Empirical likelihood for varying-coefficient semiparametric mixed-effects errors-in-variables models with longitudinal data

Abstract

In this paper, the empirical likelihood inferences for varying-coefficient semiparametric mixed-effects errors-in-variables models with longitudinal data are investigated. We construct the empirical log-likelihood ratio function for the fixed-effects parameters and the mean parameters of random-effects. The empirical log-likelihood ratio at the true parameters is proven to be asymptotically $\chi ^2_{q+r}$, where $q$ and $r$ are dimensions of the fixed and random effects respectively, and the corresponding confidence regions for them are then constructed. We also obtain the maximum empirical likelihood estimator of the parameters of interest, and prove it is the asymptotically normal under some suitable conditions. A simulation study and a real data application are undertaken to assess the finite sample performance of the proposed method.

Appendix

To illustrate our main results, the following assumptions are imposed. These assumptions are actually quite mild and can be easily satisfied.

• (a) The bandwidth satisfies $h=h_0n^{-1/5}$ for some constant $h_0>0$.

• (b) The matrices $\Upsilon (t)$, $\Phi (t)$ and $\Psi (t)$ are twice continuously differentiable on $(0,1)$, and $\Upsilon (t)$ is positive definite on $(0,1)$.

• (c) $\{\alpha _l(u),l=1,\ldots ,p\}$ has continuous second derivatives on $(0,1)$.

• (d) The Kernel $K(\cdot )$ is a symmetric density function with compact support.

• (e) The intensity function $f(t)$ of the process $N(t)$ is bounded away from 0 and infinity on $[0,1]$, and is continuously differentiable on $(0,1)$.

• (f) There is an $s>2$ such that $\sup _{0\le t\le 1}E\Vert X(t)\Vert ^{2s}<\infty$, $\sup _{0\le t\le 1}E\Vert W(t)\Vert ^{2s}<\infty$, $\sup _{0\le t\le 1}E\Vert Z(t)\Vert ^{4s}<\infty$, $\sup _{0\le t\le 1}E\Vert \mu (t)\Vert ^{2s}<\infty$, $\sup _{0\le t\le 1}E\Vert \nu (t)\Vert ^{2s}\!<\!\infty$, $E\Vert \gamma \Vert ^{4s}<\infty$, $\sup _{0\le t\le 1}E\Vert \epsilon (t)\Vert ^{2s}<\infty$, $i=1,\ldots n$, and for some $\delta <2-s^{-1}$ such that $n^{2\delta -1}h\rightarrow \infty$.

• (g) $\Gamma$ is a positive definite matrix, where $\Gamma$ is defined in Theorem 2.

Let $c_n=\left( \frac{\log (1/h)}{nh}\right) ^{1/2}+h^2$, $\kappa _l=\int u^lK(u)du$, $l=0,1,2$, and $A\otimes B$ be the Kronecker product of matrix $A$ and $B$. To prove our main results, we first give several lemmas.

Lemma 1

Suppose that assumptions (a)–(f) hold. Then it holds uniformly for $t\in \mathcal I$

$$\begin{aligned} B_t^T M_t B_t&= nf(t)\textit{diag}(1,\kappa _2)\otimes \Upsilon (t)(1+O_p(c_n)),\end{aligned}$$

(5.1)

$$\begin{aligned} B_t^T M_t W&= nf(t)(1,0)^T\otimes \Phi (t)(1+O_p(c_n)),\end{aligned}$$

(5.2)

$$\begin{aligned} B_t^T M_t Z&= nf(t)(1,0)^T\otimes \Psi (t)(1+O_p(c_n)),\end{aligned}$$

(5.3)

$$\begin{aligned} B_t^T M_t X_\alpha&= nf(t)(1,0)^T\otimes \Upsilon (t)\alpha (t)(1+O_p(c_n)). \end{aligned}$$

(5.4)

Here we omit the proof, which is similar to Lemma 2 in Fan and Huang (2005).

Lemma 2

Suppose that assumptions (a)–(f) hold. Then it holds uniformly for $t\in \mathcal I$

$$\begin{aligned} S(t)W&= \Upsilon ^{-1}(t)\Phi (t)(1+O_p(c_n)),\end{aligned}$$

(5.5)

$$\begin{aligned} S(t)Z&= \Upsilon ^{-1}(t)\Psi (t)(1+O_p(c_n)),\end{aligned}$$

(5.6)

$$\begin{aligned} S(t)X_\alpha&= \alpha (t)(1+O_p(c_n)). \end{aligned}$$

(5.7)

Proof

Noting that $S(t)=[I_p\ \ 0] \left( B_t^TM_tB_t\right) ^{-1}B_t^TM_t$, (5.5)–(5.7) can be obtained directly by Lemma 1.

Lemma 3

Suppose that assumptions (a)–(f) hold. Then it holds uniformly for $t\in \mathcal I$

$$\begin{aligned} S(t)\epsilon&= O_p(c_n),\end{aligned}$$

(5.8)

$$\begin{aligned} S(t)\omega&= O_p(c_n), \end{aligned}$$

(5.9)

where $\omega$ is $\mu$ or $\nu$.

Proof

With the similar proof of Lemmas 1 and 2, one can obtain (5.8) and (5.9).

Lemma 4

Let $e_i, i=1,\ldots ,n$, be a sequence of multi-independent random variate with $E(e_i)=0$ and $E(e_i^2)<c<\infty$. Then

$$\begin{aligned} \max _{1\le k\le n}\left| \sum \limits _{i=1}^k e_i\right| =O_p(\sqrt{n}\log n). \end{aligned}$$

Further, let $(j_1,\ldots ,j_n)$ be a permutation of $(1,\ldots ,n)$. Then we have

$$\begin{aligned} \max _{1\le k\le n}\left| \sum \limits _{i=1}^k e_{j_i}\right| =O_p(\sqrt{n}\log n). \end{aligned}$$

Proof

The proof of Lemma 4 can be found in Zhao and Xue (2009).

Let $e_{ij}=Z_{ij}^T(\gamma _i-\gamma _\mu )+\epsilon _{ij}-(\mu ^T_{ij},\nu ^T_{ij})\beta$,

$$\begin{aligned} J_{i1}&= \sum \limits _{j=1}^{n_i}\Bigg \{\left( \begin{array}{c} W_{ij}+\widetilde{\mu }_{ij}-\Lambda ^T(t_{ij})X_{ij} \\ Z_{ij}+\widetilde{\nu }_{ij}-\Pi ^T(t_{ij})X_{ij} \\ \end{array} \right) \left[ Z_{ij}^T(\gamma _i-\gamma _\mu )+\epsilon _{ij}-(\widetilde{\mu }_{ij}^T,\widetilde{\nu }_{ij}^T)\beta \right] \\&\quad +\textit{diag}(\widetilde{\Sigma }_{ij\mu },\widetilde{\Sigma }_{ij\nu })\beta \Bigg \},\\ J_{i2}&= \sum \limits _{j=1}^{n_i}\left\{ \left( \begin{array}{c} \Lambda ^T(t_{ij})X_{ij}-(S(t_{ij})W)^TX_{ij} \\ \Pi ^T(t_{ij})X_{ij}-(S(t_{ij})Z)^TX_{ij} \\ \end{array} \right) [e_{ij}+X_{ij}^TS(t_{ij})(\mu ,\nu )\beta ] \right\} ,\\ J_{i3}&= \sum \limits _{j=1}^{n_i}\left\{ \left( \begin{array}{c} W_{ij}-\Lambda ^T(t_{ij})X_{ij} \\ Z_{ij}-\Pi ^T(t_{ij})X_{ij}\\ \end{array} \right) \left[ X_{ij}^T\alpha (t_{ij})-X_{ij}^TS(t_{ij})(Y-(W,Z)\beta )\right] \right\} ,\\ J_{i4}&= \sum \limits _{j=1}^{n_i}\left\{ \left( \begin{array}{c} \mu _{ij} \\ \nu _{ij}\\ \end{array} \right) \left[ X_{ij}^T\alpha (t_{ij})-X_{ij}^TS(t_{ij})(Y-(W,Z)\beta )\right] \right\} ,\\ J_{i5}\!&= \!\sum \limits _{j=1}^{\!n_i}\left\{ \left( \begin{array}{c} \Lambda ^T(t_{ij})\!X_{ij}\!-\!(\!S(t_{ij})W^*)^TX_{ij} \\ \Pi ^T(t_{ij})\!X_{ij}\!-\!(S(t_{ij})Z^*)^TX_{ij}\\ \end{array} \right) \left[ X_{ij}^T\alpha (t_{ij})\!-\!X_{ij}^TS(t_{ij})(Y\!-\!(W,Z)\!\beta )\right] \right\} . \end{aligned}$$

and $J_\kappa =\frac{1}{\sqrt{n}}\sum \limits _{i=1}^n J_{i\kappa }, \kappa =1,\ldots ,5$.

Lemma 5

Suppose that assumptions (a)–(f) hold, we have

$$\begin{aligned} J_\kappa =o_p(1), \kappa =2,\ldots ,5. \end{aligned}$$

(5.10)

Proof

First, we prove $J_2=o_p(1)$. Let $A_{ij}=(\Lambda (t_{ij})-S(t_{ij})W,\Pi (t_{ij})-S(t_{ij})Z)^T$ be $(q+r)\times p$ matrix, and $b_{ij}=X_{ij}e_{ij}$ be $p\times 1$ vector. Then

$$\begin{aligned} J_2=\frac{1}{\sqrt{n}}\sum \limits _{i=1}^n\sum \limits _{j=1}^{n_i}A_{ij}b_{ij} +\frac{1}{\sqrt{n}}\sum \limits _{i=1}^n\sum \limits _{j=1}^{n_i}A_{ij}X_{ij}X_{ij}^TS(t_{ij})(\mu ,\nu )\beta =J_{21}+J_{22}. \nonumber \\ \end{aligned}$$

(5.11)

Further, let $a_{ij,rs}$ be the $(r,s)$ component of $A_{ij}$, $b_{ij,s}$ be the $s$th component of $b_{ij}$, $a_{ij_0,rs_0}b_{ij_0,s_0}=\max _{j,s}\{a_{ij,rs}b_{ij,s}\}$, and $(a_{l_i,r},i=1,\ldots ,n)$ be a permutation of $(a_{ij_0,rs_0},i=1,\ldots ,n)$ such that $a_{l_1,r}\ge \cdots \ge a_{l_n,r}$, corresponding permutation of $(b_{ij_0,s_0},i=1,\ldots ,n)$ are denoted by $(b_{l_i},i=1,\ldots ,n)$. Denote $J_{21r}$ be the $r$th component of $J_{21}$. By Abel’s inequality, Lemmas 2 and 4, we have

$$\begin{aligned} |J_{21r}|&= \frac{1}{\sqrt{n}}\left| \sum \limits _{i=1}^n \sum \limits _{j=1}^{n_i}\sum \limits _{s=1}^pa_{ij,rs}b_{ij,s}\right| \le \frac{pn_i}{\sqrt{n}}\left| \sum \limits _{i=1}^n a_{ij_0,rs_0}b_{ij_0,s_0}\right| \\&= \frac{pn_i}{\sqrt{n}}\left| \sum \limits _{i=1}^n a_{l_i,r}b_{l_i}\right| \le \frac{C}{\sqrt{n}}\sup _{1\le i\le n}|a_{l_i,r}|\max _{1\le k\le n}\left| \sum \limits _{i=1}^k b_{l_i}\right| \\&= \frac{C}{\sqrt{n}}O_p(c_n)O_p(\sqrt{n}\log n)=o_p(1). \end{aligned}$$

For $J_{22}$, by (5.5), (5.6) and (5.9), we have $\Vert J_{22}\Vert \le O_p(\sqrt{n}c_n^2)=o_p(1)$. Together it with (5.11), $J_2=o_p(1)$ holds.

For $J_3$ and $J_4$, we denote $A_{ij}=\alpha (t_{ij})-S(t_{ij})(Y-(W,Z)\beta )$. From (5.7) and (5.8), we obtain uniformly in $t_{ij}\in \mathcal I$

$$\begin{aligned} \Vert A_{ij}\Vert =\Vert (\alpha (t_{ij})-S(t_{ij})X_\alpha )\Vert +\Vert S(t_{ij})\epsilon \Vert =O_p(c_n). \end{aligned}$$

(5.12)

Together this with $E\left\{ \left( \begin{array}{c} W_{ij}-\Lambda ^T(t_{ij})X_{ij} \\ Z_{ij}-\Pi ^T(t_{ij})X_{ij}\\ \end{array} \right) X_{ij}^T\right\} =0$, and $E\left\{ \left( \begin{array}{c} \mu _{ij} \\ \nu _{ij}\\ \end{array} \right) X_{ij}^T\right\} =0$, and similar to the proof of $J_{21}$, we obtain $J_3=o_p(1)$ and $J_4=o_p(1)$. For $J_5$, by (5.5), (5.6), (5.9) and (5.12), we have $\Vert J_5\Vert \le O_p(\sqrt{n}c_n^2)=o_p(1)$. So the proof of Lemma 5 is completed.

Lemma 6

Suppose that assumptions (a)–(f) hold, we have

$$\begin{aligned} \frac{1}{\sqrt{n}}\sum \limits _{i=1}^n\eta _{i}(\beta ){\stackrel{\mathfrak{D }}{\longrightarrow }}N(0, B), \end{aligned}$$

where $B$ is defined in Theorem 2.

Proof

From (2.7), we have

$$\begin{aligned} \left( \begin{array}{c} W_{ij}^* \\ Z_{ij}^* \\ \end{array} \right)&= \left( \begin{array}{c} W_{ij}+\widetilde{\mu }_{ij}-\Lambda ^T(t_{ij})X_{ij} \\ Z_{ij}+\widetilde{\nu }_{ij}-\Pi ^T(t_{ij})X_{ij} \\ \end{array} \right) +\left( \begin{array}{c} \Lambda ^T(t_{ij})X_{ij}-(S(t_{ij})W)^TX_{ij}\\ \Pi ^T(t_{ij})X_{ij}-(S(t_{ij})Z)^TX_{ij} \\ \end{array} \right) \\&= \left( \begin{array}{c} W_{ij}-\Lambda ^T(t_{ij})X_{ij} \\ Z_{ij}-\Pi ^T(t_{ij})X_{ij} \\ \end{array} \right) +\left( \begin{array}{c} \mu _{ij} \\ \nu _{ij} \\ \end{array} \right) +\left( \begin{array}{c} \Lambda ^T(t_{ij})X_{ij}-(S(t_{ij})W^*)^TX_{ij} \\ \Pi ^T(t_{ij})X_{ij}-(S(t_{ij})Z^*)^TX_{ij} \\ \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} \widetilde{Y}_{ij}-\left( \widetilde{ W}^{*T}_{ij}, \widetilde{Z}^{*T}_{ij}\right) \beta&= \left[ X_{ij}^T\alpha (t_{ij})-X_{ij}^TS(t_{ij})(Y-(W,Z)\beta )\right] \\&+\left[ e_{ij}+X_{ij}^TS(t_{ij})(\mu ,\nu )\beta \right] . \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{1}{\sqrt{n}}\sum \limits _{i=1}^n \eta _{i}(\beta )=J_{1}+J_{2}+J_{3}+J_{4}+J_{5}. \end{aligned}$$

(5.13)

By directly calculating its expectation and variance, we have $E(J_1)=0$ and $\textit{Var}(J_1)=B+o(1)$. Then, by the central limit theorem, we have $J_1{\stackrel{\mathfrak{D }}{\longrightarrow }}N(0, B)$. Further, by Lemma 5 and (5.13), we compete the proof of Lemma 6.

Lemma 7

Suppose that assumptions (a)–(f) hold, we have

$$\begin{aligned} \frac{1}{\sqrt{n}}\sum \limits _{i=1}^n \eta _{i}(\beta )\eta _{i}^T(\beta ){\stackrel{\mathcal{P }}{\longrightarrow }}B. \end{aligned}$$

We omit the proof of Lemma 7, which can be obtained based on Lemmas 5 and 6 by the same arguments as for the proof of Lemma 5.6 in Zhao and Xue (2009).

Lemma 8

Suppose that assumptions (a)–(f) hold, we have

$$\begin{aligned} \max _{1\le i\le n} \Vert \eta _{i}(\beta )\Vert =o_p(n^{1/2}). \end{aligned}$$

Proof

Note that

$$\begin{aligned} \max _{1\le i\le n}\Vert \eta _{i}(\beta )\Vert \le \max _{1\le i\le n}\Vert J_{i1}\Vert \!+\!\max _{1\le i\le n}\Vert J_{i2}\Vert \!+\!\max _{1\le i\le n}\Vert J_{i3}\Vert \!+\!\max _{1\le i\le n}\Vert J_{i4}\Vert \!+\!\max _{1\le i\le n}\Vert J_{i5}\Vert . \end{aligned}$$

By Lemma 3 of Owen (1990), we have $\max _{1\le i\le n}\Vert J_{i1}\Vert =o_p(n^{1/2})$. From (5.5), (5.6), (5.9) and Lemma 3 of Owen (1990), we have

$$\begin{aligned}&\max _{1\le i\le n}\Vert J_{i2}\Vert \le O_p(c_n)\\&\quad \times \left( \max _{1\le i\le n}\left\| \int \limits _0^1b_i(t)dN_i(t)\right\| +\max _{1\le i\le n}\left\| \int \limits _0^1X_i(t)X_i^T(t)S(t)(\mu ,\nu )\beta dN_i(t)\right\| \right) \\&=O_p(c_n)\left( o_p(n^{1/2})+o_p(n^{1/2})o_p(n^{1/2})O_p(c_n)\right) =o_p(n^{1/2}). \end{aligned}$$

Similar to the arguments of $J_{i2}$, we can obtain $\max _{1\le i\le n}\Vert J_{il}\Vert =o_p(n^{1/2}) (l=3,4,5)$. This competes the proof of Lemma 8.

Proof of Theorem 1

From Lemmas 6–8, using the same arguments as were used in the proof of expression (2.14) of Owen (1990), we have

$$\begin{aligned} \Vert \lambda \Vert =O_p(n^{-1/2}). \end{aligned}$$

(5.14)

From (2.13), we have

$$\begin{aligned} 0&= \frac{1}{n}\sum \limits _{i=1}^n\frac{\eta _i(\beta )}{1+\lambda ^T\eta _i(\beta )} =\frac{1}{n}\sum \limits _{i=1}^n\eta _i(\beta )-\frac{1}{n}\sum \limits _{i=1}^n\eta _i(\beta )\eta _i^T(\beta )\lambda \\&+\frac{1}{n}\sum \limits _{i=1}^n\frac{\eta _i(\beta )(\lambda ^T\eta _i(\beta ))^2}{1+\lambda ^T\eta _i(\beta )}. \end{aligned}$$

By using (5.14) and Lemma 8, we obtain

$$\begin{aligned}&\sum \limits _{i=1}^n(\lambda ^T\eta _i(\beta ))^2=\sum \limits _{i=1}^n\lambda ^T\eta _i(\beta )+o_p(1),\end{aligned}$$

(5.15)

$$\begin{aligned}&\lambda =\left[ \sum \limits _{i=1}^n\eta _i(\beta )\eta _i^T(\beta )\right] ^{-1}\sum \limits _{i=1}^n\eta _i(\beta ) +o_p(n^{-1/2}). \end{aligned}$$

(5.16)

Applying the Taylor expansion to (2.12), and using (5.14) and Lemma 8, it follows that

$$\begin{aligned} \ell _n(\beta )=2\sum \limits _{i=1}^n\left[ \lambda ^T\eta _i(\beta )- \frac{1}{2}(\lambda ^T\eta _i(\beta ))^2\right] +o_p(1). \end{aligned}$$

(5.17)

Then, by (5.15)–(5.17), we have

$$\begin{aligned} \ell _n(\beta )=\left[ \frac{1}{\sqrt{n}}\sum \limits _{i=1}^n\eta _i(\beta )\right] ^T \left[ \frac{1}{n}\sum \limits _{i=1}^n\eta _i(\beta )\eta _i^T(\beta )\right] ^{-1} \left[ \frac{1}{\sqrt{n}}\sum \limits _{i=1}^n\eta _i(\beta )\right] +o_p(1). \end{aligned}$$

Together with Lemmas 6–8, This completes the proof of Theorem 1.

Proof of Theorem 2

Following the similar arguments as were used in the proof of Theorem 2 in Xue and Zhu (2008), we have

$$\begin{aligned} \hat{\beta }-\beta =\hat{\Gamma }^{-1}n^{-1}\sum \limits _{i=1}^n\eta _i(\beta )+o_p(n^{-1/2}). \end{aligned}$$

By Lemma 2, we can prove $\hat{\Gamma }{\stackrel{\mathcal{P }}{\longrightarrow }}\Gamma$ by the law of large numbers. Together with Lemma 6 and the Slutsky Theorem, this proves Theorem 2.