The focused information criterion for varying-coefficient partially linear measurement error models

Abstract

Under general parametric models, Claeskens and Hjort (J Am Stat Assoc 98:900–916, 2003) proposed a focused information criterion for model selection which emphasizes the accuracy of estimation for particular parameters of interest. This paper extends their framework to include a semi-parametric varying-coefficient partially linear model when covariates in both the parametric and the non-parametric parts are subject to measurement errors. We allow the covariance matrices of the measurement errors to be unknown and be estimated by replicated observations. Also, we derive the asymptotic properties of the frequentist model average estimator for the model in consideration, which generalizes the results obtained by Wang et al. (Electron J Stat 6:1017–1039, 2012). In addition to asymptotic properties, finite sample performance of the proposed methods are examined in a simulation study, and a data set obtained from Continuing Survey of Food Intakes by Individuals conducted by the U.S. Department of Agriculture’s (CSFII) is considered.

Appendix

The following conditions are required for the proof.

• 1. The random variable $T$ has bounded support $\varvec{\Omega }$, and its density $f$ is Lipschitz continuous and bounded away from 0 on its support.

• 2. For each $T\in \varvec{\Omega }$, the $r\times r$ matrix $\mathbf {E}(ZZ^{\top }|T)$ is non-singular, and each element of $\mathbf {E}(ZZ^{\top }|T)$, $\mathbf {E}(XX^{\top }|T)$ or $\mathbf {E}(ZX^{\top }|T)$ is Lipschitz continuous.

• 3. There exists some $\epsilon >2$ such that $\mathbf {E} \Vert X\Vert ^{2\epsilon }<\infty$, $\mathbf {E} \Vert Z\Vert ^{2\epsilon }<\infty$, $\mathbf {E} \Vert U\Vert ^{2\epsilon }<\infty$, $\mathbf {E} \Vert V\Vert ^{2\epsilon }<\infty$ and $\mathbf {E} \Vert \varepsilon \Vert ^{2\epsilon }<\infty$, and $\rho <2-\epsilon ^{-1}$ such that $nh^{2\rho -1}\rightarrow \infty$ and $nh^8\rightarrow 0$.

• 4. $\alpha _j(T),j=1,\ldots ,r$, is twice continuously differentiable in $T\in \varvec{\Omega }$.

• 5. $K(\cdot )$ is a symmetric density with compact support.

Proof of Theorem 1

Let $\hat{U}_i =({\bar{\psi }}_{i}\mathbf {U})^{\top }$, $\hat{\varepsilon }_i={\bar{\psi }}_{i}\varvec{\varepsilon }$ and $\bigtriangledown =\widetilde{\bar{\mathbf {W}}}^{\top }\widetilde{\bar{\mathbf {W}}}-nJ^{-1}\hat{\Sigma }_u$. Then from Eq. (4), we have

$$\begin{aligned} \hat{\theta }-\theta _{{\mathrm{true}}}&=\bigtriangledown ^{-1}\sum _{i=1}^n(\bar{W}_i-\hat{\bar{W}}_i)(Y_i-\hat{\bar{Y}}_i)-\bigtriangledown ^{-1}\bigtriangledown \theta _{{\mathrm{true}}}\\&=\bigtriangledown ^{-1}nJ^{-1}\hat{\Sigma }_u\theta _{{\mathrm{true}}}+\bigtriangledown ^{-1}\sum _{i=1}^n(\bar{W}_i-\hat{\bar{W}}_i)\left\{ Y_i-\hat{\bar{Y}}_i-(\bar{W}_i-\hat{\bar{W}}_i)^{\top }\theta _{{\mathrm{true}}}\right\} . \end{aligned}$$

From the expressions of $\hat{\bar{Y}}_i$, $\hat{\bar{W}}_i$, $\hat{\bar{U}}_i$ and $\hat{\bar{\varepsilon }}_i$, we obtain

$$\begin{aligned}&Y_i-\hat{\bar{Y}}_i-(\bar{W}_i-\hat{\bar{W}}_i)^{\top }\theta _{{\mathrm{true}}} =Z_i^{\top }\alpha (T_i)+\varepsilon _i-\bar{U}_i^{\top }\theta _{{\mathrm{true}}}-\hat{\varepsilon }_i+\hat{\bar{U}}_i^{\top }\theta _{{\mathrm{true}}}-{\bar{\psi }}_{i}\mathbf {M}. \end{aligned}$$

Hence,

$$\begin{aligned}&\sum _{i=1}^n(\bar{W}_i-\hat{\bar{W}}_i)\left\{ Y_i-\hat{\bar{Y}}_i-(\bar{W}_i-\hat{\bar{W}}_i)\theta _{{\mathrm{true}}}\right\} \\&\quad =\sum _{i=1}^n(\bar{W}_i-\hat{\bar{W}}_i)(\varepsilon _i-\bar{U}_i^{\top }\theta _{{\mathrm{true}}})\\&\qquad +\sum _{i=1}^n(\bar{W}_i-\hat{\bar{W}}_i)(\hat{\bar{U}}_i^{\top }\theta _{{\mathrm{true}}}-\hat{\varepsilon }_i) +\sum _{i=1}^n(\bar{W}_i-\hat{\bar{W}}_i)\{Z_i^{\top }\alpha (T_i)-{\bar{\psi }}_{i}\mathbf {M}\}\\&\quad =\sum _{i=1}^n\left[ \bar{W}_i-\mathbf {E}(\bar{W}_iZ_i^{\top }|T_i)\{\mathbf {E}(Z_iZ_i^{\top }|T_i)\}^{-1}{\bar{\zeta }}_{i}\right] (\varepsilon _i-\bar{U}_i^{\top }\theta _{{\mathrm{true}}})\\&\qquad +\sum _{i=1}^n\left[ \mathbf {E}(\bar{W}_iZ_i^{\top }|T_i)\{\mathbf {E}(Z_iZ_i^{\top }|T_i)\}^{-1}{\bar{\zeta }}_{i}-\hat{\bar{W}}_i\right] (\varepsilon _i-\bar{U}_i^{\top }\theta _{{\mathrm{true}}})\\&\qquad +\sum _{i=1}^n(\bar{W}_i-\hat{\bar{W}}_i)(\hat{\bar{U}}_i^{\top }\theta _{{\mathrm{true}}}-\hat{\varepsilon }_i) +\sum _{i=1}^n(\bar{W}_i-\hat{\bar{W}}_i)\{Z_i^{\top }\alpha (T_i)-{\bar{\psi }}_{i}\mathbf {M}\}\\&\quad \equiv J_1+J_2+J_3+J_4. \end{aligned}$$

Applying the method used in Fan and Huang (2005) provides that, uniformly in $T$, $\hat{\bar{W}}_i^{\top }=(\bar{\varvec{\zeta }}_{i}^{\top },\;0)\left\{ (\mathcal {D}^{\bar{\zeta }}_{t_i})^{\top }\Omega _{t_i}\mathcal {D}^{\bar{\zeta }}_{t_i}-{\bar{\phi }}_{t_i}\right\} ^{-1}(\mathcal {D}^{\bar{\zeta }}_{t_i})^{\top }\Omega _{t_i}\mathbf {W}={\bar{\zeta }}_{i}^{\top }\{\mathbf {E}(Z_iZ_i^{\top }|T_i)\}^{-1}\mathbf {E}(Z_iX_i^{\top }|T_i)\{1+O_P(c_n)\},$ where $c_n=\{\log (1/h)/(nh)\}^{1/2}+h^2$. In addition, since $\{\mathbf {E}(\bar{W}_iZ_i^{\top }|T_i)\}^{\top }=\mathbf {E}(Z_i\bar{W}_i^{\top }|T_i)$ and $\theta _{{\mathrm{true}}}=\theta _0+(0^{\top },\delta ^{\top })^{\top }/\sqrt{n}$, we have

$$\begin{aligned} J_2=\sum _{i=1}^n[\mathbf {E}(\bar{W}_iZ_i^{\top }|T_i)\{\mathbf {E}(Z_iZ_i^{\top }|T_i)\}^{-1}{\bar{\zeta }}_{i}](\varepsilon _i-\bar{U}_i^{\top }\theta _0)O_P(c_n). \end{aligned}$$

The application of the Central Limit Theorem yields $\sum _{i=1}^n[\mathbf {E}(\bar{W}_iZ_i^{\top }|T_i)\{\mathbf {E}(Z_iZ_i^{\top }|T_i)\}^{-1}Z_i](\varepsilon _i-\bar{U}_i^{\top }\theta _0)=O_P(\sqrt{n})$. Therefore, $J_2=O_P(\sqrt{n}c_n)=o_P(\sqrt{n})$. Similarly $J_3=o_P(\sqrt{n})$, and $J_4=o_P(\sqrt{n})$. Using Slutsky’s Theorem and recognizing that $\bigtriangledown /n=B_n\overset{p}{\longrightarrow }B$ as $n\rightarrow \infty$, we obtain

$$\begin{aligned} \sqrt{n}(\hat{\theta }-\theta _{{\mathrm{true}}})&= \, \bigg (\frac{\bigtriangledown }{n}\bigg )^{-1}\frac{1}{\sqrt{n}}\sum _{i=1}^{n}\bigg \{\Big (\bar{W}_i-E(\bar{W}_iZ_i^{\top }|T_i)\{E(Z_iZ_i^{\top }|T_i)\}^{-1}{\bar{\zeta }}_{i}\Big )\\&\qquad \times \left( \varepsilon _i-\bar{U}_i^{\top }\theta _{{\mathrm{true}}}\right) +\frac{\sum _{j=1}^{J}(W_{ij}-\bar{W}_i)^{\otimes 2}\theta _{{\mathrm{true}}}}{J(J-1)}\bigg \}+o_P(1)\\&= \bigg (\frac{\bigtriangledown }{n}\bigg )^{-1}\frac{1}{\sqrt{n}}\sum _{i=1}^{n}\bigg \{\Big (\bar{W}_i-E(\bar{W}_iZ_i^{\top }|T_i)\{E(Z_iZ_i^{\top }|T_i)\}^{-1}{\bar{\zeta }}_{i}\Big )\\&\qquad \times (\varepsilon _i-\bar{U}_i^{\top }\theta _0)+\frac{\sum _{j=1}^{J}(W_{ij}-\bar{W}_i)^{\otimes 2}\theta _0}{J(J-1)}\bigg \}+o_P(1)\\&\overset{d}{\longrightarrow } N(0,\;B^{-1}FB^{-1}). \end{aligned}$$

The above results together with Eq. (5) and the Continuous Mapping Theorem finish the proof. $\square$

Proofs of Theorem 2 and Theorem 3

With the result in Theorem 1, they can be proved using approaches similar to those used in the proof of Theorem 1 and Theorem 2 in Wang et al. (2012), respectively. We skip the details here to save space. $\square$