$ and the bootstrap EL ratio as $R_b\left(\mu \right)\mathrm{<mo>=</mo>}-2l_{b\text{}\mathrm{,}n}\left(\mu \right).$

To construct bootstrap CIs for $\mu ,$ we seek suitable bootstrap analogues of $\sqrt{n}\left(\widehat{\mu }-{\mu }_N\right)$ and $R\left({\mu }_N\right).$ In particular, we propose asymptotically correct bootstrap quantities based on ${\widehat{\mu }}_b$ and $R_b\left(\mu \right)$ that approximate the distributions of $\sqrt{n}\left(\widehat{\mu }-{\mu }_N\right)$ and $R\left({\mu }_N\right).$ We will further show that the usual bootstrap analogues $\sqrt{n}\left({\widehat{\mu }}_b-\widehat{\mu }\right)$ and $R_b\left(\widehat{\mu }\right),$ suggested by Shao and Sitter (1996), are asymptotically incorrect for approximating the distributions of $\sqrt{n}\left(\widehat{\mu }-{\mu }_N\right)$ and $R\left({\mu }_N\right)$ under fractional imputation with fixed $J.$

The proposed bootstrap analogues of $\sqrt{n}\left(\widehat{\mu }-{\mu }_N\right)$ and $R\left({\mu }_N\right)$ rely on a quantity which we call complete-data MELE as defined below. Let $n_k\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s}1\left(z_i\mathrm{<mo>=</mo>}k\right)}$ and $r_k\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in s}{\delta }_{i}1\left(z_i\mathrm{<mo>=</mo>}k\right)}$ for $k\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}K.$ For all $i\in s,$ define

$${\stackrel{⌣}{h}}_i\left(\mu \right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K}\frac{n_k}{r_k}{\delta }_{i}1\left(z_i\mathrm{<mo>=</mo>}k\right)d_i\left\{h\left(y_i\right)-\mu \right\}\mathrm{.}$$

Note that ${\stackrel{⌣}{h}}_i\left(\mu \right)$ does not involve imputation. Similar to (3.1), we define a profile log-EL based on ${\stackrel{⌣}{h}}_i\left(\mu \right),$

$${\stackrel{⌣}{l}}_n\left(\mu \right)\mathrm{<mo>=</mo>}\underset{\left\{q_i\right\}}{{\displaystyle \sup }}\left\{{\displaystyle \sum _{i\in s}\log q_i\mathrm{<mo>:</mo>}{q}_i\ge 0}\mathrm{,}{\displaystyle \sum _{i\in s}{q}_i\mathrm{<mo>=</mo>\; 1}}\mathrm{,}{\displaystyle \sum _{i\in s}{q}_i{\stackrel{⌣}{h}}_i\left(\mu \right)\mathrm{<mo>=</mo>\; 0}}\right\}\mathrm{.}$$

Again, the design weights $d_i$ are included in the definition of ${\stackrel{⌣}{h}}_i\left(\mu \right),$ although ${\stackrel{⌣}{l}}_n\left(\mu \right)$ does not explicitly depends on them. We then define the complete-data MELE as $\stackrel{⌣}{\mu }\mathrm{<mo>=</mo>}{\text{argmin}}_{\mu }{\stackrel{⌣}{l}}_n\left(\mu \right).$ As the profile log-EL defined in (3.1), the maximum of ${\stackrel{⌣}{l}}_n\left(\mu \right)$ is attained when $q_i\mathrm{<mo>=</mo>}1/n,$ and, as a consequence, $\stackrel{⌣}{\mu }$ is the solution to the equation $n^{-1}{\displaystyle {\sum }_{i\in s}{\stackrel{⌣}{h}}_i\left(\mu \right)\mathrm{<mo>=</mo>\; 0}},$ which is simply given by

$$\stackrel{⌣}{\mu }\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{i\in s}{\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K\left(n_k/r_k\right){\delta }_{i}1\left(z_i\mathrm{<mo>=</mo>}k\right)d_{i}h\left(y_i\right)}}}{{\displaystyle {\sum }_{i\in s}{\displaystyle {\sum }_{k\mathrm{<mo>=</mo>\; 1}}^K\left(n_k/r_k\right){\delta }_{i}1\left(z_i\mathrm{<mo>=</mo>}k\right)d_i}}}\mathrm{.}(4.1)$$

The complete-data MELE $\stackrel{⌣}{\mu }$ plays an important role in constructing asymptotically correct bootstrap quantities, as shown by Theorem 3.

Theorem 3. Let ${\mathcal{F}}_n$  denote the sample data $\left\{(y_i\mathrm{,}{z}_i\mathrm{,}{\delta }_i):i\in s\right\}.$  Under the conditions of Theorem 2,

$$\underset{t}{\sup }\left|P\left\{\sqrt{n}\left({\widehat{\mu }}_b-\stackrel{⌣}{\mu }\right)\le t|{\mathcal{F}}_n\right\}-P\left\{\sqrt{n}\left(\widehat{\mu }-{\mu }_N\right)\le t\right\}\right|=o_p\left(1\right)(4.2)$$

and

$$\underset{t}{{\displaystyle \sup }}\left|P\left\{R_b\left(\stackrel{⌣}{\mu }\right)\le t|{\mathcal{F}}_n\right\}-P\left\{R\left({\mu }_N\right)\le t\right\}\right|\mathrm{<mo>=</mo>}{o}_p\left(1\right)\mathrm{.}(4.3)$$

The proof of Theorem 3 is given in Appendix B.

Remark 1. The difference between the usual bootstrap quantity $\sqrt{n}({\widehat{\mu }}_b-\widehat{\mu })$ (or $R_b\mathrm{<mo\; stretchy="false">(</mo>}\widehat{\mu }\mathrm{<mo\; stretchy="false">)</mo>})$ and the proposed bootstrap quantity $\sqrt{n}\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{\mu }}_b-\stackrel{⌣}{\mu }\mathrm{<mo\; stretchy="false">)</mo>}$ (or $R_b\mathrm{<mo\; stretchy="false">(</mo>}\stackrel{⌣}{\mu }\mathrm{<mo\; stretchy="false">)</mo>})$ can be shown to be $O_p\left(1\right)$ instead of $o_p\left(1\right)$ when $J$ is a fixed constant. This, together with Theorem 3, shows that the usual bootstrap quantities do not have the same limiting distributions as those of $\sqrt{n}\mathrm{<mo\; stretchy="false">(</mo>}\widehat{\mu }-{\mu }_N\mathrm{<mo\; stretchy="false">)</mo>}$ and $R\mathrm{<mo\; stretchy="false">(</mo>}{\mu }_N\mathrm{<mo\; stretchy="false">)</mo>},$ and will lead to asymptotically incorrect coverage of ${\mu }_N.$ If $J$ is allowed to increase to $\infty$ as $n\to \infty ,$ then the differences between the usual bootstrap quantities and the proposed quantities becomes $o_p\left(1\right)$ and both are asymptotically correct.

Two bootstrap approaches to constructing a $\left(1-\alpha \right),$ $\alpha \in \left(\mathrm{0,}1\right),$ level CI on $\mu$ are suggested by Theorem 3. Independently generate $b\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}B$ bootstrap samples, and obtain ${\widehat{\mu }}_b$ and $R_b\left(\stackrel{⌣}{\mu }\right)$ for all $b.$ The first approach is based on the bootstrap distribution of $\sqrt{n}\left\{{\widehat{\mu }}_b-\stackrel{⌣}{\mu }\right\}.$ Find the ${\left(1-\alpha /2\right)}^{\text{th}}$ and ${\left(\alpha /2\right)}^{\text{th}}$ sample quantiles, ${\widehat{\mu }}_{b\text{}\mathrm{,}1-\alpha /2}$ and ${\widehat{\mu }}_{b\text{}\mathrm{,}\alpha /2},$ of $\left\{{\widehat{\mu }}_b\text{}\mathrm{<mo>:</mo>}b\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}B\right\}.$ An approximate $\left(1-\alpha \right)$ level CI for $\mu$ is given by

$$\left(\widehat{\mu }-\left({\widehat{\mu }}_{b\text{}\mathrm{,}1-\alpha /2}-\stackrel{⌣}{\mu }\right)\mathrm{,}\widehat{\mu }-\left({\widehat{\mu }}_{b\text{}\mathrm{,}\alpha /2}-\stackrel{⌣}{\mu }\right)\right)\mathrm{.}$$

We call the above CI the bootstrap-EL percentile (BELP) interval.

The second approach relies on the bootstrap distribution of the bootstrap EL ratio $R_b\left(\stackrel{⌣}{\mu }\right).$ Find the ${\left(1-\alpha \right)}^{\text{th}}$ sample quantile, denoted $R_{b\text{}\mathrm{,}1-\alpha }\left(\stackrel{⌣}{\mu }\right),$ of $\left\{R_b(\stackrel{⌣}{\mu }):b\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}B\right\}.$ Then an approximate $\left(1-\alpha \right)$ level CI for $\mu$ based on $R_b\left(\stackrel{⌣}{\mu }\right)$ is given by the interval defined by

$$\left\{\mu \mathrm{<mo>:</mo>}R\left(\mu \right)\le R_{b\text{}\mathrm{,}1-\alpha }\left(\stackrel{⌣}{\mu }\right)\right\}\mathrm{.}$$

We call this CI the bootstrap-EL ratio (BELR) interval.

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