4. Robustness

Jae Kwang Kim and Shu Yang

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We now discuss the robustness of the proposed method against a small departure from the assumed parametric model. The robustness feature in our proposed estimator is defined to be robust against imputation model misspecification, a small exponential tilting of the true model. For simplicity of the presentation, assume that the sampling design is simple random sampling and the realized sample is a random sample from the superpopulation model.

We assume that the true model $g\left(y|x\right)$ does not belong to $\left\{f\left(y|x;\theta \right);\theta \in \Omega \right\}.$ However, we can still specify a working model $f\left(y|x;\theta \right)$ and compute the MLE of $\theta .$ It is well known (White 1982) that the MLE converges to ${\theta }^{*},$ the minimizer of the Kullback-Leibler information

$$K\left(\theta \right)=E_g\left[\log \left\{\frac{g\left(Y|x\right)}{f\left(Y|x;\theta \right)}\right\}\right]$$

for $\theta \in \Omega .$ Sung and Geyer (2007) discussed the asymptotic properties of the Monte Carlo MLE of $\theta$ under missing data.

To formally discuss robustness, suppose that the true distribution $g\left(y|x\right)$ belongs to the neighborhood

$${\mathcal{N}}_{\epsilon }=\left\{g;D\left(g\mathrm{,}f\right)\text{ < }\frac{1}{2}{\epsilon }^2\right\} (4.1)$$

for some radius $\epsilon \text{>}\text{0},$ where

$$D\left(g\mathrm{,}f\right)={\displaystyle \int log\left(\frac{g}{f}\right)gdy}\mathrm{,} (4.2)$$

is the Kullback-Leibler distance measure. The neighborhood (4.1) can be characterized in the following way. Let $z\left(x\mathrm{,}y\mathrm{,}\theta \right)$ be a function of $x,y$ and $\theta ,$ standardized to satisfy $E_{Y|x}\left(z\right)=0$ and $Va{r}_{Y|x}\left(z\right)=1,$ and define

$$g\left(y|x\right)=f\left(y|x;\theta \right)\exp \left\{\epsilon z\left(x\mathrm{,}y\mathrm{,}\theta \right)-\kappa \left(x\mathrm{,}\theta \right)\right\}\mathrm{,} (4.3)$$

where

$$\kappa =\log \left(E_{Y|x}\left[\exp \left\{\epsilon z\left(x\mathrm{,}Y\mathrm{,}\theta \right)\right\}\right]\right)\mathrm{.}$$

For small $\epsilon \text{>}0$ it can be shown that

$$\kappa \cong D\left(g\mathrm{,}f\right)\cong \frac{1}{2}{\epsilon }^2\mathrm{.} (4.4)$$

Equation (4.3) represents an extensive set of distributions close to $f\left(y|x;\theta \right)$ created by varying $z\left(x\mathrm{,}y\mathrm{,}\theta \right)$ over different standardized functions, where $z$ and $\epsilon$ contain some geometric interpretation which represent the direction and magnitude of the misspecification respectively. For $p\text{-}$ dimension parameter $\theta ,$ we can specify the directions of the misspecification as

$${\left(z_1\mathrm{,}{z}_2\mathrm{,}\dots \mathrm{,}{z}_p\right)}^T=I_{\theta }^{-1/2}s\left(x\mathrm{,}y\mathrm{,}\theta \right)\mathrm{,}$$

where $s\left(x\mathrm{,}y\mathrm{,}\theta \right)=\partial \log f\left(y|x;\theta \right)/\partial \theta$ and $I_{\theta }$ is the information matrix for $\theta .$ Represent $z\left(x\mathrm{,}y\mathrm{,}\theta \right)$ as

$$z\left(x\mathrm{,}y\mathrm{,}\theta \right)={\lambda }^T{I}_{\theta }^{-1/2}s\left(x\mathrm{,}y\mathrm{,}\theta \right)\mathrm{,}$$

where ${\displaystyle {\sum }_{i=1}^p}{\lambda }_i^2=1,$ then $z\left(x\mathrm{,}y\mathrm{,}\theta \right)$ satisfies the standardization criterion of $E_{Y|x}\left(z\right)=0$ and $Va{r}_{Y|x}\left(z\right)=1.$ See Copas and Eguchi (2001) for further discussion of this expression.

Let $w_{ij\mathrm{,}g}^{*}$ be the fractional weight of the form (3.3) using the true density $g$ and $w_{ij\mathrm{,}f}^{*}$ be the corresponding fractional weight using the "working density" $f.$ By the special construction of the weights, we can establish

$$w_{ij\mathrm{,}g}^{*}\cong w_{ij\mathrm{,}f}^{*}+\epsilon {\lambda }^T{I}_{\theta }^{-1/2}\frac{\partial }{\partial \theta }\left(w_{ij\mathrm{,}f}^{*}\right)\mathrm{.} (4.5)$$

Proof of (4.5) is given in Appendix A.2. Thus

$$\begin{array}{ll}{\displaystyle \sum _i}{w}_i{\displaystyle \sum _j}{w}_{ij\mathrm{,}g}^{*}U\left(\eta ;x_i\mathrm{,}{y}_j\right)\cong \hfill & {\displaystyle \sum _i}{w}_i{\displaystyle \sum _j}{w}_{ij\mathrm{,}f}^{*}U\left(\eta ;x_i\mathrm{,}{y}_j\right)\hfill \\ \hfill & +\epsilon {\lambda }^T{I}_{\theta }^{-1/2}{\displaystyle \sum _i}{w}_i{\displaystyle \sum _j}\frac{\partial }{\partial \theta }\left(w_{ij\mathrm{,}f}^{*}\right)U\left(\eta ;x_i\mathrm{,}{y}_j\right)\mathrm{.}\hfill \end{array} (4.6)$$

For small $\epsilon ,$ we have

$${\displaystyle \sum _i}{w}_i{\displaystyle \sum _j}{w}_{ij\mathrm{,}g}^{*}U\left(\eta ;x_i\mathrm{,}{y}_j\right)\cong {\displaystyle \sum _i}{w}_i{\displaystyle \sum _j}{w}_{ij\mathrm{,}f}^{*}U\left(\eta ;x_i\mathrm{,}{y}_j\right)\mathrm{,}$$

and so the resulting estimator of $\eta$ from ${\displaystyle {\sum }_i}{w}_i{\displaystyle {\sum }_j}{w}_{ij\mathrm{,}f}^{*}U\left(\eta ;x_i\mathrm{,}{y}_j\right)=0$ will be close to the true value ${\eta }_0.$

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