An improvement on the efficiency of complete-case-analysis with nonignorable missing covariate data

Abstract

This paper develops a weighted composite quantile regression method for linear models where some covariates are missing not at random but the missingness is conditionally independent of the response variable. It is known that complete case analysis (CCA) is valid under these missingness assumptions. By fully utilizing the information from incomplete data, empirical likelihood-based weights are obtained to conduct the weighted composite quantile regression. Theoretical results show that the proposed estimator is more efficient than the CCA one if the probability of missingness on the fully observed variables is correctly specified. Besides, the proposed algorithm is computationally simple and easy to implement. The methodology is illustrated on simulated data and a real data set.

Appendix

The following regularity conditions are required for the asymptotic analysis. Conditions (C1, C2, C6–C8) are similar to those in Sun and Ma (2017). Condition (C3) is a necessary assumption about the missing data mechanism in this paper. Condition (C4–C5) guarantee that the asymptotic covariance matrices of the CCA-based and ACC-based CQR estimators are both positive definite.

• (C1) $\varvec{w}$ has a bounded support.

• (C2) The density function $f(\cdot )$ of $\varepsilon$ is bounded away from zero and has a continuous and uniformly bounded derivative.

• (C3) $\delta$ is independent of y given $\varvec{w}$.

• (C4) $\varvec{D}$ and $\varvec{S_\phi }$ are positive definite.

• (C5) ${\varvec{S}}_{\varvec{B}}$, $\varvec{P}_2$ and $\varvec{S_\phi }-\varvec{P}_1\varvec{P}_2^{-1}\varvec{P}_1^T$ are positive definite.

• (C6) For all $(y_i,\varvec{z}_i)$, $\pi (y_i,\varvec{z}_i,\varvec{\gamma })$ admits all third partial derivatives $\frac{\partial ^3\pi (y_i,\varvec{z}_i,\varvec{\gamma })}{\partial \gamma _k\partial \gamma _l\partial \gamma _m}$ for all $\varvec{\gamma }$ in a neighborhood of the true value $\varvec{\gamma }^*$, $\Vert \frac{\partial ^3\pi (y_i,\varvec{z}_i,\varvec{\gamma })}{\partial \gamma _k\partial \gamma _l\partial \gamma _m}\Vert$ is bounded by an integrable function for all $\varvec{\gamma }$ in this neighborhood, and $\Vert \frac{\partial \pi (y_i,\varvec{z}_i,\varvec{\gamma })}{\partial \varvec{\gamma }^T}\Vert$ is bounded by an integrable function for all $\varvec{\gamma }$ in this neighborhood.

• (C7) $\pi (y,\varvec{z},\varvec{\gamma }^*)$ is bounded away from zero, i.e. $\underset{y,z}{\inf }\,\pi (y,\varvec{z},\varvec{\gamma }^*)\ge c_0$ for some $c_0 > 0$.

• (C8) $\Vert \varvec{\xi }(y,\varvec{z},\varvec{\zeta })\Vert ^2$ is bounded by an integrable function for all $\varvec{\zeta }$ in a neighbourhood of $\varvec{\zeta }^*$ and $\varvec{\xi }(y,\varvec{z},\varvec{\zeta })$ is continuous at each $\varvec{\zeta }$ with probability one in this neighbourhood, where $\varvec{\zeta }=(\varvec{\alpha }^T,\varvec{\beta }^T,\varvec{b}^T)^T$ and $\varvec{\zeta }^*=({\varvec{\alpha }^*}^T,{\varvec{\beta }^*}^T,{\varvec{b}^*}^T)^T$. For some $c > 0$,

$$\begin{aligned} \underset{\Vert \varvec{\zeta }-\varvec{\zeta }^*\Vert \le cn^{-1/2}\,}{\sup }\bigg \Vert n^{-1/2}\sum _{i=1}^n(\delta _i-\pi (y_i,\varvec{z}_i,\varvec{\gamma }^*))(\varvec{\xi }(y_i,\varvec{z}_i,\varvec{\zeta })-\varvec{\xi }(y_i,\varvec{z}_i,\varvec{\zeta }^*))\bigg \Vert =o_p(1). \end{aligned}$$

For convenience of representation, for $i=1,\ldots ,n$ and $k=1,\ldots ,q$, write

$$\begin{aligned}&\varvec{\kappa }=\ (\varvec{\alpha }^T,\varvec{\beta }^T,{\varvec{b}}^T,\varvec{\gamma }^T)^T,\quad \varvec{\kappa }^*=({\varvec{\alpha }^*}^T,{\varvec{\beta }^*}^T,{\varvec{b}^*}^T,{\varvec{\gamma }^*}^T)^T,\quad \\&\qquad \hat{\varvec{\kappa }}=(\hat{\varvec{\alpha }}^T,\hat{\varvec{\beta }}_{\scriptscriptstyle cc}^T,\hat{\varvec{b}}_{\scriptscriptstyle cc}^T,\hat{\varvec{\gamma }}^T)^T,\\&\zeta _k(\varvec{t}_i,\varvec{\beta },\varvec{b})=\ I(y_i-\varvec{w}_i^T\varvec{\beta }\le b_k)-\tau _k, \quad \zeta _{i,k}=\zeta _k(\varvec{t}_i,\varvec{\beta }^*,\varvec{b}^*),\\&\eta (\varvec{t}_i,\varvec{\beta },\varvec{b})=\sum _{k=1}^{q}\zeta _k(\varvec{t}_i,\varvec{\beta },\varvec{b}),\quad \eta _i=\eta (\varvec{t}_i,\varvec{\beta }^*,\varvec{b}^*),\\&\varvec{\xi }_i=\ \varvec{\xi }(y_i,\varvec{z}_i,\varvec{\alpha }^*,\varvec{\beta }^*,\varvec{b}^*),\quad \varvec{h}_i=\varvec{h}(\varvec{t}_i,\varvec{\kappa }^*),\quad \varvec{g}_i=\varvec{g}(\varvec{t}_i,\varvec{\kappa }^*),\\&\varvec{S}_{\varvec{g}}=\ E(\varvec{g}_i\varvec{g}_i^T), \quad \varvec{F}_{\varvec{g}}=E(\varvec{\phi }_i\varvec{g}_i^T), \quad {\varvec{\Lambda }}({\varvec{\lambda }},\varvec{\kappa })=\frac{1}{n}\sum _{i=1}^n\frac{\varvec{g}(\varvec{t}_i,\varvec{\kappa })}{1+{\varvec{\lambda }}^T\varvec{g}(\varvec{t}_i,\varvec{\kappa })},\\&\varvec{G_\gamma }=E(\partial {\varvec{g}_i}/\partial {\varvec{\gamma }^T}), \quad \varvec{F_\gamma }=E(\partial {\varvec{\phi }_i}/\partial {\varvec{\gamma }^T}). \end{aligned}$$

Lemma 1

• (i) $det(\varvec{S}_{\varvec{g}})= det(\varvec{P}_2)det(\varvec{S}_B)$, where $det(\varvec{A})$ denotes the determinant of some matrix $\varvec{A}$.

• (ii) $\varvec{G_\gamma }=-E(\varvec{g}_i\varvec{U}_{B_i}^T)$, $\varvec{F_\gamma } =-E(\varvec{\phi }_i\varvec{U}_{B_i}^T)$. Thus $\varvec{F_\gamma }=\varvec{F_g}\varvec{S_g}^{-1}\varvec{G_\gamma }$.

• (iii) $\varvec{F}_{\varvec{g}}\varvec{S}_{\varvec{g}}^{-1}\varvec{F}_{\varvec{g}}^T=\varvec{P}_1\varvec{P}_2^{-1}\varvec{P}_1^T+E(\varvec{\phi }_i\varvec{U}_{B_i}^T)\varvec{S}_{B}^{-1}E(\varvec{U}_{B_i}\varvec{\phi }_i^T)$, $\varvec{F}_{\varvec{g}}\varvec{S}_{\varvec{g}}^{-1}\varvec{G}_{\varvec{\gamma }}=-E(\varvec{\phi }_i\varvec{U}_{B_i}^T)$, $\varvec{S}_{\varvec{g}}^{-1}\varvec{G}_{\varvec{\gamma }}= \begin{pmatrix}\varvec{0}^T, -\varvec{I}^T \end{pmatrix}^T.$

Proof of Lemma 1

The results can be derived by some calculations and the details are omitted. $\square$

Proof of Theorem 1

The proof of Theorem 1 is similar to and much easier than that of Theorem 2. The details are omitted. $\square$

Proof of Theorem 2

For fixed estimators $\hat{\varvec{\kappa }}$, the Lagrange multiplier $\hat{\varvec{\lambda }}$ satisfies the constraints equations ${\varvec{\Lambda }}(\hat{\varvec{\lambda }},\hat{\varvec{\kappa }})=0.$ According to Lemma A.1 in Sun and Ma (2017),

$$\begin{aligned} \hat{\varvec{\lambda }}={\varvec{\lambda }}(\hat{\varvec{\kappa }})=\Big (\frac{1}{n}\sum _{i=1}^n\varvec{g}(\varvec{t}_i,\hat{\varvec{\kappa }})\varvec{g}(\varvec{t}_i,\hat{\varvec{\kappa }})^T\Big )^{-1}\frac{1}{n}\sum _{i=1}^n\varvec{g}(\varvec{t}_i,\hat{\varvec{\kappa }})+o_p(n^{-\frac{1}{2}}). \end{aligned}$$

(4.1)

By Taylor expansion,

$$\begin{aligned} \frac{1}{n}\sum _{i=1}^n\varvec{g}(\varvec{t}_i,\hat{\varvec{\kappa }})= & {} \frac{1}{n}\sum _{i=1}^n\varvec{g}(\varvec{t}_i,{\varvec{\kappa }}^*) +\frac{1}{n}\sum _{i=1}^n\frac{\partial {\varvec{g}(\varvec{t}_i,\tilde{\varvec{\kappa }})}}{\partial {\varvec{\alpha }^T}}(\hat{\varvec{\alpha }}-\varvec{\alpha }^*)\\&+\frac{1}{n}\sum _{i=1}^n\frac{\partial {\varvec{g}(\varvec{t}_i,\tilde{\varvec{\kappa }})}}{\partial {\varvec{\beta }^T}}(\hat{\varvec{\beta }}_{\scriptscriptstyle cc}-\varvec{\beta }^*)\\&+\,\frac{1}{n}\sum _{i=1}^n\frac{\partial {\varvec{g}(\varvec{t}_i,\tilde{\varvec{\kappa }})}}{\partial {\varvec{b}^T}}(\hat{\varvec{b}}_{\scriptscriptstyle cc}-\varvec{b}^*) +\frac{1}{n}\sum _{i=1}^n\frac{\partial {\varvec{g}(\varvec{t}_i,\tilde{\varvec{\kappa }})}}{\partial {\varvec{\gamma }^T}}(\hat{\varvec{\gamma }}-\varvec{\gamma }^*), \end{aligned}$$

where $\tilde{\varvec{\kappa }}=(\tilde{\varvec{\alpha }}^T,\tilde{\varvec{\beta }}^T,\tilde{\varvec{b}}^T,\tilde{\varvec{\gamma }}^T)^T\in \{\varvec{\kappa }:\Vert \varvec{\kappa }-\varvec{\kappa }^*\Vert \le cn^{-1/2}\}$ with a positive constant c. According to the law of large numbers,

$$\begin{aligned} \frac{1}{n}\sum \limits _{i=1}^n\frac{\partial {\varvec{g}(\varvec{t}_i,\tilde{\varvec{\kappa }})}}{\partial {\varvec{\alpha }^T}}&{\mathop {\rightarrow }\limits ^{P}}\varvec{0},\quad \frac{1}{n}\sum \limits _{i=1}^n\frac{\partial {\varvec{g}(\varvec{t}_i,\tilde{\varvec{\kappa }})}}{\partial {\varvec{\beta }^T}} {\mathop {\rightarrow }\limits ^{P}}\varvec{0},\quad \frac{1}{n}\sum \limits _{i=1}^n\frac{\partial {\varvec{g}(\varvec{t}_i,\tilde{\varvec{\kappa }})}}{\partial {\varvec{b}^T}} {\mathop {\rightarrow }\limits ^{P}}\varvec{0},\\ \frac{1}{n}\sum \limits _{i=1}^n\frac{\partial {\varvec{g}(\varvec{t}_i,\tilde{\varvec{\kappa }})}}{\partial {\varvec{\gamma }^T}}&{\mathop {\rightarrow }\limits ^{P}} \varvec{G}_{\varvec{\gamma }},\quad \frac{1}{n}\sum \limits _{i=1}^n\varvec{g}(\varvec{t}_i,\tilde{\varvec{\kappa }})\varvec{g}(\varvec{t}_i,\tilde{\varvec{\kappa }})^T {\mathop {\rightarrow }\limits ^{P}}\varvec{S}_{\varvec{g}}. \end{aligned}$$

By combining these with (4.1) and

$$\begin{aligned} \hat{\varvec{\gamma }}-\varvec{\gamma }^*=\frac{1}{n}\varvec{S}_B^{-1}\sum _{i=1}^n\varvec{U}_{B_i}+o_p(n^{-\frac{1}{2}}), \end{aligned}$$

it holds that

$$\begin{aligned} \hat{\varvec{\lambda }}=\frac{1}{n}\varvec{S}_{\varvec{g}}^{-1}\sum _{i=1}^n(\varvec{g}_i+\varvec{G}_{\varvec{\gamma }}\varvec{S}_B^{-1}\varvec{U}_{B_i})+o_p(n^{-\frac{1}{2}}). \end{aligned}$$

(4.2)

Let $\varvec{u}=\sqrt{n}(\varvec{\beta }-\varvec{\beta }^*)$, $\hat{\varvec{u}}=\sqrt{n}(\hat{\varvec{\beta }}_{acc}-\varvec{\beta }^*)$, $v_k=\sqrt{n}(b_k-b_k^*)$ and $\hat{v}_k=\sqrt{n}(\hat{b}_{k,acc}-b_k^*)$, $k=1,\ldots ,q$. Let $\varvec{\theta }=(\varvec{u}^T,v_1,\ldots ,v_q)^T$ and $\hat{\varvec{\theta }}=(\hat{\varvec{u}}^T,\hat{v}_1,\ldots ,\hat{v}_q)^T$, then $\hat{\varvec{\theta }}$ is the minimizer of

$$\begin{aligned} \Xi _n(\varvec{\theta })&= \sum _{k=1}^q\sum _{i=1}^nn\hat{p}_i\delta _i\big \{\rho _{\tau _k}(\varepsilon _i-b_k^*-n^{-\frac{1}{2}}(v_k+\varvec{w}_i^T\varvec{u}))-\rho _{\tau _k}(\varepsilon _i-b_k^*)\big \} \nonumber \\&= \sum _{k=1}^qv_kz_{n,k}+\varvec{\Pi }_n^T\varvec{u}+\sum _{k=1}^qB_{n,k}, \end{aligned}$$

(4.3)

where

$$\begin{aligned} z_{n,k}&=\frac{1}{\sqrt{n}}\sum _{i=1}^nn\hat{p}_i\delta _i\zeta _{i,k},\quad \varvec{\Pi }_n=\frac{1}{\sqrt{n}}\sum \limits _{i=1}^nn\hat{p}_i\delta _i\varvec{w}_i\eta _i,\\ B_{n,k}&=\sum _{i=1}^nn\hat{p}_i\delta _i\int _{0}^{\frac{v_k+\varvec{w}_i^T\varvec{u}}{\sqrt{n}}}\big (I(\varepsilon _i\le b_k^*+s)-I(\varepsilon _i\le b_k^*)\big )\hbox {d}s. \end{aligned}$$

Owing to (2.4),

$$\begin{aligned} n\hat{p}_i=1-\hat{\varvec{\lambda }}^T\varvec{g}_i+o_p(1). \end{aligned}$$

(4.4)

Then by substituting (4.2) and (4.4) into $\varvec{\Pi }_n$, it holds that

$$\begin{aligned} \varvec{\Pi }_n&=\frac{1}{\sqrt{n}}\sum _{i=1}^n\delta _i\varvec{w}_i\eta _i-\Big (\frac{1}{n}\sum _{i=1}^n\delta _i\varvec{w}_i\eta _i\varvec{g}_i^T\Big )\sqrt{n}\hat{\varvec{\lambda }}+o_p(1)\nonumber \\&=\frac{1}{\sqrt{n}}\sum _{i=1}^n\varvec{\phi }_i-\Big (\frac{1}{n}\sum _{i=1}^n\varvec{\phi }_i\varvec{g}_i^T\Big )\sqrt{n}\hat{\varvec{\lambda }}+o_p(1)\nonumber \\&=\frac{1}{\sqrt{n}}\sum _{i=1}^n\varvec{\phi }_i-\varvec{F}_{\varvec{g}}\varvec{S}_{\varvec{g}}^{-1}\frac{1}{\sqrt{n}}\sum _{i=1}^n(\varvec{g}_i+\varvec{G}_{\varvec{\gamma }}\varvec{S}_B^{-1}\varvec{U}_{B_i})+o_p(1). \end{aligned}$$

(4.5)

Based on some calculations and Lemma 1 (ii),

$$\begin{aligned} \varvec{F}_{\varvec{g}}\varvec{S}_{\varvec{g}}^{-1}= & {} \begin{pmatrix} \varvec{P}_1\varvec{P}_2^{-1}&E(\varvec{\phi }_i\varvec{U}_{B_i}^T)\varvec{S}_B^{-1}-\varvec{P}_1\varvec{P}_2^{-1}E(\varvec{h}_i\varvec{U}_{B_i}^T)\varvec{S}_B^{-1} \end{pmatrix},\\ \sum _{i=1}^n(\varvec{g}_i+\varvec{G}_{\varvec{\gamma }}\varvec{S}_B^{-1}\varvec{U}_{B_i})= & {} \begin{pmatrix} \sum \limits _{i=1}^n\varvec{h}_i \\ \sum \limits _{i=1}^n\varvec{U}_{B_i} \end{pmatrix}+ \begin{pmatrix} -E(\varvec{h}_i\varvec{U}_{B_i}^T) \\ -E(\varvec{U}_{B_i}\varvec{U}_{B_i}^T) \end{pmatrix} \varvec{S}_B^{-1}\sum \limits _{i=1}^n\varvec{U}_{B_i}\\= & {} \begin{pmatrix} \sum \limits _{i=1}^n\big (\varvec{h}_i-E(\varvec{h}_i\varvec{U}_{B_i}^T)\varvec{S}_B^{-1}\varvec{U}_{B_i} \big )\\ \varvec{0} \end{pmatrix}. \end{aligned}$$

Substituting these into (4.5) leads to

$$\begin{aligned} \varvec{\Pi }_n=\frac{1}{\sqrt{n}}\sum _{i=1}^n\big (\varvec{\phi }_i-\varvec{P}_1\varvec{P}_2^{-1}\big (\varvec{h}_i-E(\varvec{h}_i\varvec{U}_{B_i}^T)\varvec{S}_B^{-1}\varvec{U}_{B_i} \big )\big )+o_p(1). \end{aligned}$$

By some calculations, it holds that

$$\begin{aligned} Var\big (\varvec{\phi }_i-\varvec{P}_1\varvec{P}_2^{-1}\big (\varvec{h}_i-E(\varvec{h}_i\varvec{U}_{B_i}^T)\varvec{S}_B^{-1}\varvec{U}_{B_i} \big )\big )= \varvec{S}_{\varvec{\phi }}-\varvec{P}_1\varvec{P}_2^{-1}\varvec{P}_1^T, \end{aligned}$$

and $\varvec{\Pi }_n{\mathop {\rightarrow }\limits ^{d}}\varvec{\Pi }_0$, where $\varvec{\Pi }_0$ is a random vector following $N(\varvec{0},\varvec{S_\phi }-\varvec{P}_1\varvec{P}_2^{-1}\varvec{P}_1^T)$.

Similarly, for $k=1,\ldots ,q$,

$$\begin{aligned} z_{n,k}= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n\delta _i\zeta _{i,k}-\Big (\frac{1}{n}\sum _{i=1}^n\delta _i\zeta _{i,k}\varvec{g}_i^T\Big )\sqrt{n}\hat{\varvec{\lambda }}+o_p(1)\\= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^n\big (\delta _i\zeta _{i,k}-E(\delta _i\zeta _{i,k}\varvec{g}_i^T)\varvec{S}_{\varvec{g}}^{-1}(\varvec{g}_i+\varvec{G}_{\varvec{\gamma }}\varvec{S}_B^{-1}\varvec{U}_{B_i})\big )+o_p(1), \end{aligned}$$

and $z_{n,k}{\mathop {\rightarrow }\limits ^{d}}z_{k}$, where $z_{k}$ follows some one-dimensional normal distribution.

For $i=1,\ldots ,n$ and $k=1,\ldots ,q$, write $A_{i,k}=\int _{0}^{(v_k+\varvec{w}_i^T\varvec{u})/\sqrt{n}}\big (I(\varepsilon _i\le b_k^*+s)-I(\varepsilon _i\le b_k^*)\big )\hbox {d}s$. Then

$$\begin{aligned} B_{n,k}&=\sum \limits _{i=1}^nn\hat{p}_i\delta _iA_{i,k}=\sum \limits _{i=1}^n\delta _iA_{i,k}-\sum \limits _{i=1}^n\delta _iA_{i,k}\varvec{g}_i^T\hat{\varvec{\lambda }}+o_p(1). \end{aligned}$$

Based on some derivations,

$$\begin{aligned} E\Big (\sum _{i=1}^n\delta _iA_{i,k}\Big )&=\sum \limits _{i=1}^nE\Big (E(\delta _i|\varvec{w}_i) \int _{0}^{\frac{v_k+\varvec{w}_i^T\varvec{u}}{\sqrt{n}}} \big (F(b_k^*+s)-F(b_k^*)\big )\hbox {d}s\Big )\\&=\frac{1}{2}f(b_k^*)v_k^2E(\delta _i)+\frac{1}{2}f(b_k^*)\varvec{u}^TE(\delta _i\varvec{w}_i\varvec{w}_i^T)\varvec{u}+o_p(1),\\ Var\Big (\sum \limits _{i=1}^n\delta _iA_{i,k}\Big )&= \sum \limits _{i=1}^nVar(\delta _iA_{i,k})=\sum \limits _{i=1}^nE\big [\big (\delta _iA_{i,k}-E(\delta _iA_{i,k})\big )^2\big ]\\&\le 4E\Big (\sum \limits _{i=1}^n\delta _iA_{i,k}\Big )\frac{|v_k|+\max _{1\le i\le n}|\varvec{w}_i^T\varvec{u}|}{\sqrt{n}}=o_p(1),\\ \Big |\sum _{i=1}^n\delta _iA_{i,k}\varvec{g}_i^T\hat{\varvec{\lambda }}\Big |&\le \underset{1\le i\le n}{\max }\left\| \varvec{g}_i\right\| \Vert \hat{\varvec{\lambda }}\Vert \Big |\sum _{i=1}^n\delta _iA_{i,k}\Big |=o_p(n^\frac{1}{2})O_p(n^{-\frac{1}{2}})O_p(1)=o_p(1). \end{aligned}$$

Therefore

$$\begin{aligned} B_{n,k}=\frac{1}{2}f(b_k^*)v_k^2E(\delta _i)+\frac{1}{2}f(b_k^*)\varvec{u}^TE(\delta _i\varvec{w}_i\varvec{w}_i^T)\varvec{u}+o_p(1). \end{aligned}$$

By substituting the expression of $B_{n,k}$ into (4.3),

$$\begin{aligned} \Xi _n(\varvec{\theta })= \sum _{k=1}^qv_kz_{n,k}+\varvec{\Pi }_n^T\varvec{u}+\frac{1}{2}\sum _{k=1}^qf(b_k^*)v_k^2E(\delta _i)+\frac{1}{2}\varvec{u}^T\varvec{D}\varvec{u}+o_p(1). \end{aligned}$$

According to the convexity lemma in Pollard (1991), it follows that

$$\begin{aligned} \hat{\varvec{u}}=-\varvec{D}^{-1}\varvec{\Pi }_n+o_p(1). \end{aligned}$$

Finally, combining this with $\varvec{\Pi }_n{\mathop {\rightarrow }\limits ^{d}}\varvec{\Pi }_0$ leads to

$$\begin{aligned} \sqrt{n}(\hat{\varvec{\beta }}_{\scriptscriptstyle acc}-\varvec{\beta }^*){\mathop {\rightarrow }\limits ^{d}}N\big (\varvec{0},\varvec{D}^{-1}(\varvec{S}_{\varvec{\phi }}-\varvec{P}_1\varvec{P}_2^{-1}\varvec{P}_1^T)\varvec{D}^{-1}\big ). \end{aligned}$$

$\square$

Proof of Theorem 3

Write $\varvec{M}_i=\varvec{M}(\varvec{t}_i,\varvec{\beta }^*,\varvec{\gamma }^*)$. According to the proof of Lemma 6 in Molanes Lopez et al. (2009), it holds that

$$\begin{aligned} \sqrt{n}\begin{pmatrix}\hat{\varvec{\beta }}_{\scriptscriptstyle el}-\varvec{\beta }^*\\ \hat{\varvec{\gamma }}_{\scriptscriptstyle el}-\varvec{\gamma }^*\end{pmatrix} {\mathop {\rightarrow }\limits ^{d}}N(\varvec{0},(\varvec{V}_1^T\varvec{V}_2^{-1}\varvec{V}_1)^{-1}), \end{aligned}$$

where

$$\begin{aligned} \varvec{V}_1=-\begin{pmatrix}\varvec{D} &{} \varvec{0} \\ \varvec{0} &{} \varvec{G}_{\varvec{\gamma }}\end{pmatrix}, \varvec{V}_2=\begin{pmatrix}\varvec{S}_{\varvec{\phi }} &{} \varvec{F}_{\varvec{g}} \\ \varvec{F}_{\varvec{g}}^T &{} \varvec{S}_{\varvec{g}}\end{pmatrix}{\mathop {=}\limits ^{\vartriangle }}\begin{pmatrix}\varvec{B}_{11} &{} \varvec{B}_{12} \\ \varvec{B}_{12}^T &{} \varvec{B}_{22}\end{pmatrix}. \end{aligned}$$

Define $\varvec{B}_{11.2}=\varvec{B}_{11}-\varvec{B}_{12}\varvec{B}_{22}^{-1}\varvec{B}_{12}^T=\varvec{S}_{\varvec{\phi }}-\varvec{F}_{\varvec{g}}\varvec{S}_{\varvec{g}}^{-1}\varvec{F}_{\varvec{g}}^T$. According to Lemma 1 (iii),

$$\begin{aligned} \varvec{B}_{11.2}&= \ \varvec{S}_{\varvec{\phi }}-\varvec{P}_1\varvec{P}_2^{-1}\varvec{P}_1^T-E(\varvec{\phi }_i\varvec{U}_{B_i}^T)\varvec{S}_{B}^{-1}E(\varvec{U}_{B_i}\varvec{\phi }_i^T),\\ \varvec{V}_1^T\varvec{V}_2^{-1}\varvec{V}_1&=\ \begin{pmatrix}\varvec{D} &{} \varvec{0} \\ \varvec{0} &{} \varvec{G}_{\varvec{\gamma }}^T\end{pmatrix} \begin{pmatrix} \varvec{B}_{11.2}^{-1} &{} -\varvec{B}_{11.2}^{-1}\varvec{F}_{\varvec{g}}\varvec{S}_{\varvec{g}}^{-1} \\ -\varvec{S}_{\varvec{g}}^{-1}\varvec{F}_{\varvec{g}}^T\varvec{B}_{11.2}^{-1} &{} \varvec{S}_{\varvec{g}}^{-1}+\varvec{S}_{\varvec{g}}^{-1}\varvec{F}_{\varvec{g}}^T\varvec{B}_{11.2}^{-1}\varvec{F}_{\varvec{g}}\varvec{S}_{\varvec{g}}^{-1} \end{pmatrix} \begin{pmatrix}\varvec{D} &{} \varvec{0} \\ \varvec{0} &{} \varvec{G}_{\varvec{\gamma }}\end{pmatrix}\\&=\ \begin{pmatrix} \varvec{D}\varvec{B}_{11.2}^{-1}\varvec{D} &{} -\varvec{D}\varvec{B}_{11.2}^{-1}\varvec{F}_{\varvec{g}}\varvec{S}_{\varvec{g}}^{-1}\varvec{G}_{\varvec{\gamma }} \\ -\varvec{G}_{\varvec{\gamma }}^T\varvec{S}_{\varvec{g}}^{-1}\varvec{F}_{\varvec{g}}^T\varvec{B}_{11.2}^{-1}\varvec{D} &{} \varvec{G}_{\varvec{\gamma }}^T(\varvec{S}_{\varvec{g}}^{-1}+\varvec{S}_{\varvec{g}}^{-1}\varvec{F}_{\varvec{g}}^T\varvec{B}_{11.2}^{-1}\varvec{F}_{\varvec{g}}\varvec{S}_{\varvec{g}}^{-1})\varvec{G}_{\varvec{\gamma }} \end{pmatrix}\\&=\ \begin{pmatrix} \varvec{D}\varvec{B}_{11.2}^{-1}\varvec{D} &{} \varvec{D}\varvec{B}_{11.2}^{-1}E(\varvec{\phi }_i\varvec{U}_{B_i}^T) \\ E(\varvec{U}_{B_i}\varvec{\phi }_i^T)\varvec{B}_{11.2}^{-1}\varvec{D} &{} \varvec{S}_B+E(\varvec{U}_{B_i}\varvec{\phi }_i^T)\varvec{B}_{11.2}^{-1}E(\varvec{\phi }_i\varvec{U}_{B_i}^T) \end{pmatrix}\\&{\mathop {=}\limits ^{\vartriangle }} \ \begin{pmatrix} \varvec{H}_{11} &{} \varvec{H}_{12} \\ \varvec{H}_{12}^T &{} \varvec{H}_{22} \end{pmatrix}. \end{aligned}$$

Then it is easy to derive that

$$\begin{aligned} (\varvec{V}_1^T\varvec{V}_2^{-1}\varvec{V}_1)^{-1} =\begin{pmatrix} \varvec{H}_{11}^{-1}+\varvec{H}_{11}^{-1}\varvec{H}_{12}\varvec{H}_{22.1}^{-1}\varvec{H}_{12}^T\varvec{H}_{11}^{-1} &{} -\varvec{H}_{11}^{-1}\varvec{H}_{12}\varvec{H}_{22.1}^{-1} \\ -\varvec{H}_{22.1}^{-1}\varvec{H}_{12}^T\varvec{H}_{11}^{-1} &{} \varvec{H}_{22.1}^{-1} \end{pmatrix}, \end{aligned}$$

where $\varvec{H}_{22.1}=\varvec{H}_{22}-\varvec{H}_{12}^T\varvec{H}_{11}^{-1}\varvec{H}_{12}=\varvec{S}_B$. Note that

$$\begin{aligned} -\varvec{H}_{11}^{-1}\varvec{H}_{12}\varvec{H}_{22.1}^{-1}&=-\varvec{D}^{-1}E(\varvec{\phi }_i\varvec{U}_{B_i}^T)\varvec{S}_B^{-1},\\ \varvec{H}_{11}^{-1}+\varvec{H}_{11}^{-1}\varvec{H}_{12}\varvec{H}_{22.1}^{-1}\varvec{H}_{12}^T\varvec{H}_{11}^{-1}&=\varvec{D}^{-1}\big (\varvec{B}_{11.2}+E(\varvec{\phi }_i\varvec{U}_{B_i}^T)\varvec{S}_B^{-1}E(\varvec{U}_{B_i}\varvec{\phi }_i^T)\big )\varvec{D}^{-1}\\&=\varvec{D}^{-1}(\varvec{S}_{\varvec{\phi }}-\varvec{P}_1\varvec{P}_2^{-1}\varvec{P}_1^T)\varvec{D}^{-1}. \end{aligned}$$

Therefore

$$\begin{aligned} \sqrt{n}(\hat{\varvec{\beta }}_{\scriptscriptstyle el}-\varvec{\beta }^*){\mathop {\rightarrow }\limits ^{d}}N\big (\varvec{0},\varvec{D}^{-1}(\varvec{S}_{\varvec{\phi }}-\varvec{P}_1\varvec{P}_2^{-1}\varvec{P}_1^T)\varvec{D}^{-1}\big ). \end{aligned}$$

$\square$