Appendix

Jae Kwang Kim and Shu Yang

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A.1 Replication variance estimation

For variance estimation, replication methods can be used. Let $w_i^{[k]}$ be the $k\text{-th}$ replication weights such that

$${\widehat{V}}_{rep}={\displaystyle \sum _{k=1}^L}{c}_k{\left({\widehat{Y}}^{[k]}-\widehat{Y}\right)}^2$$

is consistent for the variance of $\widehat{Y}={\displaystyle {\sum }_{i\in A}}{w}_i{y}_i,$ where $L$ is the replication size, $c_k$ is the $k\text{-th}$ replication factor that depends on the replication method and the sampling mechanism, and ${\widehat{Y}}^{[k]}={\displaystyle {\sum }_{i\in A}}{w}_i^{[k]}{y}_i$ is the $k\text{-th}$ replicate of $\widehat{Y}.$ In delete-1 jackknife variance estimation, $L=n$ and $c_k=\left(n-1\right)/n.$

To apply the replication method in FFI, we first apply the replication weights $w_i^{[k]}$ in (2.4) to compute ${\widehat{\theta }}^{[k]}$ Once ${\widehat{\theta }}^{[k]}$ is obtained, we use the same imputed values to compute the initial replication fractional weights

$$w_{ij}^{*[k]}\propto w_j^{[k]}{w}_j^{-1}f\left(y_j|x_i;{\widehat{\theta }}^{[k]}\right)/\left\{{\displaystyle \sum _{l\in A_R}}{w}_l^{[k]}f\left(y_j|x_l;{\widehat{\theta }}^{[k]}\right)\right\}\mathrm{,}\text{ (A}\text{.1)}$$

with ${\displaystyle {\sum }_{j\in A_R}}{w}_{ij}^{*[k]}=1.$ The variance of ${\widehat{\eta }}_{FFI},$ computed from (3.4), is then computed by

$${\widehat{V}}_{rep}={\displaystyle \sum _{k=1}^L}{c}_k{\left({\widehat{\eta }}_{FFI}^{[k]}-{\widehat{\eta }}_{FFI}\right)}^2\mathrm{,}$$

where ${\widehat{\eta }}_{FFI}^{[k]}$ comes from solving

$${\displaystyle \sum _{i\in A}}{w}_i^{[k]}\left\{{\delta }_{i}U\left(\eta ;x_i\mathrm{,}{y}_i\right)+\left(1-{\delta }_i\right){\displaystyle \sum _{j\in A_R}}{w}_{ij}^{*[k]}U\left(\eta ;x_i\mathrm{,}{y}_j\right)\right\}=0,$$

and $w_{ij}^{*[k]}$ is defined in (A.1).

We now discuss replication variance estimation of the FHDI estimator ${\widehat{\eta }}_{FHDI}$ computed from (3.8). Define $d_{ij}=1$ if $j\in D_i$ and $d_{ij}=0$ otherwise. Note that ${\widehat{\eta }}_{FHDI}$ is computed via two steps: in the first step, a systematic PPS sampling is used with the selection probability proportional to the fractional weights from the FFI method. In the second step, the calibration weighting method using the constraint (3.5) with ${\displaystyle {\sum }_{j\in A_R}}{d}_{ij}{w}_{ij\mathrm{,}c}^{*}=1$ is used. Thus, the replicate fractional weights are also computed in two steps. Firstly, the initial replication fractional weight for $w_{ij0}^{*}=1/m$ is then given by

$$w_{ij0}^{*[k]}=\frac{d_{ij}\left(w_{ij}^{*[k]}/w_{ij}^{*}\right)}{{\displaystyle {\sum }_{l\in A_R}{d}_{il}\left(w_{il}^{*[k]}/w_{il}^{*}\right)}}\mathrm{,}\text{ (A}\text{.2)}$$

where $w_{ij}^{*}$ is the fractional weight for FFI defined in (2.6) and $w_{ij}^{*[k]}$ is the $k\text{-th}$ replication fractional weight for FFI defined in (A.1). Secondly, the replication fractional weights are adjusted to satisfy the calibration constraints. The calibration equation for replication fractional weights corresponding to (3.5) is then

$${\displaystyle \sum _{i\in A}}{w}_i^{[k]}\left\{\left(1-{\delta }_i\right){\displaystyle \sum _{j\in D_i}}{w}_{ij\mathrm{,}c}^{*[k]}q\left(x_i\mathrm{,}{y}_j\right)\right\}={\displaystyle \sum _{i\in A}}{w}_i^{[k]}\left\{\left(1-{\delta }_i\right){\displaystyle \sum _{j\in A_R}}{w}_{ij}^{*[k]}q\left(x_i\mathrm{,}{y}_j\right)\right\}\text{ (A}\text{.3)}$$

and ${\displaystyle {\sum }_{j\in D_i}}{w}_{ij\mathrm{,}c}^{*[k]}=1.$ Either regression weighting or entropy weighting can be used to obtain the replication fractional weights satisfying the constraints. Once the replicate fractional weights are obtained, the replicate estimate ${\widehat{\eta }}^{[k]}$ is computed by solving

$${\displaystyle \sum _{i\in A}}{w}_i^{[k]}\left\{{\delta }_{i}U\left(\eta ;x_i\mathrm{,}{y}_i\right)+\left(1-{\delta }_i\right){\displaystyle \sum _{j\in A_R}}{w}_{ijc}^{*[k]}U\left(\eta ;x_i\mathrm{,}{y}_j\right)\right\}=0.$$

The replication variance estimator of $\widehat{\eta },$ computed from (3.8), is given by

$${\widehat{V}}_{rep}\left(\widehat{\eta }\right)={\displaystyle \sum _{k=1}^L}{c}_k{\left({\widehat{\eta }}^{[k]}-\widehat{\eta }\right)}^2\mathrm{.}$$

Because $\widehat{\eta }$ is a smooth function of $\widehat{\theta },$ the consistency of ${\widehat{V}}_{rep}(\widehat{\eta })$ follows directly from the standard argument of the replication variance estimation (Shao and Tu 1995).

A.2 Proof of Equation (4.5)

Using

$$\frac{g\left(y_j|x_i\right)}{g\left(y_j|x_k\right)}=\frac{f\left(y_j|x_i\right)}{f\left(y_j|x_k\right)}\exp \left(\epsilon {\Delta }_{ik|j}-\kappa \left(x_i\right)+\kappa \left(x_k\right)\right)$$

where ${\Delta }_{ik|j}=z\left(x_i\mathrm{,}{y}_j;\theta \right)-z\left(x_k\mathrm{,}{y}_j;\theta \right).$ Based on Taylor linearization and the fact of (4.4), we have

$$\frac{g\left(y_j|x_i\right)}{g\left(y_j|x_k\right)}\cong \frac{f\left(y_j|x_i\right)}{f\left(y_j|x_k\right)}\left\{1+\epsilon {\Delta }_{ik|j}\right\}\mathrm{.}$$

If we know the true density, the correct fractional weights in (3.3) can be expressed by

$$\begin{array}{lll}{w}_{ij\mathrm{,}g}^{\mathrm{<mo>*</mo>}}\hfill & \propto \hfill & \frac{g\left(y_j|x_i\right)}{{\displaystyle {\sum }_{k\mathrm{<mo>;</mo>}{\delta }_k\mathrm{<mo>=</mo>1}}{w}_{k}g\left(y_j|x_k\right)}}\hfill \\ \hfill & \propto \hfill & \frac{1}{{\displaystyle \sum _{k\mathrm{<mo>;</mo>}{\delta }_k\mathrm{<mo>=</mo>1}}{w}_k\left\{\frac{g\mathrm{<mo\; stretchy="false">(</mo>}{y}_j|x_k\mathrm{<mo\; stretchy="false">)</mo>}}{g\mathrm{<mo\; stretchy="false">(</mo>}{y}_j|x_i\mathrm{<mo\; stretchy="false">)</mo>}}\right\}}}\hfill \\ \hfill & \propto \hfill & \frac{1}{{\displaystyle \sum _{k\mathrm{<mo>;</mo>}{\delta }_k\mathrm{<mo>=</mo>1}}{w}_k\left\{\frac{f\left(y_j|x_k\right)}{f\left(y_j|x_i\right)}\exp \left(\epsilon {\Delta }_{ki\mathrm{<mo>|</mo>}j}-\kappa \left(x_i\right)+\kappa \left(x_k\right)\right)\right\}}}\hfill \\ \hfill & \cong \hfill & \frac{1}{{\displaystyle \sum _{k\mathrm{<mo>;</mo>}{\delta }_k\mathrm{<mo>=</mo>1}}{w}_k\left\{\frac{f\left(y_j|x_k\right)}{f\left(y_j|x_i\right)}\left(1+\epsilon {\Delta }_{ki\mathrm{<mo>|</mo>}j}\right)\right\}}}\hfill \\ \hfill & \propto \hfill & \frac{1}{{\displaystyle \sum _{k\mathrm{<mo>;</mo>}{\delta }_k\mathrm{<mo>=</mo>1}}{w}_k\left\{\frac{f\left(y_j|x_k\right)}{f\left(y_j|x_i\right)}\right\}}+\epsilon {\displaystyle \sum _{k\mathrm{<mo>;</mo>}{\delta }_k\mathrm{<mo>=</mo>1}}{w}_k\left[\frac{f\left(y_j|x_k\right)}{f\left(y_j|x_i\right)}\left\{z\left(x_k\mathrm{,}{y}_j\right)-z\left(x_i\mathrm{,}{y}_j\right)\right\}\right]}}\hfill \\ \hfill & \propto \hfill & \frac{1}{{\displaystyle \sum _{k\mathrm{<mo>;</mo>}{\delta }_k\mathrm{<mo>=</mo>1}}}{w}_k\left\{\frac{f\left(y_j|x_k\right)}{f\left(y_j|x_i\right)}\right\}+\epsilon {\lambda }^T{I}_{\theta }^{-\mathrm{1<mo>/</mo>2}}{\displaystyle \sum _{k\mathrm{<mo>;</mo>}{\delta }_k\mathrm{<mo>=</mo>1}}}{w}_k\left(\frac{f\left(y_j|x_k\right)}{f\left(y_j|x_i\right)}\left\{s\left(x_k\mathrm{,}{y}_j\right)-s\left(x_i\mathrm{,}{y}_j\right)\right\}\right)}\hfill \\ \hfill & \propto \hfill & \frac{1}{{\displaystyle \sum _{k\mathrm{<mo>;</mo>}{\delta }_k\mathrm{<mo>=</mo>1}}}{w}_k\left\{\frac{f\left(y_j|x_k\right)}{f\left(y_j|x_i\right)}\right\}+\epsilon {\lambda }^T{I}_{\theta }^{-\mathrm{1<mo>/</mo>2}}{\displaystyle \sum _{k\mathrm{<mo>;</mo>}{\delta }_k\mathrm{<mo>=</mo>1}}}{w}_k\frac{\partial }{\partial \theta }\left\{\frac{f\left(y_j|x_k\right)}{f\left(y_j|x_i\right)}\right\}}\hfill \\ \hfill & \propto \hfill & \frac{1}{{\displaystyle \sum _{k\mathrm{<mo>;</mo>}{\delta }_k\mathrm{<mo>=</mo>1}}}{w}_k\left\{\frac{f\left(y_j|x_k\right)}{f\left(y_j|x_i\right)}\right\}}\left[1-\epsilon {\lambda }^T{I}_{\theta }^{-\mathrm{1<mo>/</mo>2}}\frac{{\displaystyle \sum _{k\mathrm{<mo>;</mo>}{\delta }_k\mathrm{<mo>=</mo>1}}}{w}_k\frac{\partial }{\partial \theta }\left\{\frac{f\left(y_j|x_k\right)}{f\left(y_j|x_i\right)}\right\}}{{\displaystyle \sum _{k\mathrm{<mo>;</mo>}{\delta }_k\mathrm{<mo>=</mo>1}}}{w}_k\left\{\frac{f\left(y_j|x_k\right)}{f\left(y_j|x_i\right)}\right\}}\right]\hfill \\ \hfill & \propto \hfill & \frac{f\left(y_j|x_i\right)}{{\displaystyle \sum _{k\mathrm{<mo>;</mo>}{\delta }_k\mathrm{<mo>=</mo>1}}}{w}_{k}f\left(y_j|x_k\right)}\left[1-\epsilon {\lambda }^T{I}_{\theta }^{-\mathrm{1<mo>/</mo>2}}\frac{\frac{\partial }{\partial \theta }\left\{{\displaystyle \sum _{k\mathrm{<mo>;</mo>}{\delta }_k\mathrm{<mo>=</mo>1}}}{w}_k\frac{f\left(y_j|x_k\right)}{f\left(y_j|x_i\right)}\right\}}{{\displaystyle \sum _{k\mathrm{<mo>;</mo>}{\delta }_k\mathrm{<mo>=</mo>1}}}{w}_k\frac{f\left(y_j|x_k\right)}{f\left(y_j|x_i\right)}}\right]\hfill \\ \hfill & =\hfill & \frac{f\left(y_j|x_i\right)}{{\displaystyle \sum _{k\mathrm{<mo>;</mo>}{\delta }_k\mathrm{<mo>=</mo>1}}}{w}_{k}f\left(y_j|x_k\right)}+\epsilon {\lambda }^T{I}_{\theta }^{-\mathrm{1<mo>/</mo>2}}\frac{\partial }{\partial \theta }\left\{\frac{1}{{\displaystyle \sum _{k\mathrm{<mo>;</mo>}{\delta }_k\mathrm{<mo>=</mo>1}}}{w}_k\frac{f\left(y_j|x_k\right)}{f\left(y_j|x_i\right)}}\right\}\hfill \\ \hfill & =\hfill & a_{ij}+\epsilon {\lambda }^T{I}_{\theta }^{-\mathrm{1<mo>/</mo>2}}\frac{\partial }{\partial \theta }{a}_{ij}\mathrm{,}\hfill \end{array}$$

where $a_{ij}=f\left(y_j|x_i\right)/{\displaystyle {\sum }_{k;{\delta }_k=1}}{w}_{k}f\left(y_j|x_k\right)$ and $a_{i+}={\displaystyle {\sum }_{j;{\delta }_j=1}}{a}_{ij}.$ So, $w_{ij\mathrm{,}f}^{*}=a_{ij}/a_{i+}$ and

$$\begin{array}{lll}{w}_{ij\mathrm{,}g}^{\mathrm{<mo>*</mo>}}\hfill & \cong \hfill & \frac{a_{ij}+\epsilon {\lambda }^T{I}_{\theta }^{-\mathrm{1<mo>/</mo>2}}\frac{\partial }{\partial \theta }{a}_{ij}}{a_{i+}+\epsilon {\lambda }^T{I}_{\theta }^{-\mathrm{1<mo>/</mo>2}}\frac{\partial }{\partial \theta }{a}_{i+}}\hfill \\ \hfill & =\hfill & \frac{a_{ij}}{a_{i+}}\left(1+\frac{\epsilon {\lambda }^T{I}_{\theta }^{-\mathrm{1<mo>/</mo>2}}\frac{\partial }{\partial \theta }{a}_{ij}}{a_{ij}}\right)\left(\frac{a_{i+}}{a_{i+}+\epsilon {\lambda }^T{I}_{\theta }^{-\mathrm{1<mo>/</mo>2}}\frac{\partial }{\partial \theta }{a}_{i+}}\right)\hfill \\ \hfill & \cong \hfill & \frac{a_{ij}}{a_{i+}}\left(1+\epsilon {\lambda }^T{I}_{\theta }^{-\mathrm{1<mo>/</mo>2}}\frac{\partial }{\partial \theta }\log a_{ij}\right)\left(1-\epsilon {\lambda }^T{I}_{\theta }^{-\mathrm{1<mo>/</mo>2}}\frac{\partial }{\partial \theta }\log a_{i+}\right)\hfill \\ \hfill & \cong \hfill & \frac{a_{ij}}{a_{i+}}+\epsilon {\lambda }^T{I}_{\theta }^{-\mathrm{1<mo>/</mo>2}}\frac{a_{ij}}{a_{i+}}\left(\frac{\partial }{\partial \theta }\log a_{ij}-\frac{\partial }{\partial \theta }\log a_{i+}\right)\hfill \\ \hfill & =\hfill & \frac{a_{ij}}{a_{i+}}+\epsilon {\lambda }^T{I}_{\theta }^{-\mathrm{1<mo>/</mo>2}}\frac{\partial }{\partial \theta }\left(\frac{a_{ij}}{a_{i+}}\right)\hfill \\ \hfill & =\hfill & w_{ij\mathrm{,}f}^{\mathrm{<mo>*</mo>}}+\epsilon {\lambda }^T{I}_{\theta }^{-\mathrm{1<mo>/</mo>2}}\frac{\partial }{\partial \theta }\left(w_{ij\mathrm{,}f}^{\mathrm{<mo>*</mo>}}\right)\mathrm{,}\hfill \end{array}$$

which proves (4.5).

A.3 Extension to a non-ignorable missing case

We consider an extension of the proposed method to a non-ignorable missing case. Under the non-ignorable missing assumption, both the conditional model $f\left(y|x\right)$ and the response probability model $P\left(\delta =1|x\mathrm{,}y\right)$ are needed to evaluate the expected estimating function in (4.6). Let the response probability model be given by $Pr\left({\delta }_i=1|x_i\mathrm{,}{y}_i\right)=\pi \left(x_i\mathrm{,}{y}_i;\varphi \right)$, for some $\varphi$ with a known $\pi (\cdot )$ function. We assume that the parameters are identifiable as discussed in Wang, Shao and Kim (2013).

In PFI, according to Kim and Kim (2012), the MLE $\left(\widehat{\theta }\mathrm{,}\widehat{\varphi }\right)$ can be obtained by solving

$${\displaystyle \sum _{i\in A}}{w}_i\left\{{\delta }_{i}S\left(\theta ;x_i\mathrm{,}{y}_i\right)+\left(1-{\delta }_i\right){\displaystyle \sum _{j=1}^m}{w}_{ij}^{*}S\left(\theta ;x_i\mathrm{,}{y}_i^{*(j)}\right)\right\}=0, (A.4)$$

and

$${\displaystyle \sum _{i\in A}}{w}_i\left\{{\delta }_{i}S\left(\varphi ;x_i\mathrm{,}{y}_i\right)+\left(1-{\delta }_i\right){\displaystyle \sum _{j=1}^m}{w}_{ij}^{*}S\left(\varphi ;x_i\mathrm{,}{y}_i^{*(j)}\right)\right\}=0, (A.5)$$

where $S\left(\theta ;x\mathrm{,}y\right)=\partial \log f\left(y|x;\theta \right)/\partial \theta ,$ $S\left(\varphi ;x\mathrm{,}y\right)=\partial \log \pi \left(x\mathrm{,}y;\varphi \right)/\partial \varphi ,$ and the fractional weights are given by

$$w_{ij}^{*}\left(\theta \mathrm{,}\varphi \right)=\frac{f\left(y_i^{*(j)}|x_i;\theta \right)\left\{1-\pi \left(x_i\mathrm{,}{y}_i^{*(j)}\mathrm{,}\varphi \right)\right\}/h\left(y_i^{*(j)}|x_i\right)}{{\displaystyle {\sum }_{k=1}^m\left[f\left(y_i^{*(k)}|x_i;\theta \right)\left\{1-\pi \left(x_i\mathrm{,}{y}_i^{*(k)}\mathrm{,}\varphi \right)\right\}/h\left(y_i^{*(k)}|x_i\right)\right]}}\mathrm{.}\text{ (A}\text{.6)}$$

The solution to (A.4) and (A.5) can be obtained via the EM algorithm. In the EM algorithm, the E-step computes the fractional weights in (A.6) using the current parameter values and the M-step updates the parameter value ${\widehat{\theta }}^{(t+1)}$ and ${\widehat{\varphi }}^{(t+1)}$ by solving

$${\displaystyle \sum _{i\in A}}{w}_i\left\{{\delta }_{i}S\left(\theta ;x_i\mathrm{,}{y}_i\right)+\left(1-{\delta }_i\right){\displaystyle \sum _{j=1}^m}{w}_{ij}^{*}\left({\widehat{\theta }}^{(t)}\mathrm{,}{\widehat{\varphi }}^{(t)}\right)S\left(\theta ;x_i\mathrm{,}{y}_i^{*(j)}\right)\right\}=0,$$

and

$${\displaystyle \sum _{i\in A}}{w}_i\left\{{\delta }_{i}S\left(\varphi ;x_i\mathrm{,}{y}_i\right)+\left(1-{\delta }_i\right){\displaystyle \sum _{j=1}^m}{w}_{ij}^{*}\left({\widehat{\theta }}^{(t)}\mathrm{,}{\widehat{\varphi }}^{(t)}\right)S\left(\varphi ;x_i\mathrm{,}{y}_i^{*(j)}\right)\right\}=0.$$

In the proposed FFI method, the fractional weights are given by

$$\begin{array}{l}{w}_{ij}^{*}\propto f\left(y_j|x_i\mathrm{,}{\delta }_i=0;\theta \mathrm{,}\varphi \right)/f\left(y_j|{\delta }_j=1\right)\\ \propto f\left(y_j|x_{i;}\theta \right)\left\{1-\pi \left(x_i\mathrm{,}{y}_j;\varphi \right)\right\}/f\left(y_j|{\delta }_j=1\right)\mathrm{,}\end{array}$$

with ${\displaystyle {\sum }_{j;{\delta }_j=1}}{w}_{ij}^{*}=1.$ Because

$$\begin{array}{l}f\left(y_j|{\delta }_j=1\right)={\displaystyle \int }\pi \left(x\mathrm{,}{y}_j\right)f\left(y_j|x\right)f(x)dx\text{ (A}\text{.7)}\\ \cong {\displaystyle \sum _{k\in A}}{w}_k\pi \left(x_k\mathrm{,}{y}_j\right)f\left(y_j|x_k\right)\mathrm{.}\end{array}$$

The fractional weights can be computed from

$$w_{ij}^{*}\propto \frac{f\left(y_j|x_i;\theta \right)\left\{1-\pi \left(x_i\mathrm{,}{y}_j;\varphi \right)\right\}}{{\displaystyle {\sum }_{k\in A}{w}_k\pi \left(x_k\mathrm{,}{y}_j;\varphi \right)f\left(y_j|x_k;\theta \right)}}\mathrm{.}\text{ (A}\text{.8)}$$

with ${\displaystyle {\sum }_{j\in A_R}}{w}_{ij}^{*}=1.$

Thus, we can use the following EM algorithm to obtain the desired parameter estimates.

(I-step)  For each missing unit $i\in A_M=\left\{i\in A;{\delta }_i=0\right\},$ take $m$ imputed values as $y_i^{(1)}\mathrm{,}\dots \mathrm{,}{y}_i^{(m)}$ from $A_R,$ where $m=r.$

(E-step)  The fractional weights are given by

$$w_{ij}^{*(t)}\propto \frac{f\left(y_j|x_i\mathrm{,}{\widehat{\theta }}^{(t)}\right)\left\{1-\pi \left(x_i\mathrm{,}{y}_j;{\widehat{\varphi }}^{(t)}\right)\right\}}{{\displaystyle {\sum }_{k\in A}{w}_k\pi \left(x_k\mathrm{,}{y}_j;{\widehat{\varphi }}^{(t)}\right)f\left(y_j|x_k;{\widehat{\theta }}^{(t)}\right)}}$$

and ${\displaystyle {\sum }_{j=1}^m}{w}_{ij}^{*(t)}=1.$

(M-step)  Update the parameter ${\widehat{\theta }}^{(t+1)}$  and ${\widehat{\varphi }}^{(t+1)}$ by solving the following imputed score equations,

$${\displaystyle \sum _{i\in A}}{w}_i\left\{{\delta }_{i}S\left(\theta ;x_i\mathrm{,}{y}_i\right)+\left(1-{\delta }_i\right){\displaystyle \sum _{j\in A_R}}{w}_{ij}^{*(t)}S\left(\theta ;x_i\mathrm{,}{y}_j\right)\right\}=0,$$

and

$${\displaystyle \sum _{i\in A}}{w}_i\left\{{\delta }_{i}S\left(\varphi ;x_i\mathrm{,}{y}_i\right)+\left(1-{\delta }_i\right){\displaystyle \sum _{j\in A_R}}{w}_{ij}^{*(t)}S\left(\varphi ;x_i\mathrm{,}{y}_j\right)\right\}=0.$$

Note that the I-step does not have to be repeated in the EM algorithm. Once the final parameter estimates are computed, the fractional weights are computed by (A.8), which serve as the selection probabilities for FHDI with a small imputation size $m.$ The same systematic PPS sampling method as discussed in Section 3 can be used to obtain FHDI.

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