Estimation in partially linear varying-coefficient errors-in-variables models with missing response variables

Abstract

In this paper, a partially linear varying-coefficient model with measurement errors in the nonparametric component as well as missing response variable is studied. Two estimators for the parameter vector and nonparametric function are proposed based on the locally corrected profile least squares method. The first estimator is constructed by using the complete-case data only, and another by using an imputation technique. Both proposed estimators of the parametric component are shown to be asymptotically normal, and the estimators of nonparametric function are proved to achieve the optimal strong convergence rate as the usual nonparametric regression. Some simulation studies are conducted to compare the behavior of these estimators and the results confirm that the estimators based on the imputation technique perform better than the complete-case data estimator in finite samples. Finally, an application to a real data set is illustrated.

Appendix: Proofs of the main results

We begin with the following assumption conditions required to derive the main results. These conditions are quite mild and can be easily satisfied.

C1: The random variable u has a bounded support $\Pi$. Its probability density function f(.) is Lipschitz continuous and bounded away from 0 on its support.

C2: The $q\times q$ matrix $\mathrm{E}(\mathbf{ZZ} ^T|U)$ and $\mathrm{E}(\delta \mathbf{ZZ} ^T|U)$ are nonsingular for each $U\in \Pi$. The matrix $\mathrm{E}(\mathbf{ZZ} ^T|U)$, $\mathrm{E}(\mathbf{ZZ} ^T|U)^{-1}$, $\mathrm{E}(\delta \mathbf{ZZ} ^T|U)$, $\mathrm{E}(\delta \mathbf{ZZ} ^T|U)^{-1}$, $\mathrm{E}(\mathbf{ZX} ^T|U)$ and $\mathrm{E}(\delta \mathbf{ZX} ^T|U)$ are all Lipschitz continuous.

C3: There exists an $s>0$ such that $\mathrm{E}||\mathbf{X} ||^{2s}<\infty$,$\mathrm{E}||\mathbf{Z} ||^{2s}<\infty$ and for some $k<2-s^{-1}$ such that $n^{2k-1}h\longrightarrow \infty .$

C4: $\alpha _j(u),j=1,\ldots ,q$ have continuous second derivative for $u\in \Pi$.

C5: The Kernel K(.) is a symmetric probability density function with compact support and the bandwidth h satisfies $nh^8\longrightarrow 0$ and $nh^2/(\mathrm{log} n)^2\longrightarrow \infty$ when $n\longrightarrow \infty$.

In order to prove the main results, we first give several Lemmas. The following notations will be used in the proof of the Lemmas and Theorems. Let $c_n=(\mathrm{log}n/nh)^{1/2}$, $\mu _i=\int _0^{\infty } t^i K(t)\mathrm{d}t$, $\mathbf{M} =[\mathbf{Z} _1^T\varvec{\alpha }(U_1),\dots ,\mathbf{Z} _n^T\varvec{\alpha }(U_n)]^T$, $\mathbf{M} ^\mathbf{W }=[\mathbf{W} _1^T\varvec{\alpha }(U_1),\ldots ,\mathbf{W} _n^T\varvec{\alpha }(U_n)]^T$, $\tilde{\varepsilon }_i={\varepsilon }_i-\sum _{k=1}^n \mathbf{S} _{ik}^{c} \varepsilon _k$ and $\tilde{\mathbf{Z }}_i=\mathbf{Z }_i-\sum _{k=1}^n \mathbf{S} _{ik}^{c} \mathbf{Z} _k.$

Lemma 1

Suppose that conditions C1–C5 hold. Then the followings hold uniformly

$$\begin{aligned} (\mathbf{D} _u^\mathbf{Z} )^T \varvec{\omega }_u\mathbf{D} _u^\mathbf{Z} -\varvec{\Omega }_u=nf(u)\varvec{\Gamma }(u)\otimes \left( \begin{array}{ll} 1&{} \mu _1\\ \mu _1&{} \mu _2 \\ \end{array}\right) [1+O_p(c_n)]. \end{aligned}$$

(20)

$$\begin{aligned} (\mathbf{D} _u^\mathbf{Z} )^T \varvec{\omega }_u \mathbf{X} =nf(u)\varvec{\Phi }(u)\otimes (1, \mu _1)^{T}[1+O_p(c_n)]. \end{aligned}$$

(21)

$$\begin{aligned} (\mathbf{D} _u^\mathbf{W} )^T \varvec{\omega }_u^\delta \mathbf{D} _u^\mathbf{W} -\varvec{\Omega }_u^\delta =nf(u)\varvec{\Gamma }_c(u)\otimes \left( \begin{array}{ll} 1&{} \mu _1\\ \mu _1&{} \mu _2 \\ \end{array} \right) [1+O_p(c_n)]. \end{aligned}$$

(22)

$$\begin{aligned} (\mathbf{D} _u^\mathbf{W} )^T \varvec{\omega }_u^\delta \mathbf{X} =nf(u)\varvec{\Phi }_c(u)\otimes (1, \mu _1)^{T}[1+O_p(c_n)]. \end{aligned}$$

(23)

Proof

Equations (20) and (21) are given in Lemma 2 in Feng and Xue (2014). Similarly, Eqs. (22) and (23) can also be obtained.

Lemma 2

Suppose that conditions C1–C5 hold. Then

$$\begin{aligned} \frac{1}{{n}}\sum _{i=1}^{n}\delta _i(\tilde{\mathbf{X }}_i\tilde{\mathbf{X }}_i^T-\mathbf{X} ^T\mathbf{Q} _i^T\varvec{\Sigma }_{\xi } \mathbf{Q} _i\mathbf{X} ) \longrightarrow \varvec{\Sigma _1}, ~~a.s, \end{aligned}$$

$$\begin{aligned} \frac{1}{{n}}\sum _{i=1}^{n}(\bar{\mathbf{X }}_i\bar{\mathbf{X }}_i^T-\mathbf{X} ^T\mathbf{R} _i^T\varvec{\Sigma }_{\xi } \mathbf{R} _i\mathbf{X} ) \longrightarrow \varvec{\Sigma }, ~~a.s, \end{aligned}$$

$$\begin{aligned} \frac{1}{{ n}}\sum _{i=1}^{n}(1-\delta _i)(\bar{\mathbf{X }}_i \tilde{\mathbf{X }}_i^T-\mathbf{X} ^T\mathbf{R} _i^T\varvec{\Sigma }_{\xi } \mathbf{Q} _i \mathbf{X} )\longrightarrow \varvec{\Sigma }_2,~~a.s, \end{aligned}$$

where $\varvec{\Sigma _1}$ is defined in Theorem 1, $\varvec{\Sigma }$ and $\varvec{\Sigma }_2$ are defined in Theorem 3.

Proof

The proof of this Lemma is similar to that of Lemma 7.2 in Fan and Huang (2005). Hence, the details are omitted.

Proof of Theorem 1

Let

$$\begin{aligned} B_n=\frac{1}{{n}}\sum _{i=1}^{n}\delta _i(\tilde{\mathbf{X }}_i\tilde{\mathbf{X }}_i^T-\mathbf{X} ^T\mathbf{Q} _i^T\varvec{\Sigma }_{\xi } \mathbf{Q} _i\mathbf{X} ), \end{aligned}$$

and

$$\begin{aligned} A_n=\frac{1}{{\sqrt{n}}}\sum _{i=1}^{n}\delta _i\left[ \tilde{\mathbf{X }}_i(\tilde{Y}_i-\tilde{\mathbf{X }}_i^T \varvec{\beta }) -\mathbf{X} ^T\mathbf{Q} _i^T\varvec{\Sigma }_\xi \mathbf{Q} _i(\mathbf{Y} -\mathbf{X} \varvec{\beta })\right] . \end{aligned}$$

Then,

$$\begin{aligned} \sqrt{n}(\hat{\varvec{\beta }}_c-\varvec{\beta })=B_n^{-1}A_n. \end{aligned}$$

(24)

For $A_n$, by simple calculation and similar proof of Lemma 4 in Feng and Xue (2014), we have

$$\begin{aligned} A_n= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^{n}\delta _i\left[ \tilde{\mathbf{X }}_i(\tilde{\mathbf{Z }}_i^T\varvec{\alpha }(U_i)+\tilde{\varepsilon }_i) -\mathbf{X} ^T\mathbf{Q} _i^T\varvec{\Sigma }_\xi \mathbf{Q} _i(\mathbf{M} +\varvec{\varepsilon })\right] \nonumber \\= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^{n}\delta _i \left\{ [\mathbf{X} _i-\varvec{\Phi }_c^T(U_i)\varvec{\Gamma }_c^{-1}(U_i)\mathbf{Z} _i] [\varepsilon _i-\varvec{\xi }_i^T\varvec{\alpha }(U_i)]\right. \nonumber \\&\left. -\,\varvec{\Phi }_c^T(U_i)\varvec{\Gamma }_c^{-1}(U_i)\varvec{\xi }_i\varepsilon _i +\varvec{\Phi }_c^T(U_i)\varvec{\Gamma }_c^{-1}(U_i)(\varvec{\xi }_i \varvec{\xi }_i^T-\varvec{\Sigma }_\xi )\varvec{\alpha }(U_i)\right\} +o_p(1)\nonumber \\= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^{n} \mathbf{G} _i +o_p(1). \end{aligned}$$

(25)

It is easy to see that $\mathbf{G} _i$ is independent and identical distributed with mean zero and $\mathrm {Cov}(\mathbf{G} _i)=\varvec{\Omega }_1.$

Thus, by the Slutsky theorem, Lemma 2 and the central limit theorem, we complete the Theorem.

Proof of Theorem 2

By the definition of $\hat{\varvec{\alpha }}_c(u)$, we can obtain that

$$\begin{aligned} \hat{\varvec{\alpha }}_c(u)= & {} (\mathbf{I} _q,\mathbf 0 _{q})[(\mathbf{D} _u^\mathbf{W} )^T \varvec{\omega }_u^\delta \mathbf{D} _u^\mathbf{W} -\varvec{\Omega }_u^\delta ]^{-1}(\mathbf{D} _u^\mathbf{W} )^T\varvec{\omega }_u^\delta \mathbf{M} \\ \nonumber&+\, (\mathbf{I} _q,\mathbf 0 _{q})[(\mathbf{D} _u^\mathbf{W} )^T \varvec{\omega }_u^\delta \mathbf{D} _u^\mathbf{W} -\varvec{\Omega }_u^\delta ]^{-1}(\mathbf{D} _u^\mathbf{W} )^T\varvec{\omega }_u^\delta \varvec{\varepsilon }\\\nonumber&+\, (\mathbf{I} _q,\mathbf 0 _{q})[(\mathbf{D} _u^\mathbf{W} )^T \varvec{\omega }_u^\delta \mathbf{D} _u^\mathbf{W} -\varvec{\Omega }_u^\delta ]^{-1}(\mathbf{D} _u^\mathbf{W} )^T\varvec{\omega }_u^\delta \mathbf{X} (\varvec{\beta }-\hat{\varvec{\beta }}_c). \end{aligned}$$

By Theorem 1, similar to the proof of Theorem 3.1 in Xia and Li (1999), it is easy to show that

$$\begin{aligned} \underset{{1\le j\le p}}{\mathrm {max}}\underset{{u\in \Pi }}{\mathrm {sup}}|\hat{\alpha }_{cj}(u)-\alpha _j(u)|=O\{h_1^2+(\mathrm {log}n/{nh_1})^{1/2}\},~~~~ a.s. \end{aligned}$$

Let $h_1=cn^{-1/5}$, where c is a constant. Then it yields that

$$\begin{aligned} \underset{{1\le j\le p}}{\mathrm {max}}\underset{{u\in \Pi }}{\mathrm {sup}}|\hat{\alpha }_{cj}(u)-\alpha _j(u)|=O(n^{-2/5}+(\mathrm {log} n)^{1/2}),~~~~a.s. \end{aligned}$$

Proof of Theorem 3

Similar to Theorem 1, it can be shown that

$$\begin{aligned} \sqrt{n}(\hat{\varvec{\beta }}_I-\varvec{\beta })=D_n^{-1}E_n, \end{aligned}$$

(26)

where

$$\begin{aligned} D_n=\frac{1}{{n}}\sum _{i=1}^{n}(\bar{\mathbf{X }}_i\bar{\mathbf{X }}_i^T-\mathbf{X} ^T\mathbf{R} _i^T\varvec{\Sigma }_{\xi } \mathbf{R} _i\mathbf{X} ), \end{aligned}$$

and

For convenience, we denote $[\mathbf{S} _c(\mathbf{A} )]_i$ and $[\mathbf{S} _I(\mathbf{A} )]_i$ to respectively be the ith row of product of $\mathbf{S} _c \mathbf{A}$ and $\mathbf{S} _I\mathbf{A}$ for a given matrix $\mathbf{A}$.

By simple calculation, it is obtained that

By Lemma 1, we have

$$\begin{aligned} I_1=\frac{1}{{\sqrt{n}}}\sum _{i=1}^{n}(\mathbf{X} _i-\varvec{\Phi }(U_i)\varvec{\Gamma }^{-1}(U_i) \mathbf{W} _i)\delta _i (\varepsilon _i-\varvec{\xi }_i^T\varvec{\alpha }(U_i))+o_p(1). \end{aligned}$$

(27)

In view of Theorem 1 and the law of large numbers, it follows that

$$\begin{aligned} \nonumber I_2= & {} \frac{1}{{ n}}\sum _{i=1}^{n}(1-\delta _i) \bar{\mathbf{X }}_i \tilde{\mathbf{X }}_i^T \sqrt{n} (\hat{\varvec{\beta }}_c-\varvec{\beta })\\\nonumber= & {} \frac{1}{{ n}}\sum _{i=1}^{n}(1-\delta _i)(\bar{\mathbf{X }}_i \tilde{\mathbf{X }}_i^T-\mathbf{X} ^T\mathbf{R} _i^T\varvec{\Sigma }_{\xi } \mathbf{Q} _i \mathbf{X} ) \sqrt{n} (\hat{\varvec{\beta }}_c-\varvec{\beta })\\\nonumber&+\,\frac{1}{{\sqrt{n}}}\sum _{i=1}^{n}(1-\delta _i)\mathbf{X} ^T\mathbf{R} _i^T\varvec{\Sigma }_{\xi } \mathbf{Q} _i \mathbf{X} (\hat{\varvec{\beta }}_c-\varvec{\beta })+o_p(1)\\= & {} \varvec{\Sigma }_2 \varvec{\Sigma }_1^{-1} \frac{1}{{\sqrt{n}}}\sum _{i=1}^{n}{} \mathbf{G} _i +\frac{1}{{\sqrt{n}}}\sum _{i=1}^{n}(1-\delta _i)\mathbf{X} ^T\mathbf{R} _i^T\varvec{\Sigma }_{\xi } \mathbf{Q} _i \mathbf{X} (\hat{\varvec{\beta }}_c-\varvec{\beta })+o_p(1),\nonumber \\ \end{aligned}$$

(28)

where $\mathbf{G} _i$ is defined in Theorem 1.

$I_3$ can be written as

$$\begin{aligned} I_3= & {} \frac{1}{{\sqrt{n}}}\sum _{i=1}^{n}(1-\delta _i)\mathbf{X }_i [\mathbf{S} _{c}(\varvec{\varepsilon }-\varvec{\xi }^T\varvec{\alpha }(u))]_i -\frac{1}{{\sqrt{n}}}\sum _{i=1}^{n}(1-\delta _i)[\mathbf{S} _I(\mathbf{X })]_i^T [\mathbf{S} _{c}(\varvec{\varepsilon }-\varvec{\xi }^T\varvec{\alpha }(u))]_i\\= & {} I_{31}-I_{32}. \end{aligned}$$

By Lemma 1, it can be shown that

$$\begin{aligned} I_{31}= & {} \frac{1}{{\sqrt{n}}}\sum _{i=1}^{n}(1-\delta _i)\mathbf{X }_i \mathbf{W} _i^T(n f(U_i)\varvec{\Gamma }_c(U_i))^{-1}\sum _{j=1}^nK_{h_1}(U_j-U_i)\mathbf{W} _j(\varepsilon _j-\varvec{\xi }_j^T\varvec{\alpha }(U_j))\delta _j\\= & {} \frac{1}{{\sqrt{n}}}\sum _{i=1}^{n} \varvec{\Phi }(U_i)\varvec{\Gamma }_c^{-1}(U_i) \mathbf{W} _i \delta _i(\varepsilon _i-\varvec{\xi }_i^T\varvec{\alpha }(U_i))\\&-\,\frac{1}{{\sqrt{n}}}\sum _{i=1}^{n} \varvec{\Phi }_c(U_i)\varvec{\Gamma }_c^{-1}(U_i) \mathbf{W} _i \delta _i(\varepsilon _i-\varvec{\xi }_i^T\varvec{\alpha }(U_i))+o_p(1). \end{aligned}$$

In a similar way, we obtain that,

$$\begin{aligned} I_{32}= & {} \frac{1}{{\sqrt{n}}}\sum _{i=1}^{n} \varvec{\Phi }(U_i)\varvec{\Gamma }_c^{-1}(U_i) \mathbf{W} _i \delta _i(\varepsilon _i-\varvec{\xi }_i^T\varvec{\alpha }(U_i))\\&-\,\frac{1}{{\sqrt{n}}}\sum _{i=1}^{n} \varvec{\Phi }(U_i)\varvec{\Gamma }^{-1}(U_i) \mathbf{W} _i \delta _i(\varepsilon _i-\varvec{\xi }_i^T\varvec{\alpha }(U_i))+o_p(1) \end{aligned}$$

Therefor,

$$\begin{aligned} I_3=\frac{1}{{\sqrt{n}}}\sum _{i=1}^{n}[ \varvec{\Phi }(U_i)\varvec{\Gamma }^{-1}(U_i) \mathbf{W} _i - \varvec{\Phi }_c(U_i)\varvec{\Gamma }_c^{-1}(U_i) \mathbf{W} _i ]\delta _i(\varepsilon _i-\varvec{\xi }_i^T\varvec{\alpha }(U_i))+o_p(1).\nonumber \\ \end{aligned}$$

(29)

$I_4$ can be expressed as

$$\begin{aligned} I_4= & {} \frac{1}{{\sqrt{n}}}\sum _{i=1}^{n}\bar{\mathbf{X }}_i (\mathbf{W} _i^T\varvec{\alpha }(U_i)-[\mathbf{S} _I(\mathbf{M} ^\mathbf{W} )]_i) -\frac{1}{{\sqrt{n}}}\sum _{i=1}^{n}\bar{\mathbf{X }}_i \delta _i [\mathbf{S} _{I}(\varepsilon -\varvec{\xi }^T\varvec{\alpha }(u))]_i\\&-\,\frac{1}{{\sqrt{n}}}\sum _{i=1}^{n}\bar{\mathbf{X }}_i (1- \delta _i) [\mathbf{S} _I(\mathbf{X} )]_i^T(\hat{\varvec{\beta }}_c-\varvec{\beta }) -\frac{1}{{\sqrt{n}}}\sum _{i=1}^{n}\bar{\mathbf{X }}_i (1- \delta _i)[\mathbf{S} _I(\hat{\mathbf{M }}_c^\mathbf{W }-\mathbf{M }^\mathbf{W })]_i\\= & {} I_{41}+I_{42}+I_{43}+I_{44} \end{aligned}$$

where $\hat{\mathbf{M }}_c^\mathbf{W }=[\mathbf{W} _1^T\hat{\varvec{\alpha }}_c(U_1),\ldots ,\mathbf{W} _n^T\hat{\varvec{\alpha }}_c(U_n)]^T$. By Lemma 1, it can be shown that $I_{41}=o_p(1)$ and $I_{42}=o_p(1)$. By the fact that $\hat{\varvec{\beta }}_c-\varvec{\beta }=O_p(n^{-1/2})$ from Theorem 1 and $\frac{1}{{ n}}\sum _{i=1}^{n}\bar{\mathbf{X }}_i[\mathbf{S} _I(\mathbf{X} )]_i=o_p(1)$, $I_{43}=o_p(1)$ is obtained. $I_{44}=o_p(1)$ can also be proved similarly. Thus, we have

$$\begin{aligned} I_{4}=o_p(1). \end{aligned}$$

(30)

Similar to the calculation of $I_4$, we can show that

$$\begin{aligned} I_5= & {} \frac{1}{{\sqrt{n}}}\sum _{i=1}^{n}(1-\delta _i)\bar{\mathbf{X }}_i ([\mathbf{S} _I(\mathbf{M} ^\mathbf{W} )]_i-\mathbf{W} _i^T\varvec{\alpha }(U_i))\\\nonumber= & {} o_p(1). \end{aligned}$$

(31)

Invoking (26)–(31), it can be obtained that

$$\begin{aligned} \sqrt{n}(\hat{\varvec{\beta }}_I-\varvec{\beta })=\Sigma ^{-1}(\Sigma _1+\Sigma _2)\Sigma _1^{-1}\frac{1}{{\sqrt{n}}}\sum _{i=1}^{n} \mathbf{G} _i+o_p(1). \end{aligned}$$

Thus, by the Slutsky theorem, Lemma 2 and the central limit theorem, we concludes the theorem.

Proof of Theorem 4

The proof of Theorem 4 is similar to Theorem 2, then, we omit it.