Efficient robust estimation for single-index mixed effects models with missing observations

Abstract

In this paper, we study the efficient robust estimation and empirical likelihood for a single-index mixed effects model with a subset of covariates and response missing at random. Three efficient robust estimators and empirical likelihood ratios for index coefficients are constructed using weighted, imputed and weighted-imputed method, their asymptotic properties are proved. Our results show that the three estimators are asymptotically equivalent, and a weighted-imputed empirical log-likelihood ratio is asymptotically chi-squared. An important feature of our methods is their ability to handle missing response and/or partially missing covariates. Some simulation studies and a real data example indicate that our methods have fine performance in finite sample, and are available in practice.

Appendix. Proofs of Theorems

To facilitate analysis, we introduce several lemmas, the proofs of which are given in the supplementary material.

Lemma 1

Suppose that conditions (C1)–(C8) hold. Then

$$\begin{aligned} \widehat{Q}_\textrm{L}(\beta ^{(r)}) = Q_\textrm{WI}(\beta _0^{(r)})- A(\beta ^{(r)}-\beta _0^{(r)}) + o_P(n^{-1/2}) \end{aligned}$$

when L = W, I and WI respectively, uniformly for $\beta ^{(r)}\in \mathcal{B}_n^*$ with $\mathcal{B}_n^*=\{\beta ^{(r)}|\,\Vert \beta ^{(r)}-\beta _0^{(r)}\Vert \le cn^{-1/2}\}$ for a constant $c>0$, where $\widehat{Q}_\textrm{W}(\beta ^{(r)})$, $\widehat{Q}_\textrm{I}(\beta ^{(r)})$, $\widehat{Q}_\textrm{WI}(\beta ^{(r)})$ and A are defined in (2.6) and (2.12) and (3.2) respectively,

$$\begin{aligned} Q_\textrm{WI}(\beta ^{(r)}) = \frac{1}{n}\sum _{i=1}^n\Bigg \{\frac{\delta _i}{\pi (V_i)}\xi _i(\beta ^{(r)}) + \left( 1-\frac{\delta _i}{\pi (V_i)}\right) m(V_i;\beta ^{(r)})\Bigg \}, \end{aligned}$$

$\xi _i(\beta ^{(r)})$ is defined in (3.4) and $m(V_i;\beta ^{(r)})=E\{\xi _i(\beta ^{(r)})|V_i\}$.

Lemma 2

Suppose that conditions (C1)–(C8) hold. Then

$$\begin{aligned} \sqrt{n}\widehat{Q}_\textrm{L}(\beta _0^{(r)}){\mathop {\longrightarrow }\limits ^{D}}N(0,B) \end{aligned}$$

when L = W, I and WI respectively, where $\widehat{Q}_\textrm{W}(\beta _0^{(r)})$, $\widehat{Q}_\textrm{I}(\beta _0^{(r)})$ and $\widehat{Q}_\textrm{WI}(\beta _0^{(r)})$ are defined in (2.6) and (2.12) respectively, and B is defined in (3.3).

Lemma 3

Suppose that conditions (C1)–(C8) hold. Then

$$\begin{aligned}{} & {} \quad \mathrm{(a)}~~\frac{1}{n}\sum _{i=1}^n\hat{\eta }_{i,\mathrm L}(\beta _0^{(r)})\hat{\eta }_{i,\mathrm L}^T(\beta _0^{(r)}){\mathop {\longrightarrow }\limits ^{P}}C_\textrm{L}, \nonumber \\{} & {} \quad \mathrm{(b)}~~\frac{1}{n}\sum _{i=1}^n\hat{\eta }_{i,\mathrm WI}(\beta _0^{(r)})\hat{\eta }_{i,\mathrm WI}^T(\beta _0^{(r)}){\mathop {\longrightarrow }\limits ^{P}}B \end{aligned}$$

when L = W and I respectively, where B, $C_\textrm{W}$ and $C_\textrm{I}$ are defined in (3.3), (3.5) and (3.6) respectively, and $\hat{\eta }_{i,\mathrm W}(\cdot )$, $\hat{\eta }_{i,\mathrm I}(\cdot )$ and $\hat{\eta }_{i,\mathrm WI}(\cdot )$ are defined in (2.9), (2.13) and (2.14) respectively.

Lemma 4

Suppose that conditions (C1)–(C8) hold. Then

$$\begin{aligned} \displaystyle \max _{1\le i\le n}|\hat{\eta }_{i,\mathrm L}(\beta _0^{(r)})| = o_P(n^{1/2}) \end{aligned}$$

when L = W, I and WI respectively.

Proof of Theorem 1

We consider the asymptotic normality of $\hat{\beta }_\textrm{L}$ when L= W, I and WI respectively. The proof is divided into two steps: Step (I) provides the existence of the estimator $\hat{\beta }_\textrm{L}^{(r)}$ of $\beta _0^{(r)}$, and step (II) proves the asymptotic normality of $\hat{\beta }_\textrm{L}^{(r)}$. $\square$

(I) Proof of existence. We prove the following fact: Under conditions (C1)–(C8) and with probability one there exists an estimator of $\beta _0^{(r)}$ solving the estimating Eqs. (2.6) or (2.12) in $\mathcal{B}_n^{**}$, where $\mathcal{B}_n^{**}=\big \{\beta ^{(r)}|\, \Vert \beta ^{(r)}-\beta _0^{(r)}\Vert =Mn^{-1/2}\big \}$ for some constant M such that $0<M<\infty$. In fact, by Lemma 1, we have

$$\begin{aligned} \widehat{Q}_\textrm{L}(\beta ^{(r)}) = Q_\textrm{WI}(\beta _0^{(r)}) - A(\beta ^{(r)}-\beta _0^{(r)}) + o_P(n^{-1/2}) \end{aligned}$$

(A.1)

when L = W, I and WI respectively, uniformly for $\beta ^{(r)}\in \mathcal{B}_n^{**}$, where $Q_\textrm{WI}(\beta ^{(r)})$ is defined in Lemma 1. Therefore, we have

$$\begin{aligned}&n(\beta ^{(r)}-\beta _0^{(r)})\widehat{Q}_\textrm{L}(\beta ^{(r)}) \nonumber \\&\quad = n(\beta ^{(r)}-\beta _0^{(r)})Q_\textrm{WI}(\beta _0^{(r)}) - n(\beta ^{(r)}-\beta _0^{(r)})A(\beta ^{(r)}-\beta _0^{(r)}) + o_P(1). \end{aligned}$$

(A.2)

We note that term (A.2) is dominated by the term $\sim M^2$ because $\sqrt{n}\Vert \beta ^{(r)}-\beta _0^{(r)}\Vert =M$, whereas $|n(\beta ^{(r)}-\beta _0^{(r)})^TQ_\textrm{WI}(\beta _0^{(r)})|=MO_P(1)$, and $n(\beta ^{(r)}-\beta _0^{(r)})A(\beta ^{(r)}-\beta _0^{(r)})\sim M^2$. So, for any given $\eta >0$, if M is chosen large enough, then it will follows that $n(\beta ^{(r)}-\beta _0^{(r)})\widehat{Q}_\textrm{L}(\beta ^{(r)})<0$ on an event with probability $1-\eta$. From the arbitrariness of $\eta$, we can prove the existence of the estimator of $\beta _0^{(r)}$ in $\mathcal{B}_n^{**}$ as in the proof of Theorem 5.1 of Welsh (1989). The details are omitted.

(II) Proof of asymptotic normality. From step (I) we find that $\hat{\beta }_\textrm{L}^{(r)}$ is a solution in $\mathcal{B}_n^{**}$ to the equation $\widehat{Q}_\textrm{L}(\beta ^{(r)})=0$. That is, $\widehat{Q}_\textrm{L}(\hat{\beta }_\textrm{L}^{(r)})=0$. From (A.1), we have

$$\begin{aligned} 0=Q_\textrm{WI}(\beta _0^{(r)}) - A(\hat{\beta }_\textrm{L}^{(r)} - \beta _0^{(r)}) + o_P(n^{-1/2}), \end{aligned}$$

and hence

$$\begin{aligned} \sqrt{n}(\hat{\beta }_\textrm{L}^{(r)} - \beta _0^{(r)}) = A^{-1}\sqrt{n}Q_\textrm{WI}(\beta _0^{(r)}) + o_P(1) \end{aligned}$$

when L = W, I and WI respectively. Theorem 2 follows from this, Lemma 2, Central Limit Theorem and Slutsky’s theorem. $\square$

Proof of Corollary 1

It is easy to prove

$$\begin{aligned} \hat{\beta }_\textrm{L} - \beta _0 = J_{\hat{\beta }_\textrm{L}^{(r)}}\big (\hat{\beta }_\textrm{L}^{(r)} - \beta _0^{(r)}\big ) \end{aligned}$$

when L = W, I and WI respectively. Therefore, Corollary 1 can be derived from Theorem 1. $\square$

Proof of Corollary 2

Note that

$$\begin{aligned} \hat{\beta }_\textrm{L}^T\beta _0 = \big (\hat{\beta }_\textrm{L} - \beta _0\big )^T\beta _0 + \beta _0^T\beta _0 \end{aligned}$$

when L = W, I and WI respectively. Therefore, from Theorem 1 we obtain

$$\begin{aligned} \hat{\beta }_\textrm{L}^T\beta _0 - 1 = O_P(n^{-1/2}) \end{aligned}$$

when L = W, I and WI respectively. This, together with the above formula, proves Corollary 2. $\square$

Proof of Theorem 2

By the Lagrange multiplier method, $\mathcal{R}_\textrm{L}(\beta ^{(r)}_0)$ can be represented as

$$\begin{aligned} \mathcal{R}_\textrm{L}(\beta ^{(r)}_0)=2\sum _{i=1}^n\log (1+\lambda ^T\hat{\eta }_{i,\mathrm L}(\beta ^{(r)}_0)) \end{aligned}$$

(A.3)

when L = W, I and WI respectively, where $\lambda =\lambda (\beta ^{(r)}_0)$ is a $(p-1)\times 1$ vector given as the solution to

$$\begin{aligned} \sum _{i=1}^n\frac{\hat{\eta }_{i,\mathrm L}(\beta ^{(r)}_0)}{1+\lambda ^T\hat{\eta }_{i,\mathrm L}(\beta ^{(r)}_0)}=0. \end{aligned}$$

(A.4)

By Lemmas 2 and 3, and using the same arguments as are used in the proof of (2.15) in Owen (1990), we can show that

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

(A.5)

Applying the Taylor expansion to (A.3), and invoking Lemmas 3–4 and (A.5), we get that

$$\begin{aligned} \mathcal{R}_\textrm{L}(\beta ^{(r)}_0) = 2 \sum _{i=1}^n[\lambda ^T\hat{\eta }_{i,\mathrm L}(\beta ^{(r)}_0) - \{\lambda ^T\hat{\eta }_{i,\mathrm L}(\beta ^{(r)}_0)\}^2/2] +o_P(1) \end{aligned}$$

(A.6)

when L = W, I and WI respectively. By (A.4), it follows that

$$\begin{aligned} 0 = \sum _{i=1}^n\frac{\hat{\eta }_{i,\mathrm L}(\beta ^{(r)}_0)}{1+\lambda ^T\hat{\eta }_{i,\mathrm L}(\beta ^{(r)}_0)}= & {} \sum _{i=1}^n \hat{\eta }_{i,\mathrm L}(\beta ^{(r)}_0) - \sum _{i=1}^n\hat{\eta }_{i,\mathrm L}(\beta ^{(r)}_0)\hat{\eta }_{i,\mathrm L}^T(\beta ^{(r)}_0)\lambda \\{} & {} + \sum _{i=1}^n\frac{\hat{\eta }_{i,\mathrm L}(\beta ^{(r)}_0)\{\lambda ^T\hat{\eta }_{i,\mathrm L}(\beta ^{(r)}_0)\}^2}{1+\lambda ^T\hat{\eta }_{i,\mathrm L}(\beta ^{(r)}_0)}. \end{aligned}$$

This, together with Lemmas 3 and 4 as well as (A.5), proves that

$$\begin{aligned} \sum _{i=1}^n \{\lambda ^T\hat{\eta }_{i,\mathrm L}(\beta ^{(r)}_0)\}^2 = \sum _{i=1}^n\lambda ^T\hat{\eta }_{i,\mathrm L}(\beta ^{(r)}_0)+o_P(1) \end{aligned}$$

and

$$\begin{aligned} \lambda =\left\{ \sum _{i=1}^n\hat{\eta }_{i,\mathrm L}(\beta ^{(r)}_0)\hat{\eta }_{i,\mathrm L}^T(\beta ^{(r)}_0)\right\} ^{-1} \sum _{i=1}^n \hat{\eta }_{i,\mathrm L}(\beta ^{(r)}_0) + o_P(n^{-1/2}). \end{aligned}$$

Therefore, from (A.6) we have

$$\begin{aligned} \mathcal{R}_\textrm{L}(\beta ^{(r)}_0) =&\left\{ \frac{1}{\sqrt{n}}\sum _{i=1}^n\hat{\eta }_{i,\mathrm L}(\beta ^{(r)}_0)\right\} ^T \left\{ \frac{1}{n}\sum _{i=1}^n\hat{\eta }_{i,\mathrm L}(\beta ^{(r)}_0)\hat{\eta }_{i,\mathrm L}^T(\beta ^{(r)}_0) \right\} ^{-1}\nonumber \\&\times \left\{ \frac{1}{\sqrt{n}}\sum _{i=1}^n\hat{\eta }_{i,\mathrm L}(\beta ^{(r)}_0)\right\} +o_P(1) \end{aligned}$$

(A.7)

when L = W, I and WI respectively. This, together with Slutsky’s theorem and Lemmas 2 and 3, proves (c) of Theorem 2.

Now, we prove (a) of Theorem 2 only. Similarly, (b) of Theorem 2 can also be proved. From (A.7) and (a) of Lemma 3 we obtain

$$\begin{aligned} \mathcal{R}_\textrm{W}(\beta ^{(r)}_0) = \left\{ B^{-1/2}\frac{1}{\sqrt{n}}\sum _{i=1}^n\hat{\eta }_{i,\mathrm W}(\beta ^{(r)}_0)\right\} ^TD_1 \left\{ B^{-1/2}\frac{1}{\sqrt{n}}\sum _{i=1}^n\hat{\eta }_{i,\mathrm W}(\beta ^{(r)}_0)\right\} +o_P(1), \end{aligned}$$

where $D_1=B^{1/2}C_\textrm{W}^{-1}B^{1/2}$. Let $D_2=\text{ diag }(w_1^*,\ldots ,w_{p-1}^*)$, where $w_i^*,\ldots ,w_{p-1}^*$ are the eigenvalues of $C_\textrm{W}^{-1}B$. Note that $D_1$ and $C_\textrm{W}^{-1}B$ have the same eigenvalues. Then there exists an orthogonal matrix S such that $S^TD_2S=D_1$. Hence

$$\begin{aligned} \mathcal{R}_\textrm{W}(\beta ^{(r)}_0)= & {} \left\{ SB^{-1/2}\frac{1}{\sqrt{n}}\sum _{i=1}^n\hat{\eta }_{i,\mathrm W}(\beta ^{(r)}_0)\right\} ^TD_2\\{} & {} \times \left\{ SB^{-1/2}\frac{1}{\sqrt{n}}\sum _{i=1}^n\hat{\eta }_{i,\mathrm W}(\beta ^{(r)}_0)\right\} +o_P(1). \end{aligned}$$

From Lemma 2 we obtain that

$$\begin{aligned} SB^{-1/2}\frac{1}{\sqrt{n}}\sum _{i=1}^n\hat{\eta }_{i,\mathrm W}(\beta ^{(r)}_0){\mathop {\longrightarrow }\limits ^{{D}}}N(\textbf{0},I_{p-1}). \end{aligned}$$

Above two results together proves (a) of Theorem 2. This completes the proof of Theorem 2. $\square$

Proof of Theorem 3

It is similar to the proof of Theorem 4 in Wang et al. (2010), and hence we omit its proof. $\square$

Proof of Theorem 4

Using the standard argument, we can obtain

$$\begin{aligned} n^{-1}S_{n,l}(u;\hat{\beta }_\textrm{L})=h_2^lp(u)\int \!\!t^lK(t)dt + O_P(h_2^{l+2}),~~l=0,1,2. \end{aligned}$$

Therefore, from (2.4) it follows

$$\begin{aligned} \hat{g}(u;\hat{\beta }_\textrm{L})-g(u)=\frac{1}{2}\mu _2h_2^2g''(u) + \frac{1}{n}\sum _{i=1}^n\frac{\delta _i\varepsilon _i}{\pi (V_i)p(u)}K_{h_2}(\beta ^TX_i-u) + o_P(c_n) \end{aligned}$$

when L = W, I and WI respectively, where $\mu _2=\displaystyle \int \!\!t^2K(t)dt$, $c_n=(nh_2)^{-1/2}+h_2^2$. By Theorem 4.4 of Masry and Tjøstheim (1995) as well as Slutsky’s theorem, Theorem 4 is proved. $\square$

Proof of Theorem 5

From (A.7). we can obtain

$$\begin{aligned} \tilde{R}_\textrm{L}(\beta ^{(r)}_0) = \{\sqrt{n}\widehat{Q}_\textrm{L}^T(\beta ^{(r)}_0)\}\hat{B}^{-1}(\beta ^{(r)}_0)\{\sqrt{n}\widehat{Q}_\textrm{L}(\beta ^{(r)}_0)\} +o_P(1) \end{aligned}$$

when L = W and I respectively. This, together with Slutsky’s theorem and Lemmas 2 and 3, proves Theorem 5. $\square$