Locally efficient estimation in generalized partially linear model with measurement error in nonlinear function

Abstract

We investigate the errors in covariates issues in a generalized partially linear model. Different from the usual literature (Ma and Carroll in J Am Stat Assoc 101:1465–1474, 2006), we consider the case where the measurement error occurs to the covariate that enters the model nonparametrically, while the covariates precisely observed enter the model parametrically. To avoid the deconvolution type operations, which can suffer from very low convergence rate, we use the B-splines representation to approximate the nonparametric function and convert the problem into a parametric form for operational purpose. We then use a parametric working model to replace the distribution of the unobservable variable, and devise an estimating equation to estimate both the model parameters and the functional dependence of the response on the latent variable. The estimation procedure is devised under the functional model framework without assuming any distribution structure of the latent variable. We further derive theories on the large sample properties of our estimator. Numerical simulation studies are carried out to evaluate the finite sample performance, and the practical performance of the method is illustrated through a data example.

Appendix

1.1 A.1 Proof of Theorem 1

From the definitions of $\mathbf{S}_{\mathrm{eff}}^*(Y_i, W_i,\mathbf{Z}_i, {\varvec{\delta }}, g)$ and ${\mathbf{S}_{\mathrm{res}}}_2^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }},{\varvec{\gamma }})$, we have

$$\begin{aligned} E\{\mathbf{S}_{\mathrm{eff}}^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}_0, g)| X_i,\mathbf{Z}_i\}= & {} \mathbf{0},\\ E_a\{{\mathbf{S}_{\mathrm{res}}}_2^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }}_0)| X_i,\mathbf{Z}_i\}= & {} \mathbf{0}, \end{aligned}$$

where $_a$ here and throughout the text stands for “approximate,” and $E_a$ indicates the expectation calculated with $g(\cdot )$ replaced by the approximate model $\mathbf{B}(\cdot )^{\mathrm{T}}{\varvec{\gamma }}_0$. Taking another expectation, we get

$$\begin{aligned} E\{\mathbf{S}_{\mathrm{eff}}^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}_0, g)\}= & {} \mathbf{0}, \\ E_a\{{\mathbf{S}_{\mathrm{res}}}_2^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }}_0)\}= & {} \mathbf{0}. \end{aligned}$$

Using Condition $(\mathrm {C}6)$, we further get

$$\begin{aligned} E\{\mathbf{S}_{\mathrm{eff}}^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }}_0)\}= & {} o(1),\\ E\{{\mathbf{S}_{\mathrm{res}}}_2^*\{Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }}_0)\}= & {} o(1), \end{aligned}$$

component-wise. Condition $(\mathrm {C}7)$ ensures that $[E\{\mathbf{S}_{\mathrm{eff}}^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}, {\varvec{\gamma }})\}^{\mathrm{T}}, E\{{\mathbf{S}_{\mathrm{res}}}_2^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}, {\varvec{\gamma }})\}^{\mathrm{T}}]^{\mathrm{T}}$ is invertible near its zero $\varvec{\theta }^*$ as a vector function of $\varvec{\theta }$, and the first derivative of the inverse function is bounded in the neighborhood of $\varvec{\theta }^*$. Therefore, $\Vert \varvec{\theta }^* - \varvec{\theta }_0\Vert _2 = o_p(1)$. On the other hand, since

$$\begin{aligned} \frac{1}{n}\sum _{i=1}^{n}\mathbf{S}_{\mathrm{eff}}^*(Y_i, W_i, \mathbf{Z}_i, \widehat{{\varvec{\delta }}}_n, \widehat{{\varvec{\gamma }}}_n)= & {} \mathbf{0},\\ \frac{1}{n}\sum _{i=1}^{n}{\mathbf{S}_{\mathrm{res}}}_2^*(Y_i, W_i, \mathbf{Z}_i, \widehat{{\varvec{\delta }}}_n, \widehat{{\varvec{\gamma }}}_n)= & {} \mathbf{0}, \end{aligned}$$

we have

$$\begin{aligned} E\{\mathbf{S}_{\mathrm{eff}}^*(Y_i, W_i, \mathbf{Z}_i, \widehat{{\varvec{\delta }}}_n,\widehat{{\varvec{\gamma }}}_n)\}= & {} o(1),\\ E\{{\mathbf{S}_{\mathrm{res}}}_2^*(Y_i, W_i, \mathbf{Z}_i, \widehat{{\varvec{\delta }}}_n,\widehat{{\varvec{\gamma }}}_n)\}= & {} o(1) \end{aligned}$$

element-wise. Using exactly the same argument as above, we can also obtain $\Vert \widehat{\varvec{\theta }}_n -\varvec{\theta }^*\Vert _2=o_p(1)$. Hence, combining the two results, we get $\Vert \widehat{\varvec{\theta }}_n - \varvec{\theta }_0\Vert _2=o_p(1)$. $\square$

1.2 A.2 Proof of Theorem 2

We first write

$$\begin{aligned} \mathbf{0}= & {} n^{-1/2}\sum _{i=1}^n\mathbf{S}_{\mathrm{eff}}^*\{Y_i, W_i,\mathbf{Z}_i,\widehat{{\varvec{\delta }}}_n,\widehat{{\varvec{\gamma }}}_n(\widehat{{\varvec{\delta }}}_n)\}\\= & {} \mathbf{T}_1 + \mathbf{T}_2(\widetilde{{\varvec{\delta }}}_n)\sqrt{n}(\widehat{{\varvec{\delta }}}_n-{\varvec{\delta }}_0), \end{aligned}$$

where

$$\begin{aligned} \mathbf{T}_1= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^{n}\mathbf{S}_{\mathrm{eff}}^*\{Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}_0, \widehat{{\varvec{\gamma }}}_n({\varvec{\delta }}_0)\},\\ \mathbf{T}_2({\varvec{\delta }})= & {} \mathbf{T}_{21}({\varvec{\delta }}) +\mathbf{T}_{22}({\varvec{\delta }})\frac{\partial \widehat{{\varvec{\gamma }}}_n({\varvec{\delta }})}{\partial {\varvec{\delta }}^{\mathrm{T}}}, \end{aligned}$$

where

$$\begin{aligned} \mathbf{T}_{21}({\varvec{\delta }})= & {} \frac{1}{n}\sum _{i=1}^{n}\frac{\partial \mathbf{S}_{\mathrm{eff}}^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}, \widehat{{\varvec{\gamma }}}_n)}{\partial {\varvec{\delta }}^{\mathrm{T}}},\\ \mathbf{T}_{22}({\varvec{\delta }})= & {} \frac{1}{n}\sum _{i=1}^{n}\frac{\partial \mathbf{S}_{\mathrm{eff}}^*\{Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}, \widehat{{\varvec{\gamma }}}_n({\varvec{\delta }})\}}{\partial \widehat{{\varvec{\gamma }}}_n({\varvec{\delta }})^{\mathrm{T}}}, \end{aligned}$$

and $\widetilde{{\varvec{\delta }}}_n$ is on the line connecting ${\varvec{\delta }}_0$ and $\widehat{{\varvec{\delta }}}_n$.

We further expand $\mathbf{T}_1$ as a function of $\widehat{{\varvec{\gamma }}}_n({\varvec{\delta }}_0)$ about ${\varvec{\gamma }}_0({\varvec{\delta }}_0)$ to obtain

$$\begin{aligned} \mathbf{T}_1 = \mathbf{T}_{11} + \mathbf{T}_{12}\{\widetilde{{\varvec{\gamma }}}_n({\varvec{\delta }}_0)\}\sqrt{n}\{\widehat{{\varvec{\gamma }}}_n({\varvec{\delta }}_0) - {\varvec{\gamma }}_0({\varvec{\delta }}_0)\}, \end{aligned}$$

where

$$\begin{aligned} \mathbf{T}_{11}= & {} \frac{1}{\sqrt{n}}\sum _{i=1}^{n}\mathbf{S}_{\mathrm{eff}}^*\{Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }}_0({\varvec{\delta }}_0)\},\\ \mathbf{T}_{12}\{{\varvec{\gamma }}({\varvec{\delta }}_0)\}= & {} \frac{1}{n}\sum _{i=1}^{n}\frac{\partial \mathbf{S}_{\mathrm{eff}}^*\{Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }}({\varvec{\delta }}_0)\}}{\partial {\varvec{\gamma }}({\varvec{\delta }}_0)^{\mathrm{T}}}, \end{aligned}$$

and $\widetilde{{\varvec{\gamma }}}_n({\varvec{\delta }}_0)$ is on the line connects $\widehat{{\varvec{\gamma }}}_n({\varvec{\delta }}_0)$ and ${\varvec{\gamma }}_0({\varvec{\delta }}_0)$.

Because of the consistency of $\mathbf{B}(x)^{\mathrm{T}}\widetilde{{\varvec{\gamma }}}_n$ to g(x) derived from Condition (C6) and Theorem 1, and the weak law of large numbers, for arbitrary $d_{{\varvec{\gamma }}}\times p$ matrix $\mathbf{G}$ with $\Vert \mathbf{G}\Vert _2 = 1$, we have

$$\begin{aligned} \mathbf{T}_{12}\{\widetilde{{\varvec{\gamma }}}_n({\varvec{\delta }}_0)\}\mathbf{G}= E\left\{ \frac{\partial \mathbf{S}_{\mathrm{eff}}^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }})}{\partial {\varvec{\gamma }}^{\mathrm{T}}}\mathbf{G}\bigg \arrowvert _{ \mathbf{B}(\cdot )^{\mathrm{T}}{\varvec{\gamma }}= g(\cdot )}\right\} \{1+o_p(1)\}, \end{aligned}$$

where

$$\begin{aligned}&E\left\{ \frac{\partial \mathbf{S}_{\mathrm{eff}}^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }})}{\partial {\varvec{\gamma }}^{\mathrm{T}}} \mathbf{G}\bigg \arrowvert _{ \mathbf{B}(\cdot )^{\mathrm{T}}{\varvec{\gamma }}= g(\cdot )}\right\} \nonumber \\&\quad =\int \left\{ \frac{\partial \mathbf{S}_{\mathrm{eff}}^*(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }})}{\partial {\varvec{\gamma }}^{\mathrm{T}}} \mathbf{G}\bigg \arrowvert _{ \mathbf{B}(\cdot )^{\mathrm{T}}{\varvec{\gamma }}= g(\cdot )}\right\} f(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0, g,f_X) dy_idw_id\mathbf{z}_i\nonumber \\&\quad =\int \left\{ \frac{\partial \mathbf{S}_{\mathrm{eff}}^*(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }}_0)}{\partial {\varvec{\gamma }}_0^{\mathrm{T}}}\mathbf{G}+ O_p(h_b^q)\right\} \left\{ f(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }}_0, f_X) \right. \nonumber \\&\left. \qquad +\, O_p(h_b^q) \right\} dy_idw_id\mathbf{z}_i\nonumber \\&\quad =\int \frac{\partial \mathbf{S}_{\mathrm{eff}}^*(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }}_0)}{\partial {\varvec{\gamma }}_0^{\mathrm{T}}}\mathbf{G}f(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }}_0, f_X) dy_idw_id\mathbf{z}_i + O_p(h_b^q)\nonumber \\&\quad = \frac{\partial }{\partial {\varvec{\gamma }}_0^{\mathrm{T}}}\int \{\mathbf{S}_{\mathrm{eff}}^*(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0, g)+O_P(h_b^q)\}\mathbf{G}\{f(y_i, w_i, \mathbf{z}_i, \beta _0, g, f_X) \nonumber \\&\qquad +\,O_p(h_b^q)\}dy_idw_id\mathbf{z}_i\nonumber \\&\qquad -\, \int \{\mathbf{S}_{\mathrm{eff}}^*(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0, g)+O_P(h_b^q)\}\mathbf{G}\frac{\partial f(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }}_0, f_X)}{\partial {\varvec{\gamma }}_0^{\mathrm{T}}}dy_idw_id\mathbf{z}_i \nonumber \\&\qquad +\, O_P(h_b^q)\nonumber \\&\quad =-\int \mathbf{S}_{\mathrm{eff}}^*(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0, g)\left\{ \mathbf{G}^{\mathrm{T}}\mathbf{S}_{a, {\varvec{\gamma }}}(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }}_0) \right\} ^{\mathrm{T}}\nonumber \\&\qquad f(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0, g, f_X) dy_idw_id\mathbf{z}_i +\, O_p(h_b^p)\nonumber \\&\quad = O_p(h_b^q). \end{aligned}$$

(A.1)

Here, like before, $f(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }}, f_X)$ stands for $f(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0, g, f_X)$ with $g(\cdot )$ replaced by $\mathbf{B}(\cdot )^{\mathrm{T}}{\varvec{\gamma }}$, and $\mathbf{S}_{a, {\varvec{\gamma }}}(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }}_0)\equiv \partial \log f(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }}, f_X)/\partial {\varvec{\gamma }}$. The second equality holds by condition $(\mathrm {C}6)$.

The third equality holds because $\Vert \partial \mathbf{S}_{\mathrm{eff}}^*(y_i, w_i,\mathbf{z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }}_0)/\partial {\varvec{\gamma }}_0^{\mathrm{T}}\Vert _{\infty }$ is integrable by condition $(\mathrm {C}8)$ and $f(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0,{\varvec{\gamma }}_0,f_X)$ is absolutely integrable. The fourth equality holds also by condition $(\mathrm {C}6)$. The fifth equality holds because $E\{\mathbf{S}_{\mathrm{eff}}^*(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}, g)\} = \mathbf{0}$. For the last equality, we note that

$$\begin{aligned} \mathbf{G}^{\mathrm{T}}\mathbf{S}_{a, {\varvec{\gamma }}}(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }}_0) = E[s\{y_i,\mathbf{z}_i^{\mathrm{T}}{\varvec{\beta }}_0+\mathbf{B}(X)^{\mathrm{T}}{\varvec{\gamma }}_0,{\varvec{\alpha }}_0\} \mathbf{G}^{\mathrm{T}}\mathbf{B}(X) \mid y_i,w_i,\mathbf{z}_i]. \end{aligned}$$

By Condition $(\mathrm {C}6)$ and definitions of $\varLambda _{g}$ and $\varLambda _{a, {\varvec{\gamma }}}$, for any $d_{{\varvec{\gamma }}} \times p$ matrix $\mathbf{G}$, there exists a function $\mathbf{h}(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0, g) \equiv E[s\{y_i, \mathbf{z}_i^{\mathrm{T}}{\varvec{\beta }}_0+g(X),{\varvec{\alpha }}_0\} \mathbf{G}^{\mathrm{T}}\mathbf{B}(X) \mid y_i,w_i,\mathbf{z}_i] \in \varLambda _{g}$ such that

$$\begin{aligned}\sup |\mathbf{G}^{\mathrm{T}}\mathbf{S}_{a, {\varvec{\gamma }}}(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }}_0) - \mathbf{h}(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0, g)| = O_P(h_b^q).\end{aligned}$$

Further, $\mathbf{S}_{\mathrm{eff}}^*(y_i, w_i, \mathbf{z}_i, {\varvec{\delta }}_0, g)$ is orthogonal to any function in $\varLambda _{g}$, thus the last equality holds. Hence, we obtain $\Vert \mathbf{T}_{12}\{\widetilde{{\varvec{\gamma }}}({\varvec{\delta }}_0)\} \Vert _2= O_p(h_b^q)$.

Based on the asymptotic results of Proposition 4 in Jiang and Ma (2018), we have $\Vert \widehat{{\varvec{\gamma }}}_n({\varvec{\delta }}_0) - {\varvec{\gamma }}_0({\varvec{\delta }}_0)\Vert _2 = O_p\{(nh_b)^{-1/2}\}$. Then, we have

$$\begin{aligned} \Vert \mathbf{T}_{12}\{\widetilde{{\varvec{\gamma }}}_n({\varvec{\delta }}_0)\}\sqrt{n}\{\widehat{{\varvec{\gamma }}}_n({\varvec{\delta }}_0) - {\varvec{\gamma }}_0({\varvec{\delta }}_0)\}\Vert _2 = O_p(h_b^{q - 1/2}). \end{aligned}$$

Further, by $(\mathrm {C}6)$ we have $\mathbf{T}_{11} = n^{-1/2}\sum _{i=1}^{n}\mathbf{S}_{\mathrm{eff}}^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}_0, g)+O_p(n^{1/2}h_b^q)$. Since $h_b^{q-1/2} = o_p(n^{1/2}h_b^q)$, and $n^{1/2}h_b^q= o_p(1)$ by conditions $(\mathrm {C}4)$ and $(\mathrm {C}5)$, then

$$\begin{aligned} \mathbf{T}_1 = n^{-1/2}\sum _{i=1}^{n}\mathbf{S}_{\mathrm{eff}}^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}_0,g)+o_p(1). \end{aligned}$$

(A.2)

We next consider each term in $\mathbf{T}_2(\widetilde{{\varvec{\delta }}}_n)$. Since $\widehat{{\varvec{\gamma }}}_n(\cdot )$ satisfies

$$\begin{aligned} n^{-1}\sum _{i=1}^{n}{\mathbf{S}_{\mathrm{res}}}_2^*\{Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }},\widehat{{\varvec{\gamma }}}_n({\varvec{\delta }})\}=\mathbf{0} \end{aligned}$$

for any ${\varvec{\delta }}$,

$$\begin{aligned} \frac{1}{n} \sum _{i=1}^{n}\frac{\partial {\mathbf{S}_{\mathrm{res}}}_2^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}, \widehat{{\varvec{\gamma }}}_n)}{\partial {\varvec{\delta }}^{\mathrm{T}}} + \frac{1}{n}\sum _{i=1}^{n}\frac{\partial {\mathbf{S}_{\mathrm{res}}}_2^*\{Y_i, W_i, \mathbf{Z}_i,, {\varvec{\delta }}, \widehat{{\varvec{\gamma }}}_n({\varvec{\delta }})\}}{\partial \widehat{{\varvec{\gamma }}}_n({\varvec{\delta }})^{\mathrm{T}}}\frac{\partial \widehat{{\varvec{\gamma }}}_n({\varvec{\delta }})}{\partial {\varvec{\delta }}^{\mathrm{T}}} = \mathbf{0}. \end{aligned}$$

Then,

$$\begin{aligned} \frac{\partial \widehat{{\varvec{\gamma }}}_n({\varvec{\delta }})}{\partial {\varvec{\delta }}^{\mathrm{T}}} = -\{\mathbf{T}_{23}({\varvec{\delta }})\}^{-1}\mathbf{T}_{24}({\varvec{\delta }}), \end{aligned}$$

where

$$\begin{aligned} \mathbf{T}_{23}({\varvec{\delta }})= & {} \frac{1}{n}\sum _{i=1}^{n}\frac{\partial {\mathbf{S}_{\mathrm{res}}}_2^*\{Y_i, W_i, \mathbf{Z}_i,, {\varvec{\delta }}, \widehat{{\varvec{\gamma }}}_n({\varvec{\delta }})\}}{\partial \widehat{{\varvec{\gamma }}}_n({\varvec{\delta }})^{\mathrm{T}}}, \\ \mathbf{T}_{24} ({\varvec{\delta }})= & {} \frac{1}{n} \sum _{i=1}^{n}\frac{\partial {\mathbf{S}_{\mathrm{res}}}_2^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}, \widehat{{\varvec{\gamma }}}_n)}{\partial {\varvec{\delta }}^{\mathrm{T}}}. \end{aligned}$$

Hence,

$$\begin{aligned} \mathbf{T}_2(\widetilde{{\varvec{\delta }}}_n)=\mathbf{T}_{21}(\widetilde{{\varvec{\delta }}}_n) - \mathbf{T}_{22}(\widetilde{{\varvec{\delta }}}_n)\{\mathbf{T}_{23}(\widetilde{{\varvec{\delta }}}_n)\}^{-1}\mathbf{T}_{24}(\widetilde{{\varvec{\delta }}}_n). \end{aligned}$$

By the consistency of $\widetilde{{\varvec{\delta }}}_n$ to ${\varvec{\delta }}_0$ and $\mathbf{B}(x)^{\mathrm{T}}\widehat{{\varvec{\gamma }}}_n$ to g(x), we have

$$\begin{aligned} \mathbf{T}_{21}(\widetilde{{\varvec{\delta }}}_n)=E\left\{ \frac{\partial \mathbf{S}_{\mathrm{eff}}^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}_0,{\varvec{\gamma }})}{\partial {\varvec{\delta }}_0^{\mathrm{T}}}\bigg \arrowvert _{\mathbf{B}(\cdot )^{\mathrm{T}}{\varvec{\gamma }}= g(\cdot )} \right\} \{1+o_p(1)\}, \end{aligned}$$

and

$$\begin{aligned} \mathbf{T}_{24}(\widetilde{{\varvec{\delta }}}_n) = E\left\{ \frac{\partial {\mathbf{S}_{\mathrm{res}}}_2^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }})}{\partial {\varvec{\delta }}_0 ^{\mathrm{T}}} \bigg \arrowvert _{\mathbf{B}(\cdot )^{\mathrm{T}}{\varvec{\gamma }}= g(\cdot )}\right\} \{1+o_p(1)\}. \end{aligned}$$

From (A.1), we also have

$$\begin{aligned} \mathbf{T}_{22}(\widetilde{{\varvec{\delta }}}_n) = E\left\{ \frac{\partial \mathbf{S}_{\mathrm{eff}}^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}_0, {\varvec{\gamma }})}{\partial {\varvec{\gamma }}^{\mathrm{T}}} \bigg \arrowvert _{ \mathbf{B}(\cdot )^{\mathrm{T}}{\varvec{\gamma }}= g(\cdot )}\right\} \{1+o_p(1)\}=O_p(h_b^q). \end{aligned}$$

Based on the proof of Proposition 4 in Jiang and Ma (2018), we have $\Vert \mathbf{T}_{23}(\widetilde{{\varvec{\delta }}}_n)^{-1}\Vert _2 = O_p(h_b^{-1})$. Therefore, we have $\mathbf{T}_{22}(\widetilde{{\varvec{\delta }}}_n) \{\mathbf{T}_{23}(\widetilde{{\varvec{\delta }}}_n)\}^{-1} \mathbf{T}_{24}(\widetilde{{\varvec{\delta }}}_n)=O_p(h_b^{q-1})$, where $q>1$ by condition $(\mathrm {C}2)$. Thus,

$$\begin{aligned} \mathbf{T}_2(\widetilde{{\varvec{\delta }}}_n) = E\left\{ \frac{\partial \mathbf{S}_{\mathrm{eff}}^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}_0,{\varvec{\gamma }})}{\partial {\varvec{\delta }}_0^{\mathrm{T}}}\bigg \arrowvert _{\mathbf{B}(\cdot )^{\mathrm{T}}{\varvec{\gamma }}= g(\cdot )} \right\} \{1+o_p(1)\}+O(h_b^{q-1}). \end{aligned}$$

Therefore,

$$\begin{aligned}&\sqrt{n}(\widehat{{\varvec{\delta }}}_n - {\varvec{\delta }}_0)\\&\quad = -\left[ E\left\{ \frac{\partial \mathbf{S}_{\mathrm{eff}}^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}_0,{\varvec{\gamma }})}{\partial {\varvec{\delta }}_0^{\mathrm{T}}}\bigg \arrowvert _{\mathbf{B}(\cdot )^{\mathrm{T}}{\varvec{\gamma }}= g(\cdot )} \right\} \right] ^{-1}\frac{1}{\sqrt{n}}\sum _{i=1}^{n}\mathbf{S}_{\mathrm{eff}}^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}_0, g)\\&\qquad +\,o_p(1). \end{aligned}$$

Since $n^{-1/2}\sum _{i=1}^{n}\mathbf{S}_{\mathrm{eff}}^*(Y_i, W_i, \mathbf{Z}_i, {\varvec{\delta }}_0, g)$ is the sum of zero-mean random vectors, this will converge in distribution to a multivariate normal distribution with mean $\mathbf{0}$ and covariance matrix $\mathbf{V}$ given in Theorem 2. $\square$