Doubly robust estimation and robust empirical likelihood in generalized linear models with missing responses

Abstract

In this paper, we study doubly robust estimation and robust empirical likelihood of regression parameter for generalized linear models with missing responses. A doubly robust estimating equation is proposed to estimate the regression parameter, and the resulting estimator has consistency and asymptotic normality, regardless of whether the assumed model contains the true model. A robust empirical log-likelihood ratio statistic for the regression parameter is constructed, showing that the statistic weakly converges to the standard $\chi ^2$ distribution. The result can be directly used to construct the confidence region of the regression parameter. A method for selecting the tuning parameters of $\psi$-function is also given. Simulation studies show the robustness of the estimator of the regression parameter and evaluate the performance of the robust empirical likelihood method. A real data example shows that the proposed method is feasible.

Appendix. Proofs of Theorems

In this appendix, we prove Theorems 1 and 2. The following Lemmas 1–4 are useful for proving these Theorems. Their proofs are given in the supplementary material.

Lemma 1

Suppose that conditions (C1)–(C7) hold. Then

$$\begin{aligned} \sup _{\beta \in {{{\mathcal {B}}}}_n}\Vert {\widehat{Q}}(\beta ) - Q_n(\beta )\Vert = o_P(n^{-1/2}), \end{aligned}$$

where ${{{\mathcal {B}}}}_n=\{\beta |\,\Vert \beta -\beta _0\Vert \le cn^{-1/2}\}$ for a constant $c>0$, ${\widehat{Q}}(\beta )$ is defined by (2.6),

$$\begin{aligned} Q_n(\beta )= & {} \frac{1}{n}\sum _{i=1}^n\frac{\delta _i}{\pi (X_i)}\frac{w(\beta ^TX_i)g'(\beta ^TX_i)}{V^{1/2}(g(\beta ^TX_i))}\\{} & {} \times \psi (r_i(\beta ))\{X_i-m(\beta ^TX_i)\}, \end{aligned}$$

and $r_i(\beta )$ is defined by (2.1).

Lemma 2

Suppose that conditions (C1)–(C7) hold. Then

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

where ${\widehat{Q}}(\beta _0)$, $Q_n(\beta _0)$ and $B(\beta _0)$ are defined in (2.6), Lemma 1 and Theorem 1, respectively.

Lemma 3

Suppose that conditions (C1)–(C7) hold. Then

$$\begin{aligned} \frac{1}{n}\sum _{i=1}^n{\hat{\eta _i}}(\beta _0)\hat{\eta }_i^T(\beta _0){\mathop {\longrightarrow }\limits ^{P}}B(\beta _0) \end{aligned}$$

where ${\hat{\eta _i}}(\beta _0)$ and $B(\beta _0)$ are defined in (2.9) and Theorem 1, respectively.

Lemma 4

Suppose that conditions (C1)–(C7) hold. Then

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

where ${\hat{\eta _i}}(\beta _0)$ is defined in (2.9)

Now we turn back to prove Theorems 1 and 2.

Proof of Theorem 1

We now prove the asymptotic normality of ${{\hat{\beta }}}$. The proof is divided into two steps: step (I) provides the existence of the estimator ${{\hat{\beta }}}$, and step (II) proves the asymptotic normality of ${{\hat{\beta }}}$.

(I)  The existence of the estimator of $\beta _0$. We prove the following fact: Under conditions (C1)–(C8) and with probability one there exists an estimator of $\beta _0$ solving the estimating equation (2.6) in ${{{\mathcal {B}}}}_n^*$, where ${{{\mathcal {B}}}}_n^*=\big \{\beta |\, \Vert \beta -\beta _0\Vert =Mn^{-1/2}\big \}$ for some positive constant M. From Lemma 2 we obtain that uniformly for $\beta \in {{{\mathcal {B}}}}_n^*$,

$$\begin{aligned} {\widehat{Q}}(\beta ) = Q_n(\beta _0) -A(\beta _0)(\beta -\beta _0) + o_P(n^{-1/2}), \end{aligned}$$

(A.1)

where $A(\beta _0)$ is defined in Theorem 1. Therefore, we have, uniformly for $\beta \in {{{\mathcal {B}}}}_n^*$,

$$\begin{aligned} n(\beta -\beta _0){\widehat{Q}}(\beta )= & {} n(\beta -\beta _0)Q_n(\beta _0) - n(\beta -\beta _0)A(\beta _0)\\{} & {} \times (\beta -\beta _0) + o_P(1). \end{aligned}$$

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

(II)  The asymptotic normality. From step (I) we find that ${{\hat{\beta }}}$ is a solution in ${{{\mathcal {B}}}}_n^*$ to the equation ${\widehat{Q}}(\beta )=0$, namely ${\widehat{Q}}({{\hat{\beta }}})=0$, where ${\widehat{Q}}(\beta )$ is defined in (2.6). From (A.1) we have

$$\begin{aligned} 0=Q_n(\beta _0) - A(\beta _0)({{\hat{\beta }}}-\beta _0) + o_P(n^{-1/2}), \end{aligned}$$

and hence

$$\begin{aligned} \sqrt{n}({{\hat{\beta }}}-\beta _0) = A^{-1}(\beta _0)\sqrt{n}Q_n(\beta _0) + o_P(1). \end{aligned}$$

(A.2)

Theorem 1 follows from (A.2), Lemma 2 and Slutsky’s theorem. $\square$

Proof of Theorem 2

By the Lagrange multiplier method, ${\hat{l}}(\beta _0)$ can be represented as

$$\begin{aligned} {\hat{l}}(\beta _0)=2\sum _{i=1}^n\log \big (1+\lambda ^T{\hat{\eta }}_i(\beta _0)\big ), \end{aligned}$$

(A.3)

where $\lambda =\lambda (\beta )$ is a $d\times 1$ vector given as the solution to

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

By Lemmas 2–4, and using the same arguments as are used in the proof of (2.8) in Owen (1990), we can show that

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

(A.4)

Applying the Taylor formula to (A.3), and invoking Lemmas 2–4 and (A.4), we get

$$\begin{aligned} {\hat{l}}(\beta _0) = 2 \sum _{i=1}^n\Big [\lambda ^T{\hat{\eta }}_i(\beta _0) - \big \{\lambda ^T{\hat{\eta }}_i(\beta _0)\big \}^2/2\Big ] +o_P(1).\nonumber \\ \end{aligned}$$

(A.5)

Note that $h(\lambda )=0$. It follows that

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

This together with Lemmas 2–4 and (A.4) proves that

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

(A.6)

and

$$\begin{aligned} \lambda =\bigg (\sum _{i=1}^n{\hat{\eta }}_i(\beta _0){\hat{\eta }}_i^T(\beta _0) \bigg )^{-1} \sum _{i=1}^n {\hat{\eta }}_i(\beta _0) + o_P\big (n^{-1/2}\big ).\nonumber \\ \end{aligned}$$

(A.7)

Therefore, from (A.5)–(A.7) we have

$$\begin{aligned} {\hat{l}}(\beta _0)= & {} \{\sqrt{n}{\widehat{Q}}^{T}(\beta _0)\}\bigg (\frac{1}{n}\sum _{i=1}^n{\hat{\eta }}_i(\beta _0){\hat{\eta }}_i^T(\beta _0) \bigg )^{-1} \nonumber \\{} & {} \times \{\sqrt{n}{\widehat{Q}}(\beta _0)\} +o_P(1), \end{aligned}$$

(A.8)

where ${\widehat{Q}}(\beta _0)$ is defined in (2.6). This, together with (A.8), Lemmas 2 and 3 as well as Slutsky’s theorem, proves Theorem 2. $\square$